diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlvvv" "b/data_all_eng_slimpj/shuffled/split2/finalzzlvvv" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlvvv" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\\label{introd}\n\nNuclear collective motion, such as fission, or multipole\nvibration and rotation excitation modes,\nwas successfully studied by using several\nmicroscopic-macroscopic approximations to\nthe description of the finite Fermi systems of the strongly interacting\nnucleons \\cite{migdal,myswann69,fuhi,bohrmot,mix,belyaevzel}.\nMany significant phenomena deduced from experimental data on nuclear\nfission, vibrations and rotations were explained within the\ntheoretical approaches based\n mainly on the cranking model \\cite{inglis,bohrmotpr,valat,bohrmot}\nand its extensions to the pairing correlations\n\\cite{belyaevfirst,belyaev61,belzel,belsmitolfay87},\nincluding shell and temperature effects \\cite{brackquenNPA1981},\nand to non-adiabatic effects\n\\cite{fraupash,zelev,mix,zshimansky,marshalek,belyaevhighspin,afanas,belyaevbif},\nwhich were originally applied for the\nrotational modes (see also \\cite{sfraurev} for the review paper\nand references therein).\n\n\nFor the nuclear collective excitations\nwithin the general response-function theory\n\\cite{bohrmot,siemjen,hofbook}, the basic idea is to parametrize\nthe complex dynamical problem of the collective motion of many\nstrongly interacting particles in terms of a few collective\nvariables found from the physical\nmeaning of the considered dynamical problem, for example\nthe nuclear surface itself \\cite{strtyap,strutmagbr,strutmagden}\nor its multipole deformations \\cite{bohrmot}.\nWe can then study the response to an external field of the dynamical\nquantities describing the nuclear collective motion in terms of\nthese variables. Thus, we get important information on the transport\nproperties of nuclei. For such a theoretical description of the\ncollective motion it is very important to take into account the\ntemperature dependence of the\ndissipative nuclear characteristics as the friction coefficient, as\nshown in \\cite{hofivyam,ivhofpasyam,hofmann,hofbook}.\nThe friction depends strongly on the temperature and its\ntemperature dependence can therefore not be ignored\nin the description of the collective excitations in nuclei.\nConcerning the temperature dependence of\nthe nuclear friction, one of the most important problems is\nrelated to the properties of the static susceptibilities and\nergodicity of the Fermi systems like nuclei.\n\nHowever, the quantum description of dissipative phenomena in\nnuclei is rather complicated because we have to take into account\nthe residual interactions beyond the mean-field approximation.\nTherefore, more simple models\n\\cite{strutmagbr,kolmagpl,magkohofsh,kolmagsh}\naccounting for some\nmacroscopic properties of the many-body Fermi-system are helpful\nto understand the global average properties of the collective\nmotion.\nSuch a model is based on the Landau Fermi-liquid theory\n\\cite{landau,abrikha,pinenoz}, applied for the nuclear interior\nand simple\nmacroscopic boundary conditions on the nuclear surface\n\\cite{strutmagbr,strutmagden,magstrut,magboundcond,kolmagsh,magsangzh,BMRV}\n(see\nalso macroscopic approaches with different boundary conditions\n\\cite{bekhal,ivanov,abrditstrut,komagstrvv,abrdavkolsh}).\nIn \\cite{magkohofsh}, the response-function theory can be applied to\ndescribe collective nuclear excitations as the isoscalar quadrupole\nmode. The\ntransport coefficients, such as friction and inertia, are simply\ncalculated within\nthe macroscopic Fermi-liquid droplet model (FLDM)\n\\cite{kolmagpl,magkohofsh,kolmagsh} \nand\ntheir temperature dependence can be clearly discussed\n(see also earlier works\n\\cite{strutmagden0,galiqmodgen,galiqmod,strutmagden,magstrut,denisov}). \nThe asymmetry of\nheavy nuclei near their stability line\nand the structure of the isovector dipole resonances\nare studied in \\cite{kolmag,kolmagsh,BMV,BMR}\n(see also \\cite{abrIVGDR,abrdavpl}). In this way,\nthe giant multipole resonances were described, and, with\nincreasing temperature \\cite{kolmagpl,magkohofsh}, a transition\nfrom zero sound modes to the hydrodynamic first sound.\nThe friction in \\cite{kolmagpl,magkohofsh}\nis due to the collisions of particles, which were taken into\naccount in the relaxation-time approximation\n\\cite{abrikha,pinenoz,sykbrook,brooksyk,heipethrev,baympeth} with a\ntemperature and\nfrequency dependence (retardation effects)\n\\cite{landau,kolmagpl}.\n\n\nThe most important\nresults obtained in \\cite{magkohofsh,hofivmag} are related to the\noverdamped surface excitation mode for the low energy region and\nits dissipative characteristics as friction.\nFor the low excitation energy region these investigations can be\ncompleted by the additional sources of the friction related to a\nmore precise description of the heated Fermi liquids presented in\n\\cite{heipethrev,baympeth} for the infinite matter. Following\n\\cite{heipethrev}, we should take into account the thermodynamic\nrelations along with the dynamical Landau--Vlasov equation and\nintroduce the local equilibrium distribution instead of the one of global\nstatics, used earlier in \\cite{magkohofsh,hofivmag} for the\nlinearization procedure of this equation. These new developments\nof the Landau theory are especially important for the further\ninvestigations of the temperature dependence of the friction.\nFor the first step we have to work out in more details the theory\n\\cite{heipethrev} of the heated Fermi liquids for nuclear matter\nto apply then it for the dynamical description of the collective\nmotion in the interior of nuclei in the macroscopic FLDM\n \\cite{kolmagpl,magkohofsh}. Our purpose is also to find the\nrelations to some general points of the response function theory\nand clarify them taking the example of the analytically solved\nmodel based on the non-trivial temperature-dependent Fermi-liquid\ntheory. One of the most important questions which would be better\nto clarify is the above mentioned ergodicity property, temperature\ndependence of the friction and coupling constant.\n\nAnother important\nextension of this macroscopic theory is to study the structure\nof the isovector giant\ndipole resonance (IVGDR) as a splitting phenomenon due to the nuclear\nsymmetry interaction between neutrons and protons\n near the stability line \\cite{kolmag,abrIVGDR,abrdavpl,kolmagsh,BMV,BMRV,BMR}.\nThe neutron skin of exotic nuclei with a large excess of neutrons\nis also still one of the exciting subjects of\n nuclear physics and nuclear astrophysics\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85,danielewicz1,pearson,danielewicz2,vinas1,vinas2,vinas3,vinas4}.\nSimple and accurate solutions for the isovector particle density\ndistributions were obtained within the nuclear effective surface (ES)\napproximation\n\\cite{strtyap,strutmagbr,strutmagden,magsangzh,BMRV}.\nIt exploits the saturation of nuclear matter and a\nnarrow diffuse-edge region in finite heavy nuclei. The ES is defined as\nthe location of points of the maximum density gradient. The coordinate\nsystem, connected locally with\nthe ES, is specified by the distance from the\ngiven point to the surface and by tangent coordinates at the ES.\nThe variational condition for the nuclear energy with some\nadditional fixed integrals of motion in the local energy-density theory\n\\cite{rungegrossPRL1984,marquesgrossARPC2004} is\nsignificantly\nsimplified in these coordinates. In particular, in the extended\nThomas--Fermi (ETF) approach \\cite{brguehak,sclbook}\n(with Skyrme forces\n\\cite{chaban,reinhard,bender,revstonerein,ehnazarrrein,pastore})\n this can be done for any deformations by using an expansion in a small\nleptodermic parameter. The latter\nis of the order of the diffuse edge\nthickness of heavy enough nucleus over its mean curvature radius, or\nthe number of nucleons in power one third\nunder the distortion constraint in the case of\ndeformed nuclei. The accuracy of the ES approximation in the ETF\napproach without spin-orbit (SO) and asymmetry terms was checked\n\\cite{strutmagden} by comparing results of Hartree--Fock (HF)\n\\cite{brink,ringschuck} and ETF calculations \\cite{brguehak,sclbook} for some\nSkyrme forces. The ES approach (ESA) \\cite{strtyap,strutmagbr,strutmagden}\nwas then extended by taking SO and asymmetry effects into account\n\\cite{magsangzh,BMRV}.\nSolutions for the\nisoscalar and isovector particle densities and\nenergies at the quasi-equilibrium\nin the ESA of the ETF approach were applied to\nanalytical calculations of the neutron skin and isovector stiffness\ncoefficients in the leading order of the leptodermic parameter\nand to the derivations of the macroscopic boundary conditions \\cite{BMRV}.\nOur results are compared with the fundamental researches\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85} in the\nliquid droplet model (LDM). These analytical expressions for the\nenergy surface constants can be used\nfor IVGDR calculations within the FLDM\n\\cite{denisov,kolmag,kolmagsh,BMV,BMR}.\n\nA further interesting application\nof the semiclassical response theory would consist in the study of the\nproperties of collective rotation bands in heavy deformed nuclei.\nOne may consider nuclear collective rotations\nwithin the cranking model\nas a response to the Coriolis external-field perturbation.\nThe moment of inertia (MI)\ncan be calculated as a susceptibility with respect to this external field.\nThe rotation frequency of the rotating Fermi system in the cranking model\nis determined for a given nuclear\nangular momentum through a constraint, as\nfor any other integral of motion, as in particular the particle number\nconservation \\cite{ringschuck}. In order to simplify\nsuch a rather complicated problem, the Strutinsky shell correction method (SCM)\n\\cite{strut,fuhi} was adjusted to the collective nuclear rotations in\n\\cite{fraupash,mix}. The collective MI\nis expressed as function of the particle number and temperature\nin terms of a smooth part\nand an oscillating shell correction. The smooth component can be described\nby a suitable macroscopic model, like the dynamical ETF approach\n\\cite{bloch,amadobruekner,rockmore,jenningsbhbr,sclbook,brguehak,bartelnp,bartelpl}\n similar to the FLDM,\nwhich has proven to be both simple and precise.\nFor the definition of the MI shell correction, one can apply the Strutinsky\naveraging procedure to the single-particle (s.p.) MI, in the same way as for\nthe well-known free-energy shell correction.\n\nFor a deeper understanding of the quantum results and the correspondence\nbetween classical and quantum physics of the MI shell components, it\nis worth to analyze these shell components in terms of periodic orbits\n(POs), what is now well established as the semiclassical periodic-orbit\ntheory (POT) \\cite{gutz,bablo,strutmag,bt,creglitl,sclbook,migdalrev}\n(see also its\nextension to a given angular momentum projection along with the\nenergy of the particle \\cite{magkolstr} and to the particle densities\n\\cite{strutmagvvizv1986,brackrocciaIJMPE2010} and\npairing correlations\n\\cite{brackrocciaIJMPE2010}).\nGutzwiller was the first who developed the POT for\ncompletely chaotic Hamiltonians with only one integral of motion\n (the particle energy) \\cite{gutz}.\nThe Gutzwiller approach of the POT extended\nto potentials with continuous symmetries for the description of the\nnuclear shell structure can be found in \\cite{strutmag,smod,creglitl,sclbook}.\nThe semiclassical shell-structure corrections to the level density\nand energy have been tested for a large number of s.p.\\ Hamiltonians\nin two and three dimensions (see, for instance,\n\\cite{sclbook,magosc,ellipseptp,spheroidpre,spheroidptp,maf,magNPAE2010,magvlasar}).\nFor the Fermi gas the\n entropy shell corrections of the POT as a sum of periodic orbits were\nderived in \\cite{strutmag}, and with its help,\nsimple analytical expressions for the\nshell-structure energies in cold nuclei were obtained there following\na general semiclassical theory \\cite{sclbook}.\nThese energy shell corrections are in good agreement with the\nquantum SCM results, for\ninstance for elliptic and spheroidal cavities, including the\n superdeformed bifurcation region\n\\cite{ellipseptp,spheroidptp}.\nIn particular in three dimensions, the superdeformed bifurcation\nnanostructure leads as function of deformation to the\ndouble-humped shell-structure energy with the first and second\npotential wells in heavy enough nuclei\n\\cite{smod,migdalrev,spheroidptp,sclbook,magNPAE2010},\nwhich is well known as the double-humped fission barriers in the region\nof actinide nuclei.\nAt large deformations the second well can be understood semiclassically,\nfor spheroidal type shapes, through the bifurcation of equatorial orbits\ninto equatorial and the shortest 3-dimensional periodic orbits, because of\nthe enhancement of the POT amplitudes of the shell correction to the level\ndensity near the Fermi surface at these bifurcation deformations.\n\nFor finite heated fermionic systems, it was also shown\n\\cite{strutmag,kolmagstr,magkolstrutizv1979,richter,sclbook,brackrocciaIJMPE2010}\n within the POT that the\nshell-structure of the entropy, the thermodynamical (grand-canonical) potential\n and the free-energy shell\ncorrections can be obtained by multiplying the terms of the POT expansion\nby a temperature-dependent factor, which is exponentially decreasing\nwith temperature. For the case of the so called\n``{\\it classical rotations}$\\,$'' around the symmetry $z$ axis of the nucleus,\nthe MI shell correction is obtained, for any rotational frequency and at finite\ntemperature, within the extended Gutzwiller POT through the averaging of the\nindividual angular momenta aligned along this symmetry axis\n\\cite{magkolstr,kolmagstr,magkolstrutizv1979}.\nA similar POT problem, dealing with the magnetic susceptibility of fermionic\nsystems like metallic clusters and quantum dots, was worked out in\n\\cite{richter,fraukolmagsan}.\n\n\n\nIt was suggested in \\cite{dfcpuprc2004} to use the spheroidal cavity and\nthe classical perturbation approach to the POT by Creagh \\cite{creagh,sclbook}\nto describe the collective rotation of deformed nuclei around an axis ($x$\naxis) perpendicular to the symmetry $z$ axis.\nThe small parameter of the POT perturbation approximation turns out to be\nproportional to the rotational frequency, but also to the classical action\n(in units of $\\hbar$), which causes an additional restriction to Fermi\nsystems (or particle numbers) of small enough size, in contrast to the usual\nsemiclassical POT approach.\n\nIn \\cite{mskbg,mskbPRC2010}, the nonperturbative extended Gutzwiller POT\nwas used for the calculation of the MI shell corrections within the\nmean-field cranking model for both the collective and the alignment\nrotations.\nIn these works, for the statistical equilibrium nuclear rotations,\nthe semiclassical MI shell corrections were obtained in good agreement with the\nquantum results\nin the case of the harmonic-oscillator potential.\nWe extend this approach for collective rotations\nperpendicular to symmetry axis to the analytical calculations of\nthe MI shell corrections for the case of different mean fields,\nin particular with spheroidal\nshapes and sharp edges. The main purpose is to study semiclassically\nthe enhancement effects in the MI within the improved stationary\nphase method (improved SPM or shortly ISPM)\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar},\ndue to the bifurcations of the\nperiodic orbits in the superdeformed region.\n\n\nIn the present review in Section \\ref{eqmotion} we present some\nbasic formulas of the temperature-dependent Fermi-liquid theory\n\\cite{heipethrev}. We consider in Sec.\\ \\ref{conserveqs} the\nparticle number and momentum conservation equations and derive\nfrom them the energy conservation and general transport equations,\nin particular,\nthe expressions for the viscosity, shear modulus and thermal conductivity\ncoefficients. In Sec.\\ \\ref{respfunsec} we determine the\ndensity-density and density-temperature response functions with\nthe low temperature corrections. Section\n\\ref{longwavlim} shows the long wave-length (LWL, or hydrodynamic) limit\nfor the response functions, and the specific expressions for the\ntransport coefficients.\nIn Sec.\\ \\ref{suscept}, one obtains the static\nisolated, isothermal, and adiabatic susceptibilities to clarify\nsome important points of the general response function theory,\nmainly, the ergodicity property of the Fermi systems\n\\cite{hofmann,kubo}. We study the relaxation and correlation\nfunctions on the basis of the fluctuation-dissipation theorem and\nestablish their relations to the ergodicity of the Fermi-liquid\nsystem in section \\ref{relaxcorr}.\nGeneral aspects of the response function theory for the collective\nmotion in nuclei are presented in\nSec.\\ \\ref{basdef}\nin line with \\cite{hofmann,hofbook}. Section\n\\ref{fldm} shows the basic ingredients and\nthe collective response function of the nuclear FLDM.\nSection \\ref{transprop} is devoted to the\nderivation of the temperature dependence of the\ntransport coefficients, such as friction, inertia,\nand stiffness for the density modes\nfor slow collective motion. The numerical illustrations are given in Sec.\\\n\\ref{discuss}. In Sec.\\ \\ref{npcorivgdr}, the semiclassical theory is extended\nto neutron-proton asymmetric nuclei and applied for the\ncalculations of IVGDRs.\nIn Sec.\\ \\ref{semshellmi}, the smooth\nETF and fluctuating shell-structure components of the moments of inertia\nare derived for collective rotations of heavy nuclei.\nThe MI shell component is analytically presented in terms of the\nperiodic orbits and their bifurcations within the POT. This component\nis compared with the quantum results for the simplest case of the deformed\nharmonic oscillator Hamiltonian.\nComments and conclusions are finally given in Sec.\\ \\ref{concl}.\nSome details of the thermodynamical, FLDM\n(in the LWL limit) and POT calculations, such as\nthe analytical derivations\nof the in-compressibility, viscosity, thermoconductivity,\ncoupling, and surface symmetry-energy constants, as well as the\nsemiclassical MI\nare presented in Appendices A-E.\n\n\n\n\n\\section{The quasiparticle kinetic theory}\n\\label{kinapp}\n\n\\subsection{Equations of motion for the heated Fermi liquid}\n\n\\label{eqmotion}\n\nIn the semiclassical approximation the dynamics of a Fermi liquid\nmay be described by the distribution function $f({\\bf r},{\\bf p},t)$ in the\none body phase-space. Restricting to small deviations of particle\ndensity $\\rho({\\bf r},t)$ and temperature $T$, from their values in a\nthermodynamic equilibrium one may apply the linearized\nLandau--Vlasov equation \\cite{abrikha,heipethrev}:\n\\begin{eqnarray}\\label{landvlas}\n\\frac{\\partial}{\\partial t}\n \\delta f({\\bf r},{\\bf p},t) &+&\n \\frac{\\partial \\varepsilon_{{\\bf p}}^{\\rm g.e.}}{\\partial {\\bf p}}\n {\\bf \\nabla}_r \\delta f({\\bf r},{\\bf p},t) - \n - {\\bf \\nabla}_r \\left[\\delta \\varepsilon({\\bf r},{\\bf p},t) \\right.\\nonumber\\\\ \n &+&\\left.V_{\\rm ext}\\right]\n {\\bf \\nabla}_p f_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.}) =\n \\delta St. \n\\end{eqnarray}\nThe right hand side (r.h.s.) represents the dynamic component\nof the integral collision term $\\delta St$, and\n$V_{\\rm ext}$ stands for an external field. We introduce here\nthe Fermi distribution\n\\bel{fgeq}\nf_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.}) =\n \\left[1 + \\hbox{exp} \\left(\\frac{\\varepsilon_{\\bf p}^{\\rm g.e.}-\\mu}{T}\\right)\\right]^{-1}\n\\end{equation}\nof the {\\it global equilibrium} (g.e.), with $\\mu$ being the chemical\npotential, the temperature $T$ is given, as usually\nin nuclear physics, in the energy\n(MeV) units (without Boltzmann's constant), and\n$\\delta f({\\bf r},{\\bf p},t)$ measures the deviation\n\\bel{dfgeqrpt} \\delta f({\\bf r},{\\bf p},t) = f({\\bf r},{\\bf p},t) -\nf_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.}).\n\\end{equation}\nFor the sake of simplicity, the s.p.\\ energy $\\varepsilon_{\\bf p}^{\\rm g.e.}$\nwill be assumed to be of the form\n$~\\varepsilon_{\\bf p}^{\\rm g.e.} =\np^2\/2m^*~$ with $m^*$ being the effective nucleonic\nmass. In (\\ref{landvlas}),\n$\\delta \\varepsilon({\\bf r},{\\bf p},t)$ stands for the variation of the\nquasiparticle energy $\\varepsilon({\\bf r},{\\bf p},t)$,\n\\begin{eqnarray}\\label{deleps}\n&&\\delta\n\\varepsilon({\\bf r},{\\bf p},t) = \\varepsilon({\\bf r},{\\bf p},t)- \\varepsilon_{\\bf p}^{\\rm g.e.} \\nonumber\\\\\n&=& \\frac{1}{\\mathcal{N}(T)} \\int\n\\frac{2 d{\\bf p}^\\prime}{(2 \\pi {\\hbar})^3}\\;\n \\mathcal{F}({\\bf p}, {\\bf p}^\\prime)\\;\n \\delta f({\\bf r}, {\\bf p}^\\prime, t).\n\\end{eqnarray}\nThe quasiparticles' density of\nstates $\\mathcal{N}(T)$ at the chemical potential $\\mu$ is given\nby\n \\bel{enerdensnt}\n \\mathcal{N}(T)=\\int\n \\frac{2 d{\\bf p}^\\prime}{(2 \\pi {\\hbar})^3}\n \\left(\n-\\frac{\\partial f_{{\\bf p}^{\\prime}}} {\\partial \\varepsilon_{{\\bf p}^{\\prime}}} \\right)_{g.e}.\n\\end{equation}\nEvidently, because of our linearization the density\n$\\mathcal{N}(T)$ here is the one of equilibrium. In the sequel such a\nconvention will inherently be applied to any coefficient of\nquantities of order $\\delta f$.\nThe factor 2 accounts for the spin degeneracy.\nThe amplitude of the\nquasiparticle interaction, $\\mathcal{F}({\\bf p},{\\bf p}^\\prime)$,\ncommonly is written in terms of\nthe Landau parameters $\\mathcal{F}_0$ and $\\mathcal{F}_1$, according to\n\\bel{intampfpp}\n\\mathcal{F}({\\bf p}, {\\bf p}^\\prime) =\n \\mathcal{F}_0 + \\mathcal{F}_1 {\\hat p}\n \\cdot {\\hat p}^\\prime\\;, \\qquad {\\hat p} =\n {\\bf p}\/p.\n\\end{equation}\nThese two constants may be related to the two properties of\nnuclear matter, namely the isothermal in-compressibility $K^{T}$\n(see Appendix A.1),\n\\bel{isotherk}\nK^{T} =\n9 \\rho\\mathcal{G}_0\/\\mathcal{N}(T),\n\\end{equation}\nand the effective mass $m^*$,\n\\bel{effmass}\nm^*= \\mathcal{G}_1 m, \\quad\n\\mathcal{G}_n=\\left(1+\\frac{\\mathcal{F}_n}{2n+1}\\right)\n\\end{equation}\n($n=0, 1$). The equation for the effective\nmass $m^*$ is known \\cite{abrikha,heipethrev} to be valid\nfor systems obeying Galileo invariance, which shall be assumed\nhere.\n\nIn principle, the Landau parameters $\\mathcal{F}_0$ and $\\mathcal{F}_1$\nmight vary with the momenta $p$ and $p'$. Such a dependence will\nbe neglected henceforth. This approximation appears to be\nreasonable as we are going to stick to small excitations near the\nFermi surface and to temperatures $T$, which are small as compared to\nthe chemical potential $\\mu$. Likewise, we shall discard any\ntemperature dependence of the effective mass. Notice that\nin addition to the ratio $({T \/\\mu})^2$, this dependence would be\ngoverned by the additional factor $|{m^* \/ m}-1|$ which is small\nfor nuclear matter. These assumptions will allow us to simplify\nfurther the theory \\cite{heipethrev} and to get\nmore explicit results by making use of the temperature expansion\nfor the response functions in the small parameter $T\/\\mu$, as well\nas of the standard perturbation approach to eigenvalue problems\nneeded later for the hydrodynamic (long-wave\nlength) limit. We will follow \\cite{heipethrev} in neglecting\nhigher order terms in the expansion (\\ref{intampfpp}) in Legendre\npolynomials.\n\n\nLater on we want to study motion of the system which can be\nclassified as an excitation on top of the {\\it local equilibrium}.\nFollowing \\cite{abrikha,heipethrev}, the\ncollision term $\\delta St$ can be considered in the relaxation\ntime approximation,\n\\begin{eqnarray}\\label{intcoll}\n&&\\delta St =\n-\\frac{\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)}{\\tau}\\;, \\qquad\n f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{l.e.}\\right) = \\nonumber\\\\\n &=&\\left[1 + \\hbox{exp}\\left(\\frac{\\varepsilon_{\\bf p}^{\\rm l.e.}-\\mu({\\bf r},t)\n -{\\bf p}{\\bf u}({\\bf r},t)}{T({\\bf r},t)}\\right)\\right]^{-1}.\n\\end{eqnarray}\nHere, $f_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$ is the distribution function\nof a {\\it local equilibrium} (l.e.), and $\\varepsilon_{\\bf p}^{\\rm l.e.}$ is the associated\nquasiparticle energy. $\\mu({\\bf r},t)$ represents the chemical\npotential, ${\\bf u}({\\bf r},t)$ the mean velocity field, and $T({\\bf r},t)$\nthe temperature, all defined in the local sense. Like in\n\\cite{magkohofsh}, the relaxation time $\\tau$ is assumed to be\nindependent of the quasiparticle momentum ${\\bf p}$. However, it will be\nallowed to depend $\\tau$ on $T$ as well as on the frequency of the\nmotion (thus, accounting for retardation effects in collision\nprocesses). In\n(\\ref{intcoll}), $\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$\nis defined as\n\\bel{dfleqrpt} \\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},t) = f({\\bf r},{\\bf p},t) - f_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.}).\n\\end{equation}\nIt differs from $\\delta f({\\bf r},{\\bf p},t)$ of (\\ref{dfgeqrpt}) by the\nvariations of local quantities. For the latter, we may write\n\\bel{dfgeqdfleq} \\delta f({\\bf r},{\\bf p},t) =\n \\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t) +\n \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{l.e.}\\right)\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{dfleq}\n&&\\delta f_{\\rm l.e.} \\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right)\n = f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) -\n f_{\\rm g.e.}\\left(\\varepsilon_{\\bf p}^{\\rm g.e.}\\right) \\qquad \\nonumber\\\\\n&=&\n \\left(\\frac{\\partial f_{\\bf p}}{\\partial \\varepsilon_{\\bf p}}\\right)_{\\rm g.e.}\n \\left(\\delta \\varepsilon_{\\bf p}^{\\rm l.e.} - \\delta \\mu -{\\bf p} {\\bf u} -\n \\frac{\\varepsilon_{\\bf p}^{\\rm g.e.}-\\mu}{T}\\delta T\\right).\\qquad\n\\end{eqnarray}\nFor the l.e. quasiparticle energy $\\varepsilon_{\\bf p}^{\\rm l.e.}$, one has\n\\bel{delelocglo} \\varepsilon_{\\bf p}^{\\rm l.e.}=\\varepsilon_{\\bf p}^{\\rm g.e.}+\n \\delta \\varepsilon_{\\bf p}^{\\rm l.e.},\n\\end{equation}\nwhere $\\delta \\varepsilon_{\\bf p}^{\\rm l.e.}$ is defined like in\n(\\ref{deleps}) with only $\\delta f({\\bf r},{\\bf p},t)$ replaced by\n$\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$. According to (\\ref{dfgeqdfleq}) and\n(\\ref{dfleq}), for the simplified interaction\n(\\ref{intampfpp}), one gets\n\\bel{delepsleq} \\delta \\varepsilon_{\\bf p}^{\\rm l.e.}=\n \\delta \\varepsilon ({\\bf r},{\\bf p},t)=\n \\frac{\\mathcal{F}_0}{\\mathcal{N}(T)} \\delta \\rho({\\bf r},t)\n + \\frac{\\mathcal{F}_1 m \\rho}{\\mathcal{N}(T) p_{{}_{\\! {\\rm F}}}^2} {\\bf p}{\\bf u},\n\\end{equation}\n where $\\delta \\rho$ is the dynamical component of the particle\ndensity\n \\bel{densit} \\rho({\\bf r}, t) =\n \\int \\frac{2 d{\\bf p}}{(2 \\pi \\hbar)^3} \\;\n f({\\bf r}, {\\bf p}, t) = \\rho_\\infty + \\delta \\rho({\\bf r},t)\n\\end{equation}\n with $\\rho_\\infty$ being its g.e. value\nassociated to $f_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.})$ for the infinite Fermi liquid.\nThe vector of the mean\nvelocity ${\\bf u}$ can be expressed in terms of the first moment of the\ndistribution function (current density) and the particle density\n(\\ref{densit}),\n\\bel{veloc} {\\bf u}({\\bf r},t) = \\frac{1}{\\rho}\n \\int \\frac{2 d {\\bf p}}{(2 \\pi \\hbar)^3} \\>\n\\frac{{\\bf p}}{m} \\delta f({\\bf r}, {\\bf p}, t).\n\\end{equation}\n\nThe definition of the collision term in the form (\\ref{intcoll})\nis incomplete without posing conditions for the conservation of\nthe particle number, momentum, and energy (for simplicity\nof notations, we shall omit index $\\infty$ in\nthe static nuclear-matter density component $\\rho$ at\nsecond order terms in the energy density variations).\nNotice that to the order considered, in the equation\nfor energy conservation, $\\varepsilon$ may be replaced by\n$\\varepsilon_{\\bf p}^{\\rm g.e.}$ (see also \\cite{baympeth}).\nIncidentally, for the\nquasiparticle interaction (\\ref{intampfpp}), this substitution\neven becomes {\\it exact}, as the dynamical part $\\delta \\varepsilon$\nwould drop out of the last integral (as follows from\n(\\ref{delepsleq}), (\\ref{delelocglo}), and two first equations\nin the following set of conditions\n\\cite{heipethrev},\n\\begin{eqnarray}\n \\int {\\rm d}{\\bf p}\\; \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t)\\; = \\;0\\;, \\qquad\n \\int {\\rm d}{\\bf p}\\;{\\bf p}\\;\\delta f_{\\rm l.e.}({\\bf r},{\\bf p}, t)\n \\; = \\;0\\;, \\nonumber\\\\\n \\int {\\rm d}{\\bf p}\\;\n \\varepsilon\\;\n \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t) \\; = 0. \\qquad\\qquad\\qquad\\qquad\n\\label{consereq}\n\\end{eqnarray}\nThese equations\nmimics conservation of the\ncorresponding quantities in each collision of quasiparticles and\nensures that of the same quantities calculated for the total\nsystem (without external fields). Together with the basic equation\n(\\ref{landvlas}), one thus has 6 equations for the 6 unknown\nquantities $\\delta \\rho({\\bf r},t)$, $\\delta \\mu({\\bf r},t)$, ${\\bf u}({\\bf r},t)$ and\n$\\delta T({\\bf r},t)$. They allow one to find unique solutions as\nfunctionals of the external field $V_{\\rm ext}(t)$. Below we shall\nsolve these equations in terms of response functions.\nIt may be noted that, due\nto the conditions (\\ref{consereq}), the first variation of the\ndistribution function $\\delta f({\\bf r},{\\bf p},t)$\n(\\ref{dfgeqdfleq}) disappears from the dynamical component\n$\\delta \\rho({\\bf r},t)$ of the density $\\rho({\\bf r},t)$ and of the\nvelocity field ${\\bf u}({\\bf r},t)$. As one knows (see, e.g.,\n\\cite{heipethrev,abrikha,baympeth}), the equation for the velocity field\nreduces to an identity if one takes into account the definition of\nthe effective mass $m^*$ given by (\\ref{effmass}).\n\n\\subsection{The conserving equations}\n\\label{conserveqs}\n\nIn this section, we like to deduce conserving equations for the\nparticle number, momentum, and energy, which later on will turn out\nhelpful to find appropriate solutions of the Landau--Vlasov\nequation (\\ref{landvlas}). The procedure, which basis on a moment\nexpansion, is well known from textbooks\n\\cite{pinenoz,baympeth,forster}.\nWe will follow more\nclosely the version of \\cite{kolmagpl,magkohofsh} (see also \n\\cite{galiqmod}).\n\n\\subsubsection{THE MOMENT EXPANSION}\nWhereas particle number conservation implies to have\n \\bel{conteq}\n\\frac{\\partial \\rho}{\\partial t} +\n {\\bf \\nabla} \\left(\\rho {\\bf u}\\right) = 0,\n\\end{equation}\nthe momentum conservation is reflected by the following set of\nequations\n \\bel{momenteq}\nm \\rho\n\\frac{\\partial u_\\alpha}{\\partial t} +\n \\sum_{\\beta}\n\\frac{\\partial \\Pi_{\\alpha \\beta}}{\\partial r_{\\beta}} =\n -\\frac{\\partial V_{\\rm ext}}{\\partial r_\\alpha}.\n\\end{equation}\n Besides quantities introduced before, they involve\n\\bel{momentflux}\n\\Pi_{\\alpha\\beta}=\n \\int \\frac{2 d {\\bf p}}{(2 \\pi \\hbar)^3 } \\>\n \\frac{p_\\alpha p_\\beta}{m^*} \\delta f({\\bf r}, {\\bf p}, t)\n + \\frac{\\mathcal{F}_0}{\\mathcal{N}(T)} \\delta \\rho\\left({\\bf r},t\\right)\n \\;\\delta_{\\alpha\\beta}.\n\\end{equation}\n Substituting for $\\delta f({\\bf r}, {\\bf p}, t)$\n(\\ref{dfgeqdfleq}) into (\\ref{momentflux}), one gets\n\\bel{momentflux1} \\Pi_{\\alpha\\beta} = - \\sigma_{\\alpha\\beta} +\n\\delta {\\cal P} ~\\delta_{\\alpha\\beta}.\n\\end{equation}\nThe first component $\\sigma_{\\alpha\\beta}$,\nwhich results from the first term $\\delta\nf_{\\rm l.e.}({\\bf r}, {\\bf p}, t)$ on the right of (\\ref{dfgeqdfleq}),\ndetermines the dynamic shear stress tensor,\n\\bel{presstens}\n\\sigma_{\\alpha\\beta}({\\bf r}, t)=\n - \\int \\frac{2 d {\\bf p}}{(2 \\pi \\hbar)^3}\\;\n \\frac{p_{\\alpha} p_{\\beta}}{m^*}\n \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t)\\;,\n\\end{equation}\nwhose trace vanishes.\nFor a linearized dynamics, the non-diagonal components of the\nmomentum flux tensor $\\Pi_{\\alpha\\beta}$ equal the corresponding\nstress tensor (but with the opposite sign), with correction terms\nbeing proportional to $u_\\alpha u_\\beta$ in $\\delta f$, and thus,\nof higher order, see (\\ref{veloc}) for $u_{\\alpha}$.\n\nThe second component of the momentum flux tensor of\n(\\ref{momentflux1}) can be derived from the variation $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$ as given by (\\ref{dfleq}). It\nrepresents the compressional part of the momentum flux tensor,\n\\begin{eqnarray}\\label{pressdef}\n&&\\int \\frac{2 \\hbox{d} {\\bf p} }{(2 \\pi \\hbar)^3}~\n \\frac{p_\\alpha p_\\beta }{m^*}~\n \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) +\n\\frac{\\mathcal{F}_0}{\\mathcal{N}(T)}~\\delta \\rho~\n\\delta_{\\alpha\\beta}= \\delta {\\cal P} \\delta_{\\alpha\\beta}\n\\nonumber\\\\\n&&{\\rm with}\\quad \\delta \\rho \\equiv \\delta \\rho({\\bf r},t)= \\int\n\\frac{2\\hbox{d} {\\bf p}}{(2 \\pi \\hbar)^3}~\n \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right)\n\\end{eqnarray}\n[mind (\\ref{dfgeqdfleq}) and (\\ref{consereq})]. Notice, that here\nonly the diagonal parts survive. The only non-diagonal ones could come\nfrom the terms in (\\ref{dfleq}) involving ${\\bf p}{\\bf u}$; but they vanish\nwhen integrating over angles in momentum space. Traditionally,\n$\\delta {\\cal P}$ in (\\ref{pressdef}) is referred to as the\nscalar pressure, see \\cite{brenig}.\nUsing\n(\\ref{dfleq}) for the distribution $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$ and its properties mentioned above,\nafter some simple algebraic transformations, one\ngets\n \\begin{eqnarray}\\label{pressureq}\n\\delta {\\cal P} &=&\\frac{2}{3}~ \\int \\frac{2 \\hbox{d} {\\bf p}}{(2 \\pi \\hbar)^3}\n \\frac{p^2}{2 m^*}\n \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) +\n\\frac{\\mathcal{F}_0 }{\\mathcal{N}(T)}~\\delta \\rho \\nonumber\\\\\n&=&{K^T \\over 9}\n\\delta \\rho + \\rho\\left(\\varsigma-\n\\frac{\\mathcal{M} }{\\mathcal{N}}\\right)~\\delta T,\n\\end{eqnarray}\n with $K^T$ being the isothermal\nin-compressibility (\\ref{isotherk}). For the derivation of the\nsecond equation in (\\ref{pressureq}), one can use (i)\nthe transformation of $\\delta \\mu$ to the\nvariations of $\\delta \\rho$ and $\\delta T$\n[see (\\ref{dmu})], and (ii) the relations\n (\\ref{entropydef}), (\\ref{densstat}), and (\\ref{mcapt}) for\nthe entropy per particle $\\varsigma$, the particle density $\\rho$\nas well as for the quantity $\\mathcal{M}$ (\\ref{mcapt}),\nrespectively. Inspecting (\\ref{dpdtr}) and\n(\\ref{incomprTdef}), it becomes apparent that the expression on\nthe very right of (\\ref{pressureq}) may indeed be interpreted as\nan expansion of the static pressure to the first order in\n$\\delta \\rho$ and $\\delta T$. It is thus seen that the truly\nnon-equilibrium component $\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$ only appears\nin the shear stress tensor $ \\sigma_{\\alpha\\beta} $ given in\n(\\ref{presstens}).\n\n\nNote, here and below within\nSec.\\ \\ref{conserveqs}, we omit immaterial constants related to the\nglobal equilibrium (static) components of the moments to simplify\nthe notations and adopt them to the ones of the standard textbooks\nwhen it will not lead to misunderstanding. We should\nemphasize that the Landau quasiparticle theory which is a basis of\nour derivations is working in a self-consistence way with small\ndeviations from (small excitations near) the Fermi surface\nwhich are denoted by symbol \"$\\delta$\" and takes\nabove mentioned static components as those of the external phenomenological\n(experimental) data. Therefore, all relations discussed below in\nthis section should be understood as the ones between such\nclose-to-Fermi-surface quantities within our linearized\nLandau--Vlasov phase space dynamics after exclusion of all above\nmentioned immaterial constants. Nevertheless, we keep the symbol\n$\\delta$ with the scalar pressure $\\delta \\mathcal{P}$ to avoid\npossible misunderstanding related to the linearization procedure,\nsee more comments below after (\\ref{deformtens}).\n\n\n\\subsubsection{THE STRESS TENSOR}\n\nIt may be worthwhile to relate the stress tensor $\n\\sigma_{\\alpha \\beta}$ given in (\\ref{presstens}) to the standard\nform in terms of the\ncoefficients of the shear modulus $\\lambda$ and the viscosity\n$\\nu$,\n \\bel{prestensone}\n\\sigma_{\\alpha\\beta}=\n \\sigma_{\\alpha\\beta}^{(\\lambda)}\n + \\sigma_{\\alpha\\beta}^{(\\nu)}.\n\\end{equation}\n Here, the first term $ \\sigma_{\\alpha\\beta}^{(\\lambda)}$\nis the conservative part of the stress tensor $\n\\sigma_{\\alpha\\beta}$,\n\\bel{presslamb}\n\\sigma_{\\alpha\\beta}^{(\\lambda)}=\n \\lambda\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha \\beta}\\right)\n\\end{equation}\nwith $~{\\bf u} = \\partial {\\bf w}\/\\partial t$\nand ${\\bf w}$ being the displacement field.\nThe second term in (\\ref{prestensone}) can be written as\n\\bel{pressnu}\n\\sigma_{\\alpha\\beta}^{(\\nu)}=\n \\nu\\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right),\n\\end{equation}\n where $\\nu$ is the coefficient of the shear viscosity (or the\nfirst viscosity). For more details see Appendix A.2,\nin particular for expressions of the coefficients $\\lambda$\n(\\ref{shearmod}) and $\\nu$ (\\ref{viscos}) in terms of\nFermi liquid interaction parameters.\n\nTo obtain microscopic expressions for the shear modulus $\\lambda$\nand the viscosity $\\nu$, one needs to exploit the solution $\\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},t)$ of the Landau--Vlasov equation (\\ref{landvlas})\nfor the stress tensor $\\sigma_{\\alpha\\beta}({\\bf r},t)$ (\\ref{presstens}),\nreducing the latter to the form (\\ref{prestensone}). Such a\ncalculation of $\\lambda$ and $\\mu$ in terms of the\nLandau Fermi-liquid parameters is discussed in Appendix A.2,\nin which Fourier transforms are exploited \\cite{kolmagpl}.\nEquivalently, one may\nexpress functions of space and time by plane waves, which for the\ndistribution function reads \\cite{strutmagden0,kolmagpl}\n\\bel{planewave}\n\\delta f({\\bf r}, {\\bf p}, t)=\n\\delta {\\tilde f}\\left({\\bf q},{\\bf p},\\omega \\right)\n \\hbox{exp}\\left[{i({\\bf q}{\\bf r}-\\omega t)}\\right]\n\\end{equation}\n with ${\\bf q}$ being the wave vector and $\\omega $ the frequency of the\nvibrational modes of nuclear matter. Such a plane-wave\nrepresentation is to be applied to both sides of (\\ref{presstens})\nand (\\ref{prestensone}). The amplitudes for the velocity\n${\\bf u}$ and the displacement ${\\bf w}$ field then satisfy ${\\bf\n{\\tilde w}}={\\tilde{\\bf u}}\/(-i\\omega )$.\n\nUsing (\\ref{presstens}),\n(\\ref{prestensone}), (\\ref{presslamb}) and (\\ref{pressnu}) for\nthe stress tensor $\\sigma_{\\alpha\\beta}$\nand (\\ref{pressureq}) for the scalar pressure $\\delta \\mathcal{P}$,\n \\bel{incomprtotdef}\n \\delta {\\cal P}=\\frac{K_{\\rm tot}}{9}~\\delta \\rho ,\n\\end{equation}\none\nfinally may write down a general expression for the momentum flux\ntensor $\\Pi_{\\alpha\\beta}({\\bf r},t)$ (\\ref{momentflux1}),\n\\begin{eqnarray}\\label{momentfluxtot}\n&&\\Pi_{\\alpha\\beta}=\n -\\lambda\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha \\beta}\\right) ~~\\nonumber\\\\\n &-&\\nu\\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right)\n + \\frac{K_{\\rm tot}}{9}~\\delta \\rho~\\delta_{\\alpha\\beta}.~~~\n\\end{eqnarray}\nA total effective in-compressibility $K_{\\rm tot}$ includes the change of\nthe pressure due to\nvariations of the temperature with density.\nWith the help of (\\ref{dpdtr}), the in-compressibility $K_{\\rm tot}$\ncan be expressed through the specific\nheat per particle ${\\tt C}_{\\mathcal{V}}$\n(\\ref{specifheatpdef}),\n \\bel{incomprtot}\nK_{\\rm tot}=K^{T}+ 6 {\\tt C}_{\\mathcal{V}}\\; \\rho\n ~\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}.\n\\end{equation}\nAgain, $\\delta {\\tilde T}$ and $\\delta {\\tilde \\rho}$ are the\nFourier components of $\\delta T({\\bf r},t)$ and $\\delta \\rho({\\bf r},t)$.\nLike all other kinetic coefficients, such as\n$\\lambda$ and $\\nu$ given in (\\ref{shearmod}) and (\\ref{viscos}),\nrespectively, this effective, total in-compressibility modulus\n$K_{\\rm tot}$, too, depends on $\\omega $ and $q$. Later on we shall\ndiscuss in more detail these quantities in the LWL\nlimit. In this limit, the total in-compressibility $K_{\\rm tot}$ will\nbe seen to become identical to the adiabatic one $K^{\\varsigma}$\ngiven in (\\ref{incompradexp}).\n\n\\subsubsection{ENERGY CONSERVATION AND THE GENERAL TRANSPORT\\\\ EQUATION}\n\nSo far we have not looked at the energy conservation. For this purpose,\none needs to consider thermal aspects as they appear in equations\nfor the change of entropy and temperature. To do this we will\nfollow standard procedures. We first built the\nscalar product of the mean velocity ${\\bf u}$ with the vector equation,\nwhose component $\\alpha$ is given by (\\ref{momenteq}). Making\nuse of the continuity equation (\\ref{conteq}), after some\nmanipulations, one gets\n\\begin{eqnarray}\\label{enerconstwo}\n&&\\frac{\\partial}{\\partial t}\n \\left(\\frac{1}{2} m\\rho u^2+\\rho {\\cal E}\\right) = \\nonumber\\\\\n &=& -\\sum_{\\alpha\\beta}\\frac{\n\\partial}{\\partial r_{\\beta}} \\left[u_{\\alpha}\n \\left(\\frac{1}{2} m \\rho u^2 \\delta_{\\alpha\\beta}\n +\\rho W_{\\alpha\\beta}-\n \\sigma_{\\alpha\\beta}^{(\\nu)}\n -\\kappa \\frac{\\partial T}{\\partial r_{\\alpha}}\n \\delta_{\\alpha\\beta}\\right)\\right] \\nonumber\\\\\n &+& \\rho T \\left(\\frac{\\partial \\varsigma}{\\partial t}\n + {\\bf u}{\\bf \\nabla} \\varsigma\\right)\n -{\\bf \\nabla}\\left(\\kappa {\\bf \\nabla}T\\right) \\nonumber\\\\\n &-&\\frac{\\nu}{2}\\sum_{\\alpha\\beta}\n \\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta} }+\n \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha\\beta}\\right)^2\n -\\rho {\\bf u}{\\bf \\nabla} V_{\\rm ext}.\n\\end{eqnarray}\nOn the left hand side, there appears the mean kinetic energy\ndensity and the internal energy density $\\rho \\mathcal{E}$ per unit\nvolume (defined again up to an immaterial constant). The density\n${\\cal E}$ itself may be split in three different components,\n\\bel{enerintr}\n{\\cal E}= {\\cal E}^{(\\lambda)}+{\\cal E}_{\\rm tot}^{(K)}\n+ T \\delta \\varsigma\\;.\n\\end{equation}\n The first one,\n\\bel{shearen}\n{\\cal E}^{(\\lambda)}\n =\\frac{\\lambda}{4 \\rho}\n \\sum_{\\alpha \\beta}\\left(\\frac{\n\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha\n \\beta}\\right)^2\\;,\n\\end{equation}\n is related to shear deformations, which is known from the solid state\nphysics and for Fermi liquids as coming from\ndistortions of the Fermi surface \\cite{landau}.\nThe second one may\nbe written as\n\\bel{enerintrk}\n {\\cal E}_{\\rm tot}^{(K)}= \\frac{K_{\\rm tot}}{18} \\left(\\delta \\rho\\right)^2 ;\n\\end{equation}\n it represents the\ncompressional component, associated to the effective total\nin-compressibility $K_{\\rm tot}$,\nwhich is\nin line of the known thermodynamic relations.\nEquation (\\ref{enerintrk}) resembles the expression\nfound in \\cite{strutmagbr,strutmagden}, except for a generalization\nof the physical meaning of the in-compressibility modulus\n$K_{\\rm tot}$ as function of $\\omega $ and $q$\ngiven in (\\ref{incomprtot}), as compared to the quasistatic adiabatic\ncase. The third one in (\\ref{enerintr}) represents the\nchange of heat part resulting from a change of entropy.\nWe keep here the dynamical variation symbol $\\delta$ for the\nentropy $\\varsigma$ (also for the pressure $\\mathcal{P}$ here and\nbelow) to remember that all quantities of the Landau Fermi-liquid\ntheory are presented for small dynamical deviations near the\nFermi surface in the linear (or quadratic after multiplying\n(\\ref{momenteq}) by ${\\bf u}$) form in $\\delta f$. We avoid here a\nmisunderstanding with following transformations of the energy\n${\\cal E}$ (\\ref{enerintr}), say Legendre ones, to the\ndifferential form in line of a\ngeneral comment at the beginning of this section.\nOn the r.h.s. of (\\ref{enerconstwo})\nthe enthalpy $W_{\\alpha\\beta}$ per particle has been introduced,\n\\bel{enthalp} W_{\\alpha\\beta}=\n {\\cal E} \\delta_{\\alpha\\beta}\n + \\frac{1}{\\rho}\\left(\\sigma_{\\alpha\\beta}^{(\\lambda)} +\n \\delta {\\cal P}~ \\delta_{\\alpha\\beta} \\right)\n\\end{equation}\n (see the comment above concerning $\\delta {\\cal P}$).\nFurthermore, the thermodynamic relation for the dynamical\nvariations of the internal energy ${\\cal E}$ in\nterms of those $\\varsigma$ for the entropy per particle $\n\\varsigma$, the density $\\rho$ and the displacement tensor\n$w_{\\alpha\\beta}$, is given by\n\\bel{thermener}\n{\\rm d} {\\cal E} =T {\\rm d}\n\\varsigma+ \\frac{\\delta {\\cal P}}{\\rho^2} {\\rm d} \\rho +\n\\frac{1}{\\rho}\n \\sum_{\\alpha\\beta}\\sigma_{\\alpha\\beta}^{\\lambda}\n {\\rm d} w_{\\alpha\\beta}.\n\\end{equation}\nThe displacement tensor $w_{\\alpha\\beta}$ is defined as\n\\bel{deformtens}\nw_{\\alpha\\beta}=\n \\frac{1}{2}\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}}\n + \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}}\\right)\\;.\n\\end{equation}\nNote that equation (\\ref{incomprtotdef}) for the pressure $\\delta\n{\\cal P}$ is important to get\n(\\ref{enerintr}) by integration of (\\ref{thermener}).\nAccording to (\\ref{enthalp}), we get the standard relation of\n{\\it linearized} thermodynamics of\n\\cite{brenig}, for instance,\nbetween the enthalpy (\\ref{enthalp}) and entropy $\\varsigma$ up to the\nsecond order term in $\\delta {\\cal P}$.\n\nIn (\\ref{enerconstwo}), we also added and subtracted the term\n${\\bf \\nabla} {\\bf j}_T$ containing the heat current,\n\\bel{currheat}\n{\\bf j}_{{}_{\\! T}}= -\\kappa {\\bf \\nabla} T\\;,\n\\end{equation}\nwith the coefficient $\\kappa$ for the thermal conductivity.\nWe may now write the\nequation for energy conservation as\n\\begin{eqnarray}\\label{enerconserv}\n&&\\frac{\\partial}{\\partial t}\n \\left(\\frac{1}{2} m\\rho u^2+\\rho {\\cal E} \\right)=\n -\\sum_{\\alpha\\beta}\\frac{\n\\partial}{\\partial r_\\beta} \\left[u_\\alpha\n \\left(\n \\frac{1}{2} m \\rho u^2 \\delta_{\\alpha\\beta} \\right.\\right.\\nonumber\\\\\n &+&\\left.\\left.\\rho W_{\\alpha\\beta}\n - \\sigma_{\\alpha\\beta}^{(\\nu)}\n -\\kappa \\frac{\n\\partial T}{\\partial r_{\\alpha}} \\delta_{\\alpha\\beta}\\right)\\right]\n -\\rho {\\bf u}{\\bf \\nabla} V_{\\rm ext}.\n\\end{eqnarray}\nIn this way, it is seen that from (\\ref{enerconstwo}) and\n(\\ref{enerconserv}), together with the continuity equation\n(\\ref{conteq}) and the definition of the heat current ${\\bf j}_T$\n(\\ref{currheat}), one\ngets for the change of entropy:\n\\begin{eqnarray}\\label{entropyeq}\n \\frac{\\partial (\\rho \\delta \\varsigma)}{\\partial t}\n &=&-{\\bf \\nabla}\\left(\\rho \\delta \\varsigma\\; {\\bf u}\n + \\frac{1}{T} {\\bf j}_{{}_{\\! T}}\\right)\n + \\frac{\\kappa}{T^2}\\left({\\bf \\nabla}T\\right)^2 \\nonumber\\\\\n &+&\\frac{\\nu}{2T} \\sum_{\\alpha\\beta}\n \\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right)^2\n\\end{eqnarray}\n[again the variation $\\delta$ in $\\delta \\varsigma$ is not\nomitted because of the following derivations of the Fourier\nequation (\\ref{fouriereq}) and thermal conductivity\n(\\ref{kappadef}) in Appendix A.2].\nThese two equations have a very clear physical meaning\nfor normal liquids and amorphous solids (a very\nviscose liquids are associated to the amorphous\nsolids with some shear modulus $\\lambda$ in our notations, i.e.,\nsolids without any crystal structure).\nThe first equation (\\ref{enerconserv}) claims that the change of the\ncollective and internal energy, concentrated in unit volume per\nunit of time and presented as the sum of the collective kinetic\nand internal parts, equals the corresponding energy flux through\nits surface and work of the external field. The second equation\n(\\ref{entropyeq}) is usually called as a general\nheat transport equation.\nThis\nequation states that the change of entropy in the unit volume\nper unit of time equals the entropy flux through its surface\n(heat energy flux). Two other terms\nshow the entropy increase related to the gradient of the\ntemperature and dissipation due to the shear ($\\nu$) viscosity.\nNote that there is no explicit dependence on the\nexternal field in (\\ref{entropyeq}). This dependence is\nmanifested only through the solutions of the dynamical equations\nin terms of the moments. For zero external field (closed system)\nthe entropy is increasing because of the basic thermodynamic law.\nTherefore, according to (\\ref{entropyeq}), the\nshear viscosity $\\nu$ (\\ref{viscos}) and the thermal conductivity\n$\\kappa$ should be positive. The energy conservation equation for\nthe Fermi liquid (\\ref{enerconserv}) differs from the one for\nclassical hydrodynamics by the same\nFermi-surface distortions related to the shear modulus $\\lambda$\n(\\ref{shearmod}) as discussed above. That is similar to the\namorphous solids (in the above mentioned sense\nof very viscose liquids). However, in\ncontrast to the latter, one obtains the energy conservation\ncondition (\\ref{enerconserv}) for the dynamical variations of the\nFermi-liquid collective and internal energy with the specific\nconstants $\\lambda$ (\\ref{shearmod}), $\\nu$ (\\ref{viscos}) and\n$K_{\\rm tot}$ (\\ref{incomprtot}) found from the relation to the\nLandau--Vlasov equation (\\ref{landvlas}). Our way of the\nderivation of the energy conservation equation\n(\\ref{enerconserv}) for the Fermi liquids within the {\\it\nlinearized} Landau--Vlasov dynamics (\\ref{landvlas}),\nas for normal liquids and\nsolids, leads to a more explicit form of the energy conservation\nequation than that suggested in \\cite{heipethrev,baympeth}. In this way,\nwe get rather simple expressions for the collective and internal\ncomponents of the energy out the hydrodynamical limit.\n\n\\subsubsection{POTENTIAL FLOW: FERMI LIQUID VERSUS HYDRODYNAMICS}\n\\label{potenflow}\n\nBelow, we shall be interested in the case of a viscous potential\nflow, for which one has\n\\bel{velpoten}\n{\\bf u}={\\bf\n\\nabla}\\varphi,\\quad {\\bf w}={\\bf \\nabla}\\varphi_w\\quad\n{\\rm with} \\quad \\varphi=\\dot{\\varphi}_w\n\\end{equation}\n [cf. the second\nequation with (\\ref{presslamb})]. With the help of these\ndefinitions, the momentum equation (\\ref{momenteq}) and the flux\ntensor (\\ref{momentfluxtot}) can be brought to the following\nforms:\n\\bel{navstokeq0}\nm \\rho \\frac{\\partial \\varphi}{\\partial\nt} - \\frac{4}{3} \\nu~ \\Delta \\varphi\n - \\frac{4}{3} \\lambda~ \\Delta \\varphi_w\n+ \\frac{K_{\\rm tot}}{9} \\delta \\rho=-V_{\\rm ext}\n\\end{equation}\nand\n\\begin{eqnarray}\\label{momentfluxpot}\n\\Pi_{\\alpha\\beta}&=& -2 \\left(\n\\frac{\\lambda}{-i\\omega } +\\nu \\right)\n\\left(\\frac{\n\\partial^2\\varphi}{\\partial r_\\alpha \\partial r_\\beta}- \\triangle \\varphi \\;\n\\delta_{\\alpha\\beta} \\right) \\nonumber\\\\\n&-&\\left(m \\rho \\frac{\n\\partial \\varphi}{\\partial t}+ V_{\\rm ext}\\right) \\delta_{\\alpha\\beta}.\n\\end{eqnarray}\nThe diagonal term given on the very right of\n(\\ref{momentfluxpot}) had been used to remove $\\delta \\rho$\nwhich still appears in (\\ref{momentfluxtot}). With the continuity\nequation (\\ref{conteq}) for the plane wave solutions\n(\\ref{planewave}), one has from (\\ref{navstokeq0}) the equation\nfor the velocity potential $\\varphi$ \\cite{magkohofsh}:\n\\bel{navstokeq}\nm \\rho \\frac{\\partial^2 \\varphi}{\\partial t^2}\n-\\frac{\\rho}{9}\\left(K_{\\rm tot} +12 \\lambda\/\\rho \\right) \\Delta\n\\varphi\n - \\frac{4}{3} \\nu~ \\Delta \\frac{\\partial \\varphi}{\\partial t}\n = - \\frac{\\partial V_{\\rm ext}}{\\partial t}.\n\\end{equation}\nThe structure of\n(\\ref{navstokeq}) for the potential flow is similar to that of\nthe Navier-Stokes equation for the velocity potential $\\varphi$.\nThe difference to the case of the common classical liquid, is seen in\nthe terms proportional to $\\lambda$, viz in the presence of the\nanisotropy term (\\ref{presslamb}), which actually represents a\n{\\it reversible} motion. Such a term is known from the dynamics of\namorphous solids. We emphasize that for Fermi liquids, this term\narises only in the presence of the Fermi surface distortions, which\nsurvive even in the non-viscous limit; they will turn out\nimportant for our applications below. The shear modulus $\\lambda$\nmay be interpreted as a measure of those distortions which are\nrelated to a reversible anisotropy of the momentum flux tensor.\nThey disappear in the hydrodynamic limit, and so does\n$\\lambda$, in which case all formulas of this section turn into\nthose for normal liquids; for more details see section\n\\ref{longwavlim}.\n\nAt this place an important remark is in order. It should be noted\nthat in contrast to classical hydrodynamics\nour system of equations for the moments is {\\it not closed} to the\nfirst few ones, namely particle density $\\delta \\rho$ and velocity\nfield ${\\bf u}$. This is true in particular for (\\ref{navstokeq})\nfor the potential flow $\\varphi$. Indeed, the coefficients\n$\\lambda$, $\\nu$ and $K_{\\rm tot}$ depend on the variable $\\omega \/q$\nwhich yet is unknown. The latter is determined from a\ndispersion relation, which in turn has to be derived from the\nLandau--Vlasov equation (\\ref{landvlas}). Such a procedure goes\nback to \\cite{landau} where the dispersion relation was exploited\nfor the collisionless case at $T=0$. A collision term in the\nrelaxation approximation has been taken into account in\n\\cite{abrikha}. The extension to heated Fermi liquids and low\nexcitations, in the way, which we are going to use later on, has been\ndeveloped in \\cite{heipethrev}. It may be noted that this version\nof the dispersion relation, which we are aiming at, differs\nessentially from the one obtained in the \"truncated\" (scaling model)\nversions of\nthe Fermi-liquid theory of \\cite{holzwarth,nixsierk}, where the\nmomentum flux tensor is not influenced by higher moments of the\ndistribution function.\nWe take into account all other multipolarities (larger the quadrupole\none) of the Fermi-surface distortions when there is no\nconvergence in multipolarity expansion of the distribution\nfunction for finite and large $\\omega \\tau$ or for finite\n$K_{\\rm tot}$, for instance for nuclear matter with small $\\mathcal{F}_0$,\nin contrast to the Fermi liquid $^3$He.\n\n\n\\subsection{Response functions}\n\\label{respfunsec}\n\n\\subsubsection{DYNAMIC RESPONSE}\n\\label{dynresp}\n\nAs mentioned earlier, we want to solve the linearized equations of\nmotion in terms of response functions. We concentrate on two\nquantities, namely particle density $\\rho({\\bf r},t)$ and temperature\n$T({\\bf r},t)$ and examine how they react to the external field\n$V_{\\rm ext}({\\bf r},t)$ introduced earlier. This may be quantified by the\nfollowing two response functions: The density-density response\n$\\chi_{DD}^{\\rm coll}$ and the temperature-density response\n$\\chi_{TD}^{\\rm coll}$ defined as\n\\bel{ddrespdef}\n\\chi_{DD}^{\\rm coll}(q,\\omega )=\n - \\frac{\\delta \\rho(q,\\omega )}{V_{\\rm ext}(q,\\omega )}\n\\end{equation}\n and\n \\bel{dtrespdef} \\chi_{TD}^{\\rm coll}(q,\\omega )=\n - \\frac{\\delta T(q,\\omega )}{V_{\\rm ext}(q,\\omega )},\n\\end{equation}\nrespectively. To keep the notation simple, we will omit the\ntilde characterizing the Fourier transform of the distribution\nfunction (\\ref{planewave}) (it should suffice to only show the\narguments $q,\\omega $). The definition of the response functions is\nidentical to the one of \\cite{heipethrev}, except that we have\nintroduced the suffix \"coll\". This was done adopting a notation\nused in the literature of nuclear physics when the dynamics of a\nfinite nucleus is expressed in terms of shape variables, to which\nwe will come below. Notice, however, that $V_{\\rm ext}(q,\\omega )$ is only\nproportional to the density,\n$V_{\\rm ext}(q,\\omega )=q_{\\rm ext}(\\omega )\\rho(q,\\omega )$, with $q_{\\rm ext}(\\omega )$\nbeing some externally determined function. Often, one therefore\ndefines response functions in a slightly modified way, in that the\nfunctional derivatives are performed with respect to\n$q_{\\rm ext}(\\omega )$ instead of $V_{\\rm ext}(q,\\omega )$ (see,\ne.g., \\cite{pinenoz}).\n\nAs will be seen below, these functions only depend on the wave\nnumber $q$ but not on the angles of the wave vector ${\\bf q}$. For this\nreason, it is convenient to introduce the dimensionless quantities\n$s$ and $\\tau_q$ (with $v_{{}_{\\! {\\rm F}}}=p_{{}_{\\! {\\rm F}}}\/m^*$)\n\\bel{som}\ns= \\frac{\\omega }{v_{{}_{\\! {\\rm F}}} q},\\quad\n\\tau_q =\\tau v_{{}_{\\! {\\rm F}}} q,\n \\quad{\\rm implying}\\quad \\omega \\tau = s \\tau_q,\n\\end{equation}\n instead of the frequency $\\omega $ and the wave number $q$.\n\nTo calculate the response functions (\\ref{ddrespdef}) and\n(\\ref{dtrespdef}) we follow the procedure of \\cite{heipethrev}. As\nany further details may be found there, it may suffice to outline\nbriefly the main features. In short, the strategy is as follows.\nFirstly, one rewrites the Landau--Vlasov equation (\\ref{landvlas})\nin terms of the Fourier coefficients introduced in\n(\\ref{planewave}). Evidently, in the spirit of the separation\nspecified in (\\ref{dfgeqdfleq}), we need to evaluate explicitly\nonly the first component\n$\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$ which\nenters the conditions (\\ref{consereq}). By a\nstraightforward calculation, one may then express $\\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},\\omega )$ in terms of the unknown quantities $\\delta\n\\rho$, ${\\bf u}$, $\\delta \\mu$ and $\\delta T$ for any given external\nfield $V_{\\rm ext}$. The form is given in (\\ref{basiceq}).\nThe continuity equation (\\ref{conteq})\nin the Fourier representation through\n (\\ref{planewave}),\n${\\bf q}{\\bf u}=\\omega \\delta\n \\rho \/ \\rho$,\nmay be used to eliminate the velocity field ${\\bf u}$.\nFurthermore, the thermodynamic relation [see (\\ref{drhomt}),\n(\\ref{incomprTdef}) and (\\ref{isotherk})]\n\\bel{dmu} \\delta \\mu =\n \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T \\delta \\rho +\n \\left(\\frac{\\partial \\mu }{\\partial T}\\right)_{\\rho} \\delta T =\n \\frac{K^{T} }{9\\rho}\\;\n \\delta \\rho - \\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\;\\delta T\n\\end{equation}\nallows one to express the chemical potential $\\delta \\mu$ in\nterms of the two unknown variables $\\delta \\rho$ and $\\delta T$.\nNext, one may exploit the conditions (\\ref{consereq}). As the\nsecond (set of) equation(s) is just an identity, provided one\nuses the appropriate definition of the effective mass\n(\\ref{effmass}), it is only\nthe first and the third equation which matter. They\nmay determine the remaining two\nvariables $\\delta \\rho$ and $\\delta T$ in terms of the external\nfield,\n\\bel{drhodteqone}\n \\left(\\frac{i s \\tau_q }{i s \\tau_q -1}\n -\\wp(s) \\chi _0 \\right) \\delta \\rho\n + \\frac{1}{1-i s \\tau_q } \\chi _1 \\delta T\n =-\\chi _0 \\delta V_{\\rm eff}\n\\end{equation}\n and\n\\bel{drhodteqtwo} -\\wp(s) \\chi _1 \\delta \\rho\n + \\frac{1}{1-i s \\tau_q } \\left(\\chi _2\n -i s \\tau_q \\rho \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right) \\delta\nT\n = -\\chi _1 \\delta V_{\\rm eff}.\n\\end{equation}\n Here, the quantity\n\\bel{alphas}\n \\wp(s) = \\frac{1}{\\mathcal{N}(T)}\\frac{1}{i s \\tau_q -1}\n - \\frac{3is}{\\tau_q \\mathcal{N}(0)}\n\\end{equation}\nhas been introduced with $\\mathcal{N}(0)$ being the level density\n(\\ref{enerdensnt}) of the quasiparticles at $T=0$,\n\\bel{nzero} \\mathcal{N}(0)= \\frac{p_{{}_{\\! {\\rm F}}} m^*}{\\pi^2 \\hbar^3}=\n \\frac{3}{2} \\frac{\\rho _0}{\\varepsilon_{{}_{\\! {\\rm F}}}},\n\\end{equation}\nand $\\varepsilon_{{}_{\\! {\\rm F}}}=p_{\\rm F}^2\/2m^*$.\nThe functions $\\chi _n$ are given by\n\\bel{chinfun}\n\\chi _n = -\\mathcal{N}(T) \\left\\langle\n\\frac{{\\bf q}{{\\bf v}_{\\bf p}}}{\\mathcal{D}_{\\bf p}}\n\\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu}{T} -\n\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right)^n\\; \\right\\rangle\n\\end{equation}\n with $n=0,1,2,...\\;$,\n\\begin{equation}\\label{domindp}\n\\mathcal{D}_{\\bf p}=\n \\omega -{\\bf q}{\\bf v}_{\\bf p}+ i\/\\tau, \\qquad{\\bf v}_{\\bf p}={\\bf p}\/m^*.\n\\end{equation}\n Furthermore, in (\\ref{drhodteqone}) and\n(\\ref{drhodteqtwo}) a short hand notation $\\delta V_{\\rm eff}$ has\nbeen used for the sum of two terms, namely\n\\bel{delueff} \\delta\nV_{\\rm eff}=\n V_{\\rm ext} + k(\\omega ,T) \\delta \\rho\n\\end{equation}\n with\n\\bel{couplconst} k(\\omega ,T) =\\frac{1}{\\mathcal{N}(T)}\n \\left[\\mathcal{F}_0 + \\frac{\\mathcal{F}_1}{\\mathcal{G}_1}\n \\left(\\frac{\\omega }{v_{{}_{\\!{\\rm F}}} q}\\right)^2\\right].\n\\end{equation}\nIn (\\ref{delueff}), $\\delta V_{\\rm eff}$ may be considered as an\n{\\it effective} field which includes the true external field\n$V_{\\rm ext}$ and the \"screened\" field $k\\delta \\rho$\n\\cite{heipethrev}. Our notation follows the one often used for\nfinite nuclei: The second term in (\\ref{delueff}) plays the\nrole of the collective variable and $k$ of (\\ref{couplconst})\nrepresents the \"coupling\" constant (see, e.g.,\n\\cite{bohrmot,hofbook,hofmann}).\n\nThe response function $\\chi_{DD}^{\\rm coll}$ of (\\ref{ddrespdef})\ncan be now obtained\nfrom (\\ref{drhodteqone}) and (\\ref{drhodteqtwo}),\n\\bel{ddresp} \\chi_{DD}^{\\rm coll}( \\tau_q ,s)=\n \\frac{\\aleph( \\tau_q ,s)}{D( \\tau_q ,s)},\n\\end{equation}\nwhere\n\\bel{despfunc} D( \\tau_q ,s)=\n D_0( \\tau_q ,s)+ k( \\tau_q ,s) \\aleph( \\tau_q ,s)\n\\end{equation}\n with\n\\bel{chidomin} D_0( \\tau_q ,s)=\n \\left(\\frac{i s \\tau_q }{i s \\tau_q-1}\n -\\wp(s)\\chi _0 \\right)\n \\left(\\chi _2 - i s \\tau_q \\rho\n\\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right)\n + \\wp \\chi _1 ^2.\n\\end{equation}\nIn (\\ref{ddresp}), $\\aleph(\\tau_q ,s)$ finally is given by\n\\bel{chinumer}\n\\aleph(\\tau_q ,s)=\n \\chi _0 \\left(\\chi _2\n -i s \\tau_q \\rho\n\\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right)- \\chi _1 ^2.\n\\end{equation}\n\nIt is worth noticing that the collective response function for the\ndensity-density mode, as given by (\\ref{ddrespdef}) or\n(\\ref{ddresp}), can be expressed as\n\\bel{ddrespchiin}\n\\chi^{\\rm coll}(q,\\omega )=\n \\frac{\\chi (q,\\omega )}{1+k(\\omega ,T) \\chi (q,\\omega )}.\n\\end{equation}\nThis form is analogous to the form used to describe the\ndynamics of shape variables \\cite{bohrmot,hofbook,hofmann}. We omit\nhere the suffix $DD$ because the $TD$ response function takes on a\nsimilar form\n(with some modification of the numerator). It is here where the\n\"coupling constant\" $k$ appears, as defined in\n(\\ref{couplconst}), together with the \"intrinsic\" (or\n\"un-screened\" (see \\cite{pinenoz,heipethrev}) response\nfunction $\\chi $,\n\\bel{chiindef} \\chi ( \\tau_q ,s)=\n - \\frac{\\delta \\rho}{\\delta V_{\\rm eff}}\n = \\frac{\\aleph( \\tau_q ,s)}{D_0( \\tau_q ,s)}.\n\\end{equation}\n Both expressions can be found already in \\cite{heipethrev}.\nHowever, later we will find the form (\\ref{ddresp}) more\nconvenient for our applications, in particular for the discussion\nof the low frequency limit $\\omega \\tau \\ll 1$.\nWhen we shall expand first $\\chi $ (\\ref{chiindef}) in\n(\\ref{ddrespchiin}) in small $\\omega \\tau$ near the poles of\n$\\chi ^{\\rm coll}$ (\\ref{ddresp}) (see next section), one should\nassume that the singularities of $\\chi$ related to zeros of $D_0$\nin (\\ref{chiindef}) are far away from zeros of $D$ in\n(\\ref{ddresp}), i.e., a smoothness of $\\chi$ as function of\n$\\omega \\tau$ near these poles. After the cancellation of a\npossible singularity source $D_0$ in (\\ref{ddresp}) we are free\nfrom such an assumption.\n\n\nFinally, let us turn to the temperature-density response function\n$\\chi_{TD}^{\\rm coll}$ (\\ref{dtrespdef}). It is determined by the same\nsystem of equations (\\ref{drhodteqone}) and (\\ref{drhodteqtwo})\nand can be written in the form (\\ref{ddrespchiin}) but with\nanother ``intrinsic'' response function $\\chi_{{}_{\\! TD}}$ appearing in the\nnumerator,\n\\bel{chitindef} \\chi_{{}_{\\! TD}}( \\tau_q ,s)=\n - \\frac{\\delta T}{\\delta V_{\\rm eff}}.\n\\end{equation}\n From (\\ref{drhodteqone}), (\\ref{drhodteqtwo}) and\n(\\ref{chitindef}), one obtains\n\\bel{chitbar}\n\\chi_{{}_{\\! TD}}(\\tau_q,s)=\n -\\frac{is\\tau_q~\\chi _1}{D_0(\\tau_q ,s)},\n\\end{equation}\nwhere $D_0( \\tau_q ,s)$ is given by (\\ref{chidomin}).\n As compared to the one printed in \\cite{heipethrev},\nthis expression contains an additional factor $i s\\tau_q\/\\chi_1$,\nwhich later on will turn out important, for instance, when\ncalculating susceptibilities and the in-compressibility $K_{\\rm tot}$\n(\\ref{incomprtot}). (We are grateful to H.\nHeiselberg for confirming this misprint.)\nSubstituting (\\ref{chitbar}) into the\nnumerator of (\\ref{ddrespchiin})\ninstead of $\\chi$, one gets the temperature-density response\nfunction (\\ref{dtrespdef}) in the form similar to (\\ref{ddresp}),\n\\bel{dtresp} \\chi_{TD}^{\\rm coll}( \\tau_q,s)=\n -\\frac{is\\tau_q~\\chi _1}{D( \\tau_q ,s)}.\n\\end{equation}\n Notice that according to (\\ref{ddresp}) and\n(\\ref{dtresp}), both response functions (\\ref{ddrespdef}) and\n(\\ref{dtrespdef}) have the same set of poles, which lie at the\nroots of the equation\n\\bel{despeq} D( \\tau_q ,s)= 0.\n\\end{equation}\nThis is identical to the condition of zero determinant\nfor the system of the linear equations (\\ref{drhodteqone})\nand (\\ref{drhodteqtwo}).\n\n\n\\bigskip\n\n\\subsubsection{LOW TEMPERATURE LIMIT}\n\\label{lowtemlim}\n\nThe expressions for the collective response functions become much\nsimpler at low temperatures $T \\ll \\mu$. In\nthis case, one may calculate $\\chi _n$ of (\\ref{chinfun}) by\nexpanding in powers of ${T \/ \\mu}$. For those applications to\nnuclear physics we have in mind the temperature is sufficiently\nsmall such that it suffices to mainly stick to order two. Fourth\norder terms shall be shown only when necessary.\n\nA basic element for the quantities which we need to evaluate is\nthe derivative $\\partial f_{{\\bf p}} \/ {\\partial \\varepsilon_{{\\bf p}}}$ taken at\nglobal equilibrium:\n\\bel{dfpdep} \\left(\\frac{\n\\partial f_{{\\bf p}}}{\\partial \\varepsilon_{{\\bf p}}}\\right)_{\\rm g.e.}=\n -\\left[\n4 T \\hbox{cosh}^2\\left(\\frac{\\varepsilon_{\\bf p}-\\mu}{2T}\\right)\\right]^{-1}_{\\rm g.e.}.\n\\end{equation}\n It appears in $\\mathcal{N}(T)$ of (\\ref{enerdensnt}) [see\nalso (\\ref{averag})], which in turn is needed for $\\chi _n$\nof (\\ref{chinfun}). For small $T$, this derivative is a sharp\nbell-shaped function of $\\varepsilon_{\\bf p}^{\\rm g.e.}$, such that one may evaluate\nthe averaging integrals (\\ref{chinfun}) and (\\ref{averag}) by\nexpanding the smooth functions in terms of $\\varepsilon_{\\bf p}^{\\rm g.e.}$ near\n$\\varepsilon_{\\bf p}^{\\rm g.e.}=\\mu_{\\rm g.e.}$. In this way, the Fourier-Bernoulli\nintegrals over the dimensionless variable\n$\\left[(\\varepsilon_{\\bf p}-\\mu)\/T\\right]_{\\rm g.e.}$ appear (see, e.g.,\n\\cite{heipethrev})\nwhich lead to\n\\begin{eqnarray}\\label{chitemzero}\n\\chi_0&=&\n \\left[-Q_1(\\zeta)+ \\frac{\\pi^2 {\\bar T}^2}{12}\n \\left(Q_1(\\zeta)-\\zeta Q_1^{\\prime}(\\zeta) \\right.\\right. \\nonumber\\\\\n &-& \\left.\\left. \\frac{1}{2} \\zeta^2 Q_1^{\\prime \\prime}(\\zeta)\\right)\n + \\mathcal{O}\\left({\\bar T}^4\n\\right)\\right]\n \\mathcal{N}(0),\n\\end{eqnarray}\n \\bel{chitemone} \\chi_1=\n \\left[\\frac{\\pi^2 {\\bar T}}{6} \\zeta Q_1^{\\prime}(\\zeta)\n + \\mathcal{O}\\left({\\bar T}^3\n\\right)\\right]\n \\mathcal{N}(0),\n\\end{equation}\n and\n\\begin{eqnarray}\\label{chitemtwo}\n\\chi_2 &=&\n \\left\\{-\\frac{\\pi^2}{3} \\left[Q_1(\\zeta)+\n\\frac{\\pi^2 {\\bar T}^2}{120}\n \\left(36 Q_1(\\zeta)\n - 46 \\zeta Q_1^{\\prime}(\\zeta) \\right.\\right.\\right. \\nonumber\\\\\n &-&\\left.\\left.\\left.\n 21 \\zeta^2 Q_1^{\\prime \\prime}(\\zeta)\\right)\\right]\n + \\mathcal{O}\\left({\\bar T}^4\n\\right) \\right\\}\n \\mathcal{N}(0).\n\\end{eqnarray}\n Here $Q_1(\\zeta)$ is the Legendre function of second kind with\n$\\zeta= s+i\/\\tau_q$,\n and\n${\\bar T}=T\/\\varepsilon_{{}_{\\! {\\rm F}}}$ is used also in Appendix A.3.\nThese quantities may now be used to calculate the response\nfunctions (\\ref{ddrespdef}) and (\\ref{dtrespdef}), [or more\nspecifically (\\ref{ddresp}) and (\\ref{dtresp})]. For zero\ntemperature, one gets the standard solutions\n\\cite{abrikha,heipethrev}. So far no assumption has been made\nconcerning the parameter $\\omega \\tau$ which specifies the importance\nof collision in various regimes of the collective motion\n\\cite{abrikha}. In particular, the formulas obtained in this\nsection are valid both for the regimes of zero sound ($\\omega \\tau\n\\gg 1$) and hydrodynamics ($\\omega \\tau \\ll 1$). For $\\omega \\tau \\gg\n1$ our solutions agree with those of \\cite{abrikha,heipethrev}.\nHowever, below we shall be interested mainly in collective\nexcitations of low frequencies. The notion \"low frequencies\" is\nmeant to indicate that the corresponding excitation energies are\nsmaller than those of the\ngiant resonances. Next we will turn to the hydrodynamic regime\nwhere $\\omega \\tau \\to 0$. As we shall see, at low temperatures our\nsolutions approach the ones of normal classical liquids, in\nagreement with \\cite{forster,brenig}.\n\n\\bigskip\n\n\\subsection{Hydrodynamic regime}\n\\label{longwavlim}\n\n\\medskip\n\n\\subsubsection{DISPERSION RELATION}\n\\label{disrel}\n\nThe response functions can be\nsimplified significantly in the long-wave length\nlimit. Using\n$\\tau_q$ introduced in (\\ref{som}), this (LWL) limit may be defined as\n$\\tau_q \\ll 1$. It can be reached in two ways, namely for small\nwave numbers $q$ and finite collision time $\\tau$ or for small\n$\\tau$ but finite $q$. Both cases imply that the dimensionless\nparameter $\\omega \\tau=s \\tau_q$, which determines the collision\nrate in comparison to the frequency of the modes, becomes small\nfor any finite value $s$ of (\\ref{som}) ($|s| \\siml 1$).\nAs will be shown below for nuclear matter at low temperatures, this\nquantity $s$ is not enough large, in distinction to the\ncase of liquid $^3He$. Therefore, a small $\\tau_q$ implies\nhydrodynamic behavior, in contrast to the zero sound regime; where\n$\\tau_q \\gg 1$, or $\\omega \\tau \\gg 1$.\n\nThe Landau--Vlasov equation (\\ref{landvlas}) is an integral\nequation. Its solution may be sought for in terms of an eigenvalue\nproblem with the distribution function $\\delta f$ being the\neigenfunctions and the sound velocity $s$ (\\ref{som}) being the\neigenvalues, see also \\cite{abrikha,pinenoz}. This eigenvalue\nproblem may be solved perturbatively with $\\tau_q$ being the\nsmallness parameter\n\\cite{sykbrook,brooksyk}.\nIt may be noted in passing that this method may be applied to some extent\nas well\nto the eigenvalue problem of the Schr\\\"odinger equation.\nWe shall use it to get the hydrodynamic sound\nvelocities from the kinetic equation, see\n\\cite{sykbrook,brooksyk}. To this end, we expand the solutions for\n$s$ and $\\delta f$ into power series with respect to $\\tau_q$,\nbut restricted to linear order. Thus, we may write\n\\bel{somexp} s =\ns_0 + i s_1 \\tau_q,\n\\end{equation}\n where $s_0$ and $s_1$ are independent of\nthe expansion parameter $\\tau_q$. In Appendix A.2,\nit is\nshown how the density-density response function may be calculated\nin the LWL limit.\nThere two non-linear equations for the\ncoefficients $s_0$ and $s_1$ are obtained from the dispersion\nrelation (\\ref{despeq}), namely (\\ref{eqzero}) and (\\ref{eqone})).\nThe first equation [see (\\ref{eqzero})] has one obvious solution\n$s_0=s_0^{(0)}=0$ and two others $s_0=\\pm s_0^{(1)}$ with the\nsame modulus,\n\\begin{eqnarray}\\label{sfirst0}\n s_0^{(1)} &=& \\sqrt{\\frac{\\mathcal{G}_0 \\mathcal{G}_1\n\\mathcal{N}(0)}{3 \\mathcal{N}(T)}\n \\left(1 + \\frac{\\pi^2 \\tbar^2}{3 \\mathcal{G}_0}\\right)} \\nonumber\\\\\n&\\approx&\n \\sqrt{\\frac{\\mathcal{G}_0 \\mathcal{G}_1}{3}\n \\left[1 +\n \\frac{\\pi^2 \\tbar^2 \\left(4 + \\mathcal{G}_0\\mathcal{G}_1\\right)}{12 \\mathcal{G}_0}\n\\right]}.\n\\end{eqnarray}\nSubstituting $s_0^{(0)}$ and $s_0^{(1)}$ into the second equation (\\ref{eqone}),\none finds the two solutions for $s_1$, $s_1^{(0)}$ and $s_1^{(1)}$,\nrespectively. These solutions for $s$ (\\ref{somexp})\ncan be written in terms of the dimensional\nfrequency $\\omega $ by means of (\\ref{som})\nin the following form:\n\\begin{eqnarray}\\label{shp}\n\\omega ^{(0)} &=& -i\n\\frac{{\\Gamma}^{(0)}}{2}\\;, \\qquad {\\Gamma}^{(0)} = s_1^{(0)}v_{{}_{\\! {\\rm F}}} q \\qquad\\nonumber\\\\\n&=&\n\\frac{2 \\tau_q v_{{}_{\\! {\\rm F}}} q}{3}\n\\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right]\\qquad\n\\end{eqnarray}\nand\n\\bel{sfirst}\n\\omega _{\\pm}^{(1)} =\n \\pm \\omega _0^{(1)} -i \\frac{{\\Gamma}^{(1)}}{2}\n\\quad {\\rm with}\\quad\n \\omega _0^{(1)}=s_0^{(1)} v_{{}_{\\! {\\rm F}}} q ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{gambarone}\n{\\Gamma}^{(1)} &=&\n s_1v_{{}_{\\! {\\rm F}}} q =\n \\frac{4}{15} \\tau_q v_{{}_{\\! {\\rm F}}} q \\mathcal{G}_1\n \\left[1 \\!+\\! \\frac{5 \\pi^2 \\tbar^2}{6}\n \\left(\\frac{1}{\\mathcal{G}_0 \\mathcal{G}_1}\\right)\\right] \\nonumber\\\\\n &\\approx& \\frac{4}{15} \\tau_q v_{{}_{\\! {\\rm F}}} q \\mathcal{G}_1\n \\left[1 \\! +\\! \\frac{5 \\pi^2 \\tbar^2}{12}\n \\left(1\\!+\\! \\frac{1}{\\mathcal{G}_0 \\mathcal{G}_1}\\right)\\right].\n\\end{eqnarray}\nThe first root $\\omega ^{(0)}$ given in (\\ref{shp}) is purely\nimaginary and corresponds to the overdamped excitations of the\nhydrodynamic Raleigh mode \\cite{kubo,forster}. The second\nand third ones $\\omega _{\\pm}^{(1)}$ correspond to the usual first\nsound mode, expressed in terms of the (macroscopic) parameters of\nviscosity and thermal conductivity of normal liquids\n \\cite{forster,brenig}.\nIn (\\ref{sfirst0}) and\n(\\ref{gambarone}), small corrections of the order of the product\nof the two small quantities ${\\bar T}^2$ and $\\mathcal{F}_0$ have been\nneglected, along with ${\\bar T}^2 |(m^*-m)\/m|={\\bar T}^2|\\mathcal{F}_1|\/3$.\nThis procedure should be valid for nuclear matter; where\nthe relevant parameters are small, both $\\mid\\mathcal{F}_0\\mid$ and\n$|(m^*-m)\/m|$ being of order $ \\approx 0.2$. Discarding such\nsmall corrections, our results for the sound frequencies\n$\\omega _\\pm^{(1)}$ (\\ref{sfirst}) are in agreement with\n\\cite{heipethrev}. In particular, up to these small corrections,\nthe volume (or second) viscosity disappears, as it is the case in\n\\cite{heipethrev}. In the expressions (\\ref{shp}) to\n(\\ref{gambarone}) more explicit temperature corrections are given\nfor $\\omega ^{(0)}$ and $\\omega ^{(1)}$ than those discussed in\n\\cite{heipethrev}. This will turn out important for the thermal\nconductivity $\\kappa$, which we shall address in\nSec.\\ \\ref{viscosthermcond} [see (\\ref{kappaexp})]. The \"widths\"\n${\\Gamma}^{(0)}$ and ${\\Gamma}^{(1)}$ are proportional to $\\tau_q$, and thus, to\nthe relaxation time $\\tau$ which represents the effects of\ntwo-body collisions. For nuclear matter, the Landau parameters\n$\\mathcal{F}_0$ and $\\mathcal{F}_1$ are small [$\\mathcal{G}_0$ and\n$\\mathcal{G}_1$ are close to unity, see (\\ref{effmass})]. For this reason,\naccording to\nthe last equation in (\\ref{som}), the sound velocities cannot be\nlarge [see the approximation\n(\\ref{sfirst0})]. So, the LWL limit\n($\\tau_q \\ll 1$)\nmay be identified with the hydrodynamic collision regime\n$\\omega \\tau =s \\tau_q \\siml \\tau_q \\ll 1$.\nNote that for the Fermi liquid $^3He$, for instance, the parameters\n$\\mathcal{F}_0$ and $\\mathcal{F}_1$ are large and second order equation of\n(\\ref{sfirst0}) can not be applied. Moreover, according to the\nfirst line in (\\ref{sfirst0}), the sound velocity is large.\nTherefore, in this case a smallness $\\tau_q$ does not mean yet that\n$\\omega \\tau$ is also small, i.e., the LWL\ncondition is not\nenough for the hydrodynamical collision regime.\n\n\n\\bigskip\n\n\\subsubsection{RESPONSE FOR INDIVIDUAL MODES}\n\\label{respmodes}\n\n\nIn the following, we are going to examine the collective response\nfunction $\\chi_{DD}^{\\rm coll}$ (\\ref{ddresp}), in particular its\nbehavior in the neighborhood of the individual modes given by\n(\\ref{shp}) and (\\ref{sfirst}). To simplify the notation, we shall\nat times omit the lower index \"DD\" and move down the upper index\n\"coll\". Near any of the sound poles $\\omega _{\\pm}^{(1)}$ given in\n(\\ref{sfirst}), the collective response function $\\chi_{\\rm coll}$\n(\\ref{ddresp}) may be written as\n\\begin{eqnarray}\\label{chicollone}\n\\chi_{\\rm coll}^{(1)}\\left(q,\\omega \\right) &=&\n a^{(1)} \\left( \\frac{1}{\\omega -\\omega _{-}^{(1)}} -\n \\frac{1}{\\omega -\\omega _{+}^{(1)}} \\right)\\quad {\\rm with} \\nonumber\\\\\na^{(1)} &=&\n \\frac{\n\\omega _0^{(1)} \\mathcal{N}(T)}{2 \\mathcal{G}_0\n\\left[1 + \\pi^2 \\tbar^2 \/(3 \\mathcal{G}_0) \\right]}.\n\\end{eqnarray}\n Here, we have made use of (\\ref{ampsexp}), (\\ref{dszexpzero}),\n(\\ref{couplconst}) as well as of (\\ref{somexp}).\nIt will turn out convenient to present separately the dissipative\nand reactive parts, $\\chi_{\\rm coll}^{(1)\\;\\prime\\prime}$ and\n$\\chi_{\\rm coll}^{(1)\\;\\prime}$, respectively,\n\\begin{eqnarray}\\label{chiconeqompp}\n\\chi_{\\rm coll}^{(1)\\;\\prime\\prime}(q,\\omega ) &=&\n \\frac{1}{2} a^{(1)} \\left[\n\\frac{\\Gamma^{(1)}}{\\left(\\omega -\\omega _0^{(1)} \\right)^2 +\n \\left(\\Gamma^{(1)}\\right)^2 \/ 4 } \\; \\right.\\nonumber\\\\\n &-&\\left. \\frac{\\Gamma^{(1)}}{\\left(\\omega + \\omega _0^{(1)} \\right)^2 +\n \\left(\\Gamma^{(1)}\\right)^2 \/ 4} \\right]\n\\end{eqnarray}\n and\n\\begin{eqnarray}\\label{chiconeqomp}\n\\chi_{\\rm coll}^{(1)\\;\\prime}(q,\\omega )&=&\n a^{(1)} \\left[\\frac{\n\\omega + \\omega _0^{(1)}}{\\left(\\omega + \\omega _0^{(1)}\\right)^2 +\n \\left(\\Gamma^{(1)}\\right)^2 \/ 4} \\right.\\nonumber\\\\\n &-&\\left.\\frac{\\omega -\\omega _0^{(1)}}{\\left(\\omega -\\omega _0^{(1)}\\right)^2 +\n \\left(\\Gamma^{(1)}\\right)^2 \/ 4}\\right].\n\\end{eqnarray}\n Notice that for $\\tau_q=+0$ the Lorentzians in (\\ref{chiconeqompp}) turn into\n$\\delta$-functions.\n\nThe relaxation time $\\tau$, which determines the dimensionless\nquantity $\\tau_q =\\tau \\, v_{{}_{\\! {\\rm F}}} q$, might depend on temperature and\nfrequency. A useful form is found in\n\\bel{tautom} \\tau = \\frac{\n\\tau_o}{T^2 + c_o \\left(\\hbar \\omega \\right)^2} \\quad\n \\approx \\quad \\frac{\\tau_o}{T^2} \\quad {\\rm for} \\quad\nc_o(\\hbar \\omega )^2 \\ll T^2,\n\\end{equation}\n with some parameters $\\tau_o$ and\n$c_o$ independent of $T$ and $\\omega $; see, e.g., \\cite{magkohofsh}. As\nindicated on the very right, for our present purpose we may\nneglect the frequency dependence, simply because we are interested\nin describing low frequency modes at larger temperatures (with\nrespect to $\\hbar \\omega $). Indeed,\nit is such a condition which\nhelps justifying the assumption of local equilibrium. We shall\nreturn to this question later, when we are going to apply the Landau\ntheory to a finite Fermi-liquid drop. Substituting (\\ref{tautom})\ninto the damping coefficient ${\\Gamma}^{(1)}$ (\\ref{gambarone}),\none has\n \\bel{gambaronet} {\\Gamma}^{(1)} =\n \\frac{4 \\tau_o v_{{}_{\\! {\\rm F}}}^2 q^2 \\mathcal{G}_1}{15 T^2}\n \\left[1 +\n \\frac{5 \\pi^2 \\tbar^2}{12}\\left(1+ \\frac{1}{\\mathcal{G}_0\n\\mathcal{G}_1}\\right)\\right].\n\\end{equation}\n To leading order, this gives the expected\ndependence on temperature commonly associated to hydrodynamics,\nnamely ${\\Gamma}^{(1)} \\propto 1\/T^2$.\n\nFinally, we may note that in the long-wave limit the effective,\ntotal in-compressibility $K_{\\rm tot}$ (\\ref{incomprtot}) becomes\nidentical to the adiabatic in-compressibility $K^{\\varsigma}$\n(\\ref{incompraddef}) specified in Appendix A,\n\\bel{Kadiabat}\nK^{\\varsigma}=\nK^{T}\\left(1 + \\frac{4T{\\tt C}_{\\mathcal{V}}}{K^{T}}\\right),\n\\end{equation}\nsee (\\ref{isotherk}) for the isothermal in-compressibility $K^{T}$\nand (\\ref{incompradexp}) at small temperatures.\nFor the derivation\nof this identity, it is more easy to\nconsider variations $\\delta \\tilde{T}$ and $\\delta \\tilde{\\rho}$\nas caused formally by some external field $V_{\\rm ext}$. Then,\none can represent $\\delta {\\tilde T}\/\\delta {\\tilde \\rho}$ in\n(\\ref{incomprtot}) in terms of the ratio of the\ntemperature-density $\\chi_{{}_{\\! TD}}(\\tau_q,s)$ (\\ref{chitbar}) to\ndensity-density $\\chi_{{}_{\\! DD}}(\\tau_q,s)$ (\\ref{chiindef}),\n\\bel{dtdrhoexp0}\n\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}\n\\equiv \\frac{\\delta {\\tilde T}\/V_{\\rm ext}}{\\delta {\\tilde\n\\rho}\/V_{\\rm ext}}= \\frac{\n\\chi_{{}_{\\! TD}}(\\tau_q,s)}{\\chi_{{}_{\\! DD}}(\\tau_q,s)}\\;.\n\\end{equation}\n Using then the LWL\nexpansions\n(\\ref{chiexpone}) and (\\ref{ampsexp}) up to the third order\nterms in $\\tau_q$, from (\\ref{dtdrhoexp0}) we get\n\\bel{dtdrhoexp}\n\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}\n\\approx \\frac{{\\bar T}}{\\mathcal{N}(0)}\\;\\left(1-\n\\frac{i \\tau_q}{3\ns_0^{(1)}}\\right) \\quad {\\rm for}\\quad \\tau_q \\rightarrow 0,\n\\end{equation}\n where $s_0^{(1)}$ was defined in (\\ref{sfirst0}). As the\nspecific heat ${\\tt C}_{\\mathcal{V}}$ (\\ref{specifheatv}) is\nproportional to ${\\bar T}$, we need in (\\ref{dtdrhoexp}) only\nlinear terms to get the temperature correction of the second\norder in ${\\bar T}$ in the total in-compressibility $K_{\\rm tot}$\n(\\ref{incomprtot}). Substituting (\\ref{dtdrhoexp}),\n(\\ref{specifheatv}) and (\\ref{isotherkexp}) into\n(\\ref{incomprtot}) for the total in-compressibility $K_{\\rm tot}$,\none obtains identically the same as in (\\ref{incompradexp}) for\nthe adiabatic in-compressibility $K^\\varsigma$. The same result\n(\\ref{dtdrhoexp}) in the LWL limit can be obtained\nalso from (\\ref{dtdrho}).\n\n\nLet us address now the pole at $\\om^{(0)}$ [see (\\ref{shp})]. Near\nthe latter, the collective response function $\\chi_{\\rm coll}$\n(\\ref{ddresp}) becomes [as may be checked with the help of\n(\\ref{ampsexp}), (\\ref{dszexpzero}), (\\ref{couplconst}) and\n(\\ref{somexp})]\n\\begin{eqnarray}\\label{chicollhp} \\chi_{\\rm coll} ^{(0)}\n&=&\n \\frac{i a^{(0)}}{\\omega -\\omega ^{(0)}}=\n \\frac{i a^{(0)}}{\\omega + \\frac{i {\\Gamma}^{(0)}}{2}}\n\\qquad {\\rm with} \\nonumber\\\\\n a^{(0)} &=&\n \\frac{\n\\pi^4 {\\bar T}^2 \\tau_q (8-3 \\mathcal{G}_0)}{108\n\\mathcal{G}_0^2}~v_{{}_{\\! {\\rm F}}} q ~\\mathcal{N}(0),\n\\end{eqnarray}\nand ${\\Gamma}^{(0)}$ being defined in (\\ref{shp}). It may\nbe rewritten in a more traditional form, see \\cite{kubo},\n\\cite{forster} and \\cite{brenig}. Introducing the \"diffusion\ncoefficient\"\n\\bel{diffusd}\n{\\tt D}_T = \\frac{\\kappa}{{\\tt C}_{\\cal P} \\rho}\n =\\frac{\\tau v_{{}_{\\! {\\rm F}}}^2}{3}\n\\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right]\n\\end{equation}\n$[{\\tt D}_T =\n\\Gamma^{(0)}\/(2 q^2)]$, one gets\n\\bel{chichpqom} \\chi_{\\rm coll}^{(0)} =\n\\frac{i {\\tt D}_T q^2}{\\omega + i {\\tt D}_T q^2} ~\\chi_{\\rm coll}^{(0)} (q, \\omega = 0).\n\\end{equation}\nNote that according to (\\ref{shp}) and (\\ref{tautom}), the\ntemperature dependence of $\\Gamma^{(0)}$ becomes similar to the\none found in (\\ref{gambaronet}),\n\\bel{gambarhpt} {\\Gamma}^{(0)}\n= \\frac{2 \\tau_o v_{\\rm F}^2 q^2}{3 T^2} \\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29\n\\mathcal{G}_0\\right)}{120 \\mathcal{G}_0}\\right].\n\\end{equation}\n For the dissipative and reactive parts of the response function\n$\\chi_{\\rm coll}^{(0)}$ (\\ref{chichpqom}), from (\\ref{chichpqom}) one gets\n\\begin{eqnarray}\\label{chichpqompp}\n\\chi_{\\rm coll}^{(0)\\;\\prime\\prime}(q,\\omega )&=&\n a^{(0)} \\frac{\\omega }{\\omega ^2 + \\left(\\Gamma^{(0)}\\right)^2 \/ 4},\\nonumber\\\\\n\\chi_{\\rm coll}^{(0)\\;\\prime}(q,\\omega )&=&\n a^{(0)} \\frac{\n\\Gamma^{(0)}\/2}{\\omega ^2 + \\left(\\Gamma^{(0)} \\right)^2 \/ 4}.\n\\end{eqnarray}\n The strength distribution $\\chi_{\\rm coll}^{(0)\\;\\prime\\prime}$\nhas a maximum at $\\omega =\\Gamma^{(0)}\/2$ and a width $\\Gamma^{(0)}\/2\n\\propto \\tau_q$. In the LWL\nlimit $\\tau_q \\ll 1$\nthis distribution becomes quite sharp with the maximum lying close\nto $\\omega =0$. As may be inferred with the help of (\\ref{chicollhp})\nand (\\ref{shp}), the maximal value does not depend on $\\tau_q$\nand is proportional to ${\\bar T}^2$. It will be demonstrated\nshortly that the pole at $\\om^{(0)}$ (\\ref{shp}) is related to the heat\nconduction, for which reason it sometimes is called \"heat pole\".\nNotice that the reactive response function\n$\\chi_{\\rm coll}^{(0) \\; \\prime}$ is finite at $\\omega =0$, with a value\nindependent of $\\tau_q$.\n\n\nIn the hydrodynamic regime with $\\tau_q \\ll 1$, the response\nfunction $\\chi_{\\rm coll}$ found for the {\\it Fermi liquid} becomes\nidentical to the one for {\\it normal liquids}\n\\cite{forster,brenig}. This can be made more apparent after\nintroducing the dimensional sound velocity $c$, a width parameter\n$\\Gamma$, determined as\n\\bel{liqparam} c=v_{{}_{\\! {\\rm F}}} s_0^{(1)}\\;, \\qquad\\qquad\n\\Gamma=\\Gamma^{(1)}\/q^2 \\;,\n\\end{equation}\n as well as the diffusion\ncoefficient ${\\tt D}$ (\\ref{diffusd}) and the specific heats. The\nsum of the two contributions discussed above may then be written\nas\n\\begin{eqnarray}\\label{forstereq} \\chi_{\\rm coll}^{\\prime\\prime} &=&\n \\rho \\left(\\frac{\\partial \\rho}{\\partial {\\cal P}}\\right)_T\n \\left[\\frac{\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\n c^2 q^4 \\Gamma\\; \\omega }{\\left(\\omega ^2-c^2q^2\\right)^2\n+ \\left(\\omega q^2\n\\Gamma\\right)^2} \\right.\\nonumber\\\\\n &+& \\left.\\frac{\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T\\; \\omega }{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2}\\right].\n\\end{eqnarray}\n Traditionally, the peaks related to the first and second terms\nare called Brillouin and Rayleigh (or Landau--Placzek) peak,\nrespectively. The ratio of the specific heats ${\\tt C}_{\\cal P}$ and\n${\\tt C}_{\\mathcal{V}}$ per particle is discussed in Appendix A.1,\nsee (\\ref{cpcvkakt}) and (\\ref{cpcvexp}). Note\nthat the sound speed $s_0^{(1)}$, see (\\ref{sfirst0}), is identical\nto the adiabatic sound velocity found in Appendix A.1,\nsee\n(\\ref{velocad}) ($c$ in dimensional units for normal liquids), as\nit should be for normal liquids\n\\cite{brenig,forster}.\nThe structure of (\\ref{forstereq}) is identical to that discussed\nin the literature (see, e.g., (4.44a) of \\cite{forster}), if one\nonly expresses the quantities appearing here in terms of viscosity\nand thermal conductivity. As a matter of fact, the alert reader\nmight expect a third term (as in (4.44a) of \\cite{forster}), but\nthis one is of the order of $\\tau_q^2$ and thus is neglected here.\nThe specific temperature dependence of these parameters (in the\nLWL limit) will be discussed in the next subsection,\nwith respect to the specific heats, see also Appendix A.1.\n\n\nNote that in the derivation of the both amplitudes $a^{(0)}$\n(\\ref{chicollhp}) and $a^{(1)}$ (\\ref{chicollone}) we took\n$D(s)$ (\\ref{despfunc}) at low temperatures using\n(\\ref{chitemzero}) to (\\ref{chitemtwo}); and then, expand first\nit near the poles (\\ref{shp}) and (\\ref{sfirst0}), respectively;\nand second, in small $\\tau_q$ of the LWL\nlimit. This way\nof the calculation is much more simpler because the two last\noperations can be exchanged only when we shall take into account\nnext order terms in $\\tau_q$, that takes much hard work. If we\nexchange the last two operations, expanding first in $\\tau_q$ in\nthe {\\it linear} LWL\napproximation (\\ref{somexp}), and then,\ndoing expansion near the poles, some important terms will be lost.\n\n\n\\subsubsection{SHEAR MODULUS, VISCOSITY AND THERMAL CONDUCTIVITY}\n\\label{viscosthermcond}\n\nAs explained in Appendix A.2,\nthese coefficients may be\nobtained by applying expansions to $\\chi _n$ within the\nperturbation theory mentioned above, for low temperatures (with\n${\\bar T} \\ll 1$); see in particular (\\ref{chiexpzero}), (\\ref{chiexpone}) and\n(\\ref{chiexptwo}).\nThey specify the stress tensor $\\sigma_{\\alpha\\beta}$\n(\\ref{prestensone})-(\\ref{pressnu}) and the heat current ${\\bf\nj}_T$ (\\ref{currheat}).\n\nThe shear\nmodulus $\\lambda $ (\\ref{shearmod}) in the time-reversible part $\n\\sigma_{\\alpha\\beta}^{(\\lambda)}$ (\\ref{presslamb}) of the\nstress tensor $ \\sigma_{\\alpha\\beta}$ (\\ref{prestensone})\nturns into zero in the long wave-length approximation linear in\n$\\tau_q$ as in \\cite{heipethrev}\nup to immaterial corrections of the order of ${\\bar T}^4$. By\nanother words, in this case, $\\lambda $ is a small quantity of\nthe order of $\\tau_q^2$ because such corrections were neglected\neverywhere. It means a disappearance of the Fermi-surface\ndistortions in our linear approach (\\ref{somexp}) which are the\nmain peculiarity of Fermi liquids compared to the normal ones.\n\n\nFor the shear viscosity $\\nu $\n(\\ref{viscos}) taken at the first sound frequency\n$\\omega =\\omega _0^{(1)}$ (\\ref{sfirst}), one obtains\n\\bel{shearvisonehp} \\nu = \\nu^{(1)}+\\nu^{(2)}\\;,\n\\end{equation}\n where\n\\bel{shearvisone} \\nu^{(1)} = \\frac{2}{5} \\rho \\varepsilon_{{}_{\\! {\\rm F}}} \\tau\n \\left(1 + \\frac{5 \\pi^2 \\tbar^2}{12} \\right)~\n\\end{equation}\n and\n\\bel{shearvishp} \\nu^{(2)}=\\frac{13 \\pi^4}{720} \\frac{\\rho\n\\varepsilon_{{}_{\\! {\\rm F}}} {\\bar T}^4 }{v_{{}_{\\! {\\rm F}}}^2 q^2 \\tau}.\n\\end{equation}\n The first term\n$\\nu^{(1)}$ (\\ref{shearvisone}) in (\\ref{shearvisonehp}) is proportional to the\nrelaxation time $\\tau$ and coincides mainly with that obtained\nearlier for mono-atomic gases and\nfor a Fermi liquid by using another method\n\\cite{heipethrev},\nexcept for the specific explicit dependence on temperature\npresented here. The temperature dependence of the shear viscosity\n$\\nu^{(1)}$ (\\ref{shearvisone}) is mainly the same as for the\nrate of the sound damping $\\Gamma ^{(1)}$ (\\ref{gambaronet}),\n$\\nu^{(1)} \\propto {1\/T^2}$, with the temperature dependence of the\nrelaxation time $\\tau$ (\\ref{tautom}). Although the viscosity component\n $\\nu^{(2)}$,\ntoo, is related to the first sound solution $\\omega _0^{(1)}$, it is\nproportional to $1\/\\tau$, similar to the viscosity of zero sound\nbut in contrast to the standard first sound viscosity\n(\\ref{shearvisone}). The $\\nu^{(2)}$ component (\\ref{shearvishp})\nof the viscosity (\\ref{shearvisonehp}) increases with temperature\nas $T^6$, see also (\\ref{tautom}) for the relaxation time $\\tau$.\nAlthough the second component $\\nu^{(2)}$ of the shear viscosity\nis proportional to ${\\bar T}^4$, and thus may be considered small\nunder usual conditions, it may become important for small wave\nnumbers $q$ (or frequencies $\\omega $) [for more details, see the\ndiscussion to come below in Sec.\\ \\ref{heatcorrfun}].\nThis component of the viscosity was not discussed\nin \\cite{heipethrev}.\n\n\nLet us finally turn to the thermal conductivity $\\kappa$ which\nshows up in the equation for variations of temperature $T({\\bf r},t)$\nwith ${\\bf r}$ and $t$ (see Appendix A.2).\nThe form (\\ref{kappadef}) [for\nthe heat mode $\\omega =\\omega ^{(0)}$ of (\\ref{shp})] may be rewritten as\n\\bel{kappaexp} \\kappa =\n \\rho \\frac{{\\tt C}_{\\cal P} \\Gamma^{(0)}}{2 q^2} \\approx\n \\frac{1}{3} \\rho {\\tt C}_{\\cal P} v_{{}_{\\! {\\rm F}}}^2 \\tau\n \\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right].\n\\end{equation}\n We present here also explicitly the temperature\ncorrections up to the terms of the order of ${\\bar T}^2$. Our\nexpression for the thermal conductivity $\\kappa$ (\\ref{kappaexp})\ndiffers from the one found in \\cite{forster} and\n\\cite{heipethrev} by small ${\\bar T}^2$ corrections. However, they are not\nimportant in the calculations of the damping coefficient\n$\\Gamma^{(1)}$ for the first sound mode defined in\n\\cite{forster}, and \\cite{brenig},\nsee also the comment before (\\ref{forstereq}),\n\\bel{gammaland}\n\\Gamma ^{(1)}=\n \\frac{q^2}{m \\rho }\n \\left[\n \\frac{4}{3} \\nu^{(1)}\n + \\frac{m \\kappa}{{\\tt C}_{\\cal P}}\n \\left(\\frac{{\\tt C}_{\\cal P}}{{\\tt C}_{\\mathcal{V}}} -1\\right)\n \\right],\n\\end{equation}\n Here, $\\nu^{(1)}$ is the part of the shear viscosity\ncoefficient related to the first sound mode, see\n(\\ref{shearvisone}); ${\\tt C}_{\\cal P}\/ {\\tt C}_{\\mathcal{V}}$ is the\nadiabatic ratio of the specific heats, see\n(\\ref{specifheatpdef}) and (\\ref{cvcpkSkT}). We omitted here\ncorrections related to the second viscosity in line of the second\napproximation in (\\ref{gambarone}). In\n(\\ref{gammaland}), $\\kappa $ is multiplied by a small quantity of the\norder of the ${\\bar T}^2$ as follows from (\\ref{cpcvexp}) and the\ntemperature corrections to $\\kappa $ written explicitly in\n(\\ref{kappaexp}) can be neglected in (\\ref{gammaland}) . The\nexpression for the damping coefficient $\\Gamma^{(1)}$\n(\\ref{gammaland}) with the viscosity coefficient $\\nu^{(1)}$\n(\\ref{shearvisone}), thermal conductivity $\\kappa$\n(\\ref{kappaexp}) and specific heats from (\\ref{cpcvexp}) and\n(\\ref{specifheatp}) for viscose normal liquids is in agreement\nwith our result for $\\Gamma^{(1)}$ (\\ref{gambarone}) including the\ntemperature corrections.\n\n\nThus, up to\nthe temperature corrections discussed above, we have agreement\nwith the results of\n\\cite{heipethrev} for the dispersion equation, viscosity and\nthermal conductivity coefficients in the hydrodynamic limit. Our\nderivations are more strict and direct within the perturbation\ntheory for the eigenvalue problem. We have the transition to the\nhydrodynamics of normal liquids discussed in\n\\cite{forster,brenig,kubo}\nin terms of\nthe macroscopic parameters mentioned above.\n\n\\subsection{Susceptibilities}\n\\label{suscept}\n\nIn this section, we want to address the calculation of the static\nsusceptibilities, for which one distinguishes isolated, isothermal\nand adiabatic ones \\cite{kubo,forster,brenig}. Their comparison is\nrelevant for ergodicity properties, see \\cite{kubo,brenig}. Here\nwe will concentrate on the density mode of nuclear matter\nconsidered as an infinite Fermi-liquid system.\n\n\\subsubsection{ADIABATIC AND ISOTHERMAL SUSCEPTIBILITIES}\n\\label{susceptadt}\n\nThe isolated susceptibility $\\chi_{{}_{\\! DD}}(0)$ is defined as the\nstatic limit of the response function $\\chi_{{}_{\\! DD}}(q,\\omega )$ (or\n$\\chi_{{}_{\\! DD}}(\\tau_q,s)$ of (\\ref{chiindef}) in dimensionless\nvariables), for which one first has to take the limit $q \\to 0$\n(or $\\tau_q \\to 0$), and then, $\\omega \\to 0$ (or $s \\to 0$) (see, e.g.,\n\\cite{forster})\n\\bel{statresponse} \\chi_{{}_{\\! DD}}(0)=\n \\lim_{\\omega \\rightarrow 0}\n \\left[\\lim_{q \\rightarrow 0}\\;\\chi_{{}_{\\! DD}} \\left(q,\\omega \\right)\\right]=\n \\lim_{s \\rightarrow 0}\n \\left[\\lim_{\\tau_q \\rightarrow 0}\\;\\chi_{DD} \\left(\\tau_q,s\\right)\\right].\n\\end{equation}\n Apparently, $\\chi_{{}_{\\! DD}}(0)$ satisfies the relation\n\\bel{vardenschi0}\n\\delta \\rho \\equiv -\\chi_{{}_{\\! DD}}(0) \\delta V_{\\rm eff},\n\\end{equation}\n where $\\delta V_{\\rm eff}$ and $\\delta \\rho$ are quasistatic\nvariations. They can be considered as independent of time, in\ncontrast to the ones discussed in Sec.\\ \\ref{dynresp}, see\n(\\ref{delueff}).\n\n\nThe isothermal susceptibility $\\chi_{DD}^{T}$ is defined as the\ndensity-density response at constant temperature $T$, and the\nadiabatic one, $\\chi_{DD}^{\\varsigma}$, as that at constant\nentropy (per particle $\\varsigma$). Suitable variables for\nstudying the variations of the density $\\rho$ are therefore\npressure ${\\cal P}$ and temperature $T$ in the first case, and\npressure ${\\cal P}$ and entropy per particle $\\varsigma$ in the\nsecond one. These two representations of $\\delta \\rho$ can be\nwritten as\n\\bel{vardens} \\delta \\rho \\equiv\n \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_T \\delta {\\cal P}\n + \\left(\\frac{\\partial \\rho }{\\partial T} \\right)_{\\cal P} \\delta T \\equiv\n \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_\\varsigma\n \\delta {\\cal P}\n + \\left(\\frac{\\partial \\rho }{\\partial \\varsigma} \\right)_{\\cal P}\n \\delta \\varsigma.\n\\end{equation}\n For the isothermal and adiabatic susceptibilities\n$\\chi_{DD}^{T}$ and $\\chi_{DD}^{\\varsigma}$, one thus gets the\nfollowing two relations:\n\\bel{vardenschiT} \\delta \\rho \\equiv\n \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_T \\delta {\\cal P}\n =-\\chi_{DD}^T \\delta V_{\\rm eff}\n\\end{equation}\n and\n\\bel{vardenschiS} \\delta \\rho \\equiv\n \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_\\varsigma\n \\delta {\\cal P}\n =-\\chi_{DD}^{\\varsigma} \\delta V_{\\rm eff}.\n\\end{equation}\n The variations of the density with pressure are related to the\n(in-)compressibilities, see (\\ref{incompraddef}) and\n(\\ref{incomprTdef}). As shown in Appendix A.1,\ntheir ratio\ncan be expressed by that of the corresponding specific heats, see\n(\\ref{cvcpkSkT}). Building the ratio, one therefore gets\nfrom (\\ref{vardenschiT}) and (\\ref{vardenschiS})\n\\bel{chitchiskSkT}\n\\frac{\\chi_{DD}^{T}}{\\chi_{DD}^{\\varsigma}}=\n\\frac{K^{\\varsigma}}{K^{T}}=\n\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}.\n\\end{equation}\n This is a\ngeneral relation from thermodynamics where we only have replaced\nthe system's total entropy \\cite{brenig}\nby the entropy per particle $\\varsigma$ applied\nfor the intensive systems as normal and Fermi liquids.\n\n\nWe are interested more in the calculation of the differences\nbetween the isothermal susceptibility $\\chi_{DD}^{T}$ defined by\nthe relations in (\\ref{vardenschiT}) (or adiabatic one\n$\\chi_{DD}^{\\varsigma}$, see (\\ref{vardenschiS})) and isolated\n(static) susceptibility $\\chi_{{}_{\\! DD}}(0)$ presented by\n(\\ref{vardenschi0}) \\cite{hofmann}. For this purpose, we\nfind first the ratio of the isothermal-to-isolated\nsusceptibilities $\\chi_{DD}^{T}\/\\chi_{{}_{\\! DD}}(0)$ in terms of the\nratio of the static \"intrinsic\" temperature-density response\nfunction to the isolated one $\\chi_{{}_{\\! DD}}(0)$ (\\ref{chiindef}). The\nstatic temperature-density susceptibility $\\chi_{{}_{\\! TD}}(0)$ is\ndefined in the same way (\\ref{statresponse}) as the static limit\nof the \"intrinsic\" temperature-density response function\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ given by (\\ref{chitindef}). Note that\nthe limits $\\omega \\rightarrow 0$ (or $s \\rightarrow 0$) and $q\n\\rightarrow 0$ (or $\\tau_q \\rightarrow 0$) which we consider to\nget the static response functions are not commutative \\cite{forster}.\nTaking the second equations in (\\ref{vardenschiT}) and\n(\\ref{vardenschi0}) for the intensive systems as liquids, one\ngets\n\\bel{chitchi0}\n\\frac{\\chi_{DD}^{T}}{\\chi_{{}_{\\! DD}}(0)}= 1 +\n\\left[\\varsigma \\left(\n\\frac{\\partial \\rho }{\\partial \\mu}\\right)_T\n- \\left(\\frac{\\partial \\rho }{\\partial T}\\right)_\\mu\\right]\n\\frac{\\chi_{{}_{\\! TD}}(0)}{\\chi_{{}_{\\! DD}}(0)}.\n\\end{equation}\n We used here\nthe definitions (\\ref{chiindef}) and (\\ref{chitindef}) for the\ndensity-density and temperature-density response functions and\n(\\ref{statresponse}) for their static limits $\\chi_{{}_{\\! DD}}(0)$\nand $\\chi_{{}_{\\! TD}}(0)$. We then applied the thermodynamic relations of\nAppendix A.1 for the transformations\nof the derivative\n$\\left(\\partial \\rho \/ \\partial {\\cal P}\\right)_T$. This derivative\n appears from the\ndefinition of the isothermal susceptibility $\\chi_{DD}^{T}$ in\n(\\ref{vardenschiT}) to another simpler thermodynamic\nderivatives for the application to Fermi liquids, see below.\nFor this aim, we transform the variables $(T,{\\cal P})$ to the new\nones $(T,\\mu)$. The derivatives of pressure ${\\cal P}$ over\nthese two new variables can be then reduced to the ones of the\ndensity $\\rho$ shown in the r.h.s. of (\\ref{chitchi0})\nwith the help of (\\ref{gibbsduh}).\n\nSo, the calculations of the susceptibilities are resulted\nin the\nderivation of the static limits defined by\n(\\ref{statresponse}) for the temperature-density\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ and density-density $\\chi_{{}_{\\! DD}}(\\tau_q,s)$\nresponse functions, see (\\ref{chiindef}) and\n(\\ref{chitindef}), and their ratio $\\chi_{{}_{\\! TD}}(0)\/\\chi_{{}_{\\! DD}}(0)$ for\nthe case of a heated Fermi liquid. We can then calculate the two\nratios of the susceptibilities (\\ref{chitchi0}) and\n(\\ref{chitchiskSkT}) which both determine separately each\nconsidered susceptibilities.\n\n\\subsubsection{FERMI-LIQUID SUSCEPTIBILITIES}\n\\label{flsuscept}\n\nThe expression for the ratio of the isothermal-to-static\nsusceptibilities (\\ref{chitchi0}) can be simplified my making use\nof the specific properties of Fermi liquids given by\n(\\ref{drhomt}) and second equation in (\\ref{dpdtr}),\n\\bel{chitchi0fl}\n\\frac{\\chi_{DD}^{T} }{\\chi_{{}_{\\! DD}}(0)} = 1\n+\\left(\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}-1\\right)\n\\frac{\\chi_{{}_{\\! TD}}(0) }{{\\bar T} \\chi_{{}_{\\! DD}}(0)}\\;\\mathcal{N}(0).\n\\end{equation}\nAccording to the definition (\\ref{statresponse}) of the static\nresponse functions applied to the ones in the ratio\n$\\chi_{{}_{\\! TD}}(0)\/\\chi_{{}_{\\! DD}}(0)$ of (\\ref{chitchi0fl}), we shall\nuse (\\ref{chitbar}) and (\\ref{chiindef}) for the corresponding\nintrinsic susceptibilities ($\\mathcal{F}_{0}=\\mathcal{F}_{1}=0$ there).\nThe static limit (\\ref{statresponse}) of the\nresponse functions $\\chi_{{}_{\\! DD}}(\\tau_q,s)$ (\\ref{chiindef}) and\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ (\\ref{chitbar}) in (\\ref{chitchi0fl})\ncan be found by using the LWL\nexpansions over a\nsmall parameter $\\tau_q \\ll 1$ at low temperatures, see\nSec.\\ \\ref{lowtemlim} and Appendix A.2 for the\nfirst limit ($\\tau_q \\rightarrow 0$)\nin (\\ref{statresponse}). We substitute now the\nperturbation theory expansions for small $\\tau_q$ for the\nquantities $s$ (\\ref{somexp}), $\\chi_1$\n(\\ref{chiexpone}),\n$\\aleph$ (\\ref{ampsexp}), and $D_0$ (\\ref{dszexpzero}) into\n(\\ref{chiindef}) and (\\ref{chitbar}). We get this limit as\nfunctions of $s_0$ and $s_1$, and then, we shall take the second limit of\n$s_{0} \\to 0$ and $s_{1} \\to 0$ [ $s \\rightarrow 0$ in (\\ref{statresponse})].\nFinally, we\narrive then at the very simple result\n \\bel{chitchid}\n\\frac{\\chi_{{}_{\\! TD}}(0) }{\\chi_{{}_{\\! DD}}(0)}=\n\\frac{{\\bar T} }{\\mathcal{N}(0)}\n\\end{equation}\nneglecting small cubic terms in ${\\bar T}$, which correspond to ${\\bar T}^4$\ncorrections in susceptibilities and do not matter in this section.\nNote that the sequence of the limit transitions defined in\n(\\ref{statresponse}) and recommended in \\cite{forster} is\nimportant for the calculation of this ratio: We get zero for this\nratio if we take first $s \\rightarrow 0$, and then, $\\tau_q\n\\rightarrow 0$.\n\nSubstituting now the ratio (\\ref{chitchid}) of the\nsusceptibilities into (\\ref{chitchi0fl}),\none obtains\n\\bel{chitchi0s}\n\\frac{\\chi_{DD}^{T} }{\\chi_{{}_{\\! DD}}(0)} =\n\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}\n = 1 + \\frac{\\pi^2 \\tbar^2 }{3 \\mathcal{G}_0},\n\\end{equation}\nsee also (\\ref{cpcvexp}) for the second equation. We compare\nnow this result with (\\ref{chitchiskSkT}) and get that our\nFermi-liquid system satisfies the ergodicity property:\n\\bel{ergodicity}\n\\chi_{DD}^{(\\varsigma)}=\\chi_{{}_{\\! DD}}(0) \\;.\n\\end{equation}\n This ergodicity property was proved at low temperatures, for which the\nLandau Fermi-liquid theory can be applied. It is related to\nthe adiabaticity of the velocity of the sound mode $s_0^{(1)}$,\nsee (\\ref{sfirst0}) and discussion\nafter (\\ref{forstereq}). Moreover, we got the normal liquid\n(hydrodynamic) limit from the Fermi-liquid dynamics, and therefore, the\nergodicity property is general for heated Fermi liquids and\nnormal (classical) ones.\n\nAnother aspect of the discussed ergodicity property might be the\nrelation to the non-degeneracy of the excitation spectrum in the\ninfinite Fermi liquids beside of the spin degeneracy. We have only\nthe two-fold degenerate quasiparticle states, due to the spin\ndegeneracy. However, it does not influence on our results concerning the\nergodicity relations because we consider the density-density\nexcitations, which do not disturb the spin degree of freedom. We\nhave only the multiplication factor two in all susceptibilities,\ndue to the spin degeneracy, that does not change the ratios of the\nsusceptibilities which are only important for the ergodicity\ndiscussed here.\n\nOur susceptibilities obtained above satisfy the Kubo relations,\nsee (4.2.32) of \\cite{kubo}:\n\\bel{kuborel} \\chi ^{T} \\geq \\chi\n^{\\varsigma} \\geq \\chi(0)\n\\end{equation}\n with the equal sign for the second\nrelation because of the ergodicity property. To realize this, we\nshould take into account that\n${\\tt C}_{\\cal P} > {\\tt C}_{\\mathcal{V}}$ (or $K^{\\varsigma}>K^{T}$),\naccording to (\\ref{cpcvkakt}),\nbecause all quantities on the r.h.s. of this equation are positive\nfor the stable modes $\\mathcal{G}_0=1+\\mathcal{F}_0>0$. The equal sign\nfor the first relation in (\\ref{kuborel}) becomes true in the\ntwo limit cases: For the temperature $T$ going to 0 or for the\nin-compressible matter when the interaction constant $\\mathcal{F}_0$\ntends to $\\infty$. In both limit cases we made obvious equality\n${\\tt C}_{\\cal P}={\\tt C}_{\\mathcal{V}}$ and all susceptibilities are\nidentical [equal signs in the both relations of\n(\\ref{kuborel})].\n\nNote now that namely the specific Fermi-liquid expression of the\nstatic susceptibility $\\chi_{DD}(0)$, see (\\ref{chiindef}) with\n$\\mathcal{F}_0=\\mathcal{F}_1=0$ for the case of the intrinsic response\nfunctions, depends on the sequence of the limit transitions\ndiscussed near\n(\\ref{statresponse}), (\\ref{chitchi0}), (\\ref{chitchid}) above\nand in \\cite{forster}.\nFor the definition\n(\\ref{statresponse}) of \\cite{forster}, one gets\n\\bel{statrespfl}\n\\chi_{{}_{\\! DD}}(0)=\\left(1 - \\frac{5 \\pi^2 \\tbar^2 }{12}\\right)\n\\mathcal{N}(0),\\quad \\chi_{DD}^{T}=\\mathcal{N}(T).\n\\end{equation}\n In the last\nequation, we used also (\\ref{chitchi0s}). Taking the opposite\nsequence of the limit transitions, first $s \\rightarrow 0$, one\nhas the result $\\mathcal{N}(T)$ (\\ref{enerdensnt})\nfor the isolated susceptibility\n$\\chi_{{}_{\\! DD}}(0)$ like for the isothermal one $\\chi_{DD}^{T}$. The\ndifference is in ${\\bar T}^2$ corrections. Ignoring them, the both\nversions of the limit transitions coincide, and we come to the\nresult independent on temperature discussed in \\cite{pinenoz}~.\nThe ergodicity property (\\ref{ergodicity}), Kubo's relations\n(\\ref{kuborel}) and relation of the isothermal susceptibility to\nadiabatic one (\\ref{chitchiskSkT}) do not depend on the specific\npeculiarities of the static limit of the response function\ndiscussed here in connection to Fermi liquids.\n\n\n\\subsection{Relaxation and correlation functions}\n\\label{relaxcorr}\n\n\\medskip\n\n\\subsubsection{RELAXATION FUNCTION}\n\\label{relaxfun}\n\nComing back to the dynamical problem, we note that\nthe dissipative part of the response function\n$\\chi^{\\prime\\prime}(\\omega )$ is related to the relaxation function\n$\\Phi^{\\prime\\prime} (\\omega )$ \\cite{kubo} by\n\\bel{chiimpsi}\n\\chi\n^{\\prime\\prime} =\n \\omega \\Phi ^{\\prime\\prime} (\\omega )\\;.\n\\end{equation}\nWe follow the notations of \\cite{hofmann,hofbook} and omit the\nindex \"coll\" in this section: For the comparison with the\nmicroscopic results of \\cite{hofmann} we need really the\nrelaxation and correlation functions related to the {\\it\nintrinsic} response functions. According to\n(\\ref{ddrespchiin}) and (\\ref{couplconst}), all these intrinsic\nfunctions can be formally obtained from the collective ones at the\nzero Landau constants $\\mathcal{F}_0$ and $\\mathcal{F}_1$. Taking into\naccount also (\\ref{chiconeqompp}) and (\\ref{chichpqompp}), one has\n\\begin{eqnarray}\\label{relaxcom}\n \\Phi^{\\prime\\prime}(\\omega ) &=&\n \\frac{a^{(1)} }{2 \\omega _0^{(1)}}\n \\left[\\frac{\n\\Gamma^{(1)} }{\\left(\\omega -\\omega _0^{(1)} \\right)^2\n + \\left(\\Gamma^{(1)}\\right)^2\/4} \\right. \\nonumber\\\\\n &+&\\left.\\frac{\\Gamma^{(1)} }{\\left(\\omega + \\omega _0^{(1)}\\right)^2\n + \\left(\\Gamma^{(1)}\\right)^2 \/4}\\right] \\nonumber\\\\\n &+&\n \\frac{a^{(0)} }{{\\Gamma}^{(0)}}\\;\n \\frac{\\Gamma^{(0)} }{\\omega ^2 + \\left(\\Gamma^{(0)} \\right)^2 \/ 4}.\n\\end{eqnarray}\n This equation can be re-written in the same way like to\n(\\ref{forstereq}) in terms of the parameters $c$, $\\Gamma$ and\n${\\tt D}_T$, see (\\ref{liqparam}) and (\\ref{diffusd}),\n\\begin{eqnarray}\\label{relaxbrenig}\n\\Phi^{\\prime\\prime}(\\omega ) &=&\n \\chi^T\n \\left[\\frac{\n\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right) {\\tt C}^2 q^4\n\\Gamma}{\\left(\\omega ^2-{\\tt C}^2q^2\\right)^2 +\\left(\\omega q^2 \\Gamma\\right)^2}\n\\right.\\nonumber\\\\\n &+&\\left. \\frac{\n\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T}{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2}\\right].\n\\end{eqnarray}\n We used here the Jacobian relations\nand (\\ref{incomprTdef}), (\\ref{isotherk})\nfor the transformation of the coefficient in front of the square\nbrackets in (\\ref{forstereq}) to the one, the {\\it intrinsic}\nisothermal susceptibility $\\chi^T$ (\\ref{statrespfl})\n($\\mathcal{F}_0=0$). We also neglected\nterms of the order of $\\tau_q^2$ as in the derivation of\n(\\ref{forstereq}). Equation (\\ref{relaxbrenig}) for the relaxation\nfunction $\\Phi^{\\prime\\prime}(\\omega )$ is identical to the imaginary\npart of the r.h.s. of (28.29) in \\cite{brenig}\nwith the transparent physical meaning as (\\ref{forstereq}). The\nfirst term in the square brackets of (\\ref{relaxcom}) and\n(\\ref{relaxbrenig}) is the\nfirst sound Brillouin component with the poles (\\ref{sfirst})\nassociated to the finite frequencies $\\pm\\omega _0^{(1)}$ of the\ntime-dependent relaxation-function oscillations and their damping\nrate $1\/\\Gamma^{(1)}$ ($\\pm\\omega _s$ and $1\/\\gamma_s$ in the notation\nof \\cite{brenig}, respectively, see more complete discussion of\nproperties of the time-dependent relaxation function as a Fourier\ntransform of the relaxation function $\\Phi(\\omega )$ in\n\\cite{brenig}). The second term in (\\ref{relaxcom}) and\n(\\ref{relaxbrenig}) describes the pure\ndamped Raleigh mode corresponding to the overdamped pole\n$\\omega ^{(0)}$ (\\ref{shp}) defined by the diffuseness coefficient\n${\\tt D}_T \\propto \\Gamma^{(0)}$ (or $ \\propto\\gamma_{{}_{\\! T}}$ in the\nnotation of \\cite{brenig}). As noted in \\cite{brenig}, the\nstrength of this peak is a factor $1-{\\tt C}_{\\mathcal{V}}\/{\\tt\nC}_{\\cal P}$ smaller than for the two first sound peaks. According\nto (\\ref{cpcvexp}), in the zero temperature limit $T\n\\rightarrow 0$, the Raleigh peak disappears but the\nBrillouin ones become dominating because of $\\Gamma \\propto\n\\Gamma^{(1)} \\propto 1\/T^2$; see the second equation of\n(\\ref{liqparam}) for the relation of $\\Gamma$ to\n$\\Gamma^{(1)}$ and (\\ref{gambaronet}). Note also that the\ncoefficient in front of the square brackets in\n(\\ref{relaxcom}) is finite in the limit $T \\rightarrow 0$.\n\n\n\\subsubsection{CORRELATION FUNCTION}\n\\label{corfun}\n\nWe like to present also the correlation function, partly for the\nsake of completeness and partly to allow for comparisons with\ncalculations of the function in the nuclear SM approach of\n\\cite{hofivyam,ivhofpasyam}, see also\n\\cite{hofmann,hofbook}, to the collective motion of finite nuclei. Let us\nuse now the fluctuation-dissipation theorem \\cite{kubo} to get the\ncorrelation function $\\psi^{\\prime\\prime}(\\omega )$,\n\\bel{fludiptheor}\n\\psi^{\\prime\\prime}(\\omega ) \\rightarrow\n \\hbar \\omega \\coth \\left(\\frac{\\hbar \\omega }{2T} \\right)\n \\Phi ^{\\prime\\prime}(\\omega )=\n \\hbar \\coth \\left(\\frac{\\hbar \\omega }{2T}\\right)\n \\chi ^{\\prime\\prime}(\\omega )\\;.\n\\end{equation}\n In the semiclassical limit $\\hbar \\rightarrow 0$ considered\nhere, one has\n\\bel{corrfun}\n\\psi^{\\prime\\prime}(\\omega ) =\n ~\\frac{2 T }{\\omega }~ \\chi ^{\\prime\\prime}(\\omega )=~\n 2T~ \\Phi^{\\prime\\prime}\\left(\\omega \\right).\n\\end{equation}\n According to (\\ref{forstereq}),(\\ref{relaxbrenig}), this\ncorrelation function can be split into the two components as in\n\\cite{hofivyam,hofmann},\n\\bel{corrfunhof}\n\\psi^{\\prime\\prime}(\\omega ) =\n \\psi_0^{\\prime\\prime}(\\omega ) + \\psi_R^{\\prime\\prime}(\\omega )\\;.\n\\end{equation}\n Here, $\\psi_0^{\\prime\\prime}$ is the heat pole part,\n\\begin{eqnarray}\\label{corrfunhp}\n\\psi_0^{\\prime\\prime}(\\omega ) &=&\n \\frac{2 T }{\\omega } \\chi ^{(0)\\;\\prime\\prime}(\\omega ) \\nonumber\\\\\n &=& 2T \\chi^T~\n\\frac{\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T}{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2},\n\\end{eqnarray}\n $\\chi^{(0)\\;\\prime\\prime}$ is given by the first equation in\n(\\ref{chichpqompp}) and is related to the second heat pole\nterms in the square brackets of (\\ref{forstereq}) and\n(\\ref{relaxbrenig}) [through (\\ref{chiimpsi})]. This part is\nsingular at the zero frequency point $\\omega = 0$ for $\\tau_q\n\\rightarrow 0$, see (\\ref{diffusd}) and (\\ref{som}). The\nother term $\\psi_R^{\\prime\\prime}$ in (\\ref{corrfunhof}) is\nassociated with the first sound component in the square brackets\nof (\\ref{forstereq}), (\\ref{relaxbrenig}),\n\\begin{eqnarray}\\label{corrfun1}\n\\psi_R^{\\prime\\prime}(\\omega ) &=&\n ~\\frac{2 T }{\\omega }~ \\chi ^{(1)\\;\\prime\\prime}(\\omega ) \\nonumber\\\\\n &=& ~2T~ \\chi^{T}~\\frac{\n\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\n c^2 q^4 \\Gamma}{\\left(\\omega ^2-c^2q^2\\right)^2 +\\left(\\omega q^2 \\Gamma\\right)^2}.\n\\end{eqnarray}\n This component has no such singularity at $\\omega = 0$ for $\\tau_q \\to 0$, as\nseen from (\\ref{liqparam}), (\\ref{sfirst}) and\n(\\ref{gambaronet}) [see (\\ref{chiconeqompp}) for\n$\\chi^{(1)\\;\\prime\\prime}(\\omega )$ in the middle of\n(\\ref{corrfun1})]. According to the second equation in\n(\\ref{corrfunhp}), the heat pole part\n$\\psi_0^{\\prime\\prime}(\\omega )$ of (\\ref{corrfunhof}) for the {\\it\nintrinsic} correlation function can be written as in\n\\cite{hofivyam,hofmann},\n\\bel{corrfunhphof}\n\\psi_0^{\\prime\\prime}(\\omega )= \\psi^{(0)}\n \\frac{\\hbar {\\it \\Gamma}_T }{(\\hbar \\omega )^2 + {\\it \\Gamma}_T^2 \/ 4},\n\\end{equation}\n where\n\\bel{GammaT}\n{\\it \\Gamma}_T=\n2 \\hbar q^2 {\\tt D}_T= \\hbar\n\\Gamma^{(0)},\n\\end{equation}\nand\n\\bel{psi0}\n(1 \/ T) \\psi^{(0)}=\\chi^T-\\chi^\\varsigma\n = \\chi^T-\\chi(0).\n\\end{equation}\n We applied here (\\ref{chitchiskSkT}) in the first equation\nof (\\ref{psi0}) and ergodicity condition (\\ref{ergodicity})\nfor the second one. The specific expressions for the quantities\n$\\Gamma^{(0)}$, $\\chi^T$ and $\\chi(0)$ in the last two equations\n(\\ref{GammaT}) and (\\ref{psi0}) can be found in (\\ref{shp}),\n(\\ref{gambarhpt}) and (\\ref{statrespfl}). Note that the\ncorrelation function (\\ref{corrfunhphof}), corresponding to the\nheat pole, has the Lorentzian multiplier. This multiplier approaches the\n$\\delta(\\omega )$ function in the hydrodynamic limit $\\tau_q\n\\rightarrow 0$ because of $\\Gamma_T \\rightarrow 0$, according to\n(\\ref{GammaT}) and (\\ref{shp}) ($\\Gamma^{(0)} \\rightarrow\n0$), i.e.,\n \\bel{corrdeltafunlim}\n \\psi^{(0)}(\\omega )\n \\rightarrow 2 \\pi \\psi^{(0)} \\delta (\\omega )\\qquad\\mbox{for}\\qquad\n \\tau_q \\rightarrow 0.\n\\end{equation}\n The relations (\\ref{corrfunhphof}), (\\ref{psi0}) and\n(\\ref{corrdeltafunlim}) confirm the discussion in\n\\cite{hofmann} concerning the heat pole contribution to the\ncorrelation function. The specific property of the Fermi liquid is\nthat this system is exactly ergodic, see (\\ref{ergodicity}), as\nused in the second equation of (\\ref{psi0}).\n\n\n\\section{Nuclear response within the Fermi-liquid droplet model}\n\\label{respfuntheor}\n\n\\subsection{Basic definitions}\n\\label{basdef}\n\nSo far we considered the Fermi-liquid theory for study of the\ncollective excitations at finite temperatures much smaller than\nthe Fermi energy $\\varepsilon_{{}_{\\! {\\rm F}}}$ in the {\\it infinite} nuclear matter.\nThis theory can be also helpful for investigation of the\ncollective modes and transport properties of heavy heated nucleus\nconsidered as a {\\it finite} Fermi system within the macroscopic FLDM\n\\cite{strutmagbr,strutmagden0,galiqmodgen,galiqmod,magstrut,denisov,magboundcond,kolmagpl,magkohofsh,kolmagsh}.\nSuch a semiclassical nuclear model applied\nearlier successfully to the giant multipole resonance description\n\\cite{strutmagden0,galiqmod,strutmagden,denisov,magpl,kolmagpl,kolmagsh,magkohofsh} is expected\nto be also\nincorporated in practice as an asymptotic high temperature limit\nof the quantum transport theory \\cite{hofmann} based on the shell model.\nThis theory takes into account the residue\ninteractions like particle collisions for study of the low energy\nexcitations in nuclei. The latter application of the FLDM is very\nimportant for understanding itself the dissipative processes\nlike nuclear fission at finite temperatures\n(see, g.e., \\cite{hofivyam,hofmann,hofbook,hofivmag,hofmag})\n\n\nFollowing \\cite{hofmann,hofbook}, let us describe the many-body\nexcitations of nuclei in terms of the response to an external\nperturbation\n\\bel{extfield}\nV_{\\rm ext}=q_{\\rm ext}(t)~{\\hat F},\n\\end{equation}\nwhere\n${\\hat F}$ is some one-body operator,\n\\bel{qext}\nq_{\\rm ext}(t)=q_{\\rm ext}^{\\omega }\\hbox{exp}[-i(\\omega + i\\epsilon)],\\qquad (\\epsilon=+0)\n\\end{equation}\n The linear\nresponse function can be determined through the Fourier transform\n$\\langle {\\hat F} \\rangle_\\omega $ of the time-dependent quantum\naverage $\\langle {\\hat F} \\rangle_t$ by\n\\bel{defresp}\n\\langle\n{\\hat F} \\rangle_\\omega = -\\chi_{FF}^{\\rm coll}(\\omega )~q_{\\rm ext}^\\omega .\n\\end{equation}\n Here\nand below we omit an unperturbed average value $\\langle {\\hat F}\n\\rangle_0$ and use the same notation as in \\cite{hofmann}. In the\nfollowing, we shall consider the operators ${\\hat F}$ neglecting the\nmomentum dependence in a phase space representation in the linear\napproximation for an external field $V_{\\rm ext}$ and writing\n${\\hat F}={\\hat F}({\\bf r})$. According to (\\ref{defresp}), one can\nthen express explicitly $\\chi_{FF}^{\\rm coll}(\\omega )$ in terms of the\nFourier transform $\\delta \\rho_\\omega ({\\bf r})$ of the transition density\n$\\delta \\rho({\\bf r},t)$ \\cite{magkohofsh} as\n\\bel{chicollrho}\n\\chi_{FF}^{\\rm coll}(\\omega )= -\\frac{1 }{q_{\\rm ext}^\\omega } \\int {\\rm\nd}{\\bf r}~{\\hat F}({\\bf r})~\\delta \\rho_\\omega ({\\bf r}).\n\\end{equation}\n Note that in a\nmacroscopic picture the transition density is the dynamical part\n$\\delta \\rho({\\bf r},t)$ of the particle density,\n\\bel{partdens}\n\\rho({\\bf r},t)=\\rho_{\\rm qs}+ \\delta \\rho({\\bf r},t).\n\\end{equation}\n Here, $\\rho_{\\rm qs}$ is\nthe quasistatic equilibrium particle density. We define now\n${\\hat F}$ as related to the variation of the self-consistent mean\nfield $V$ in the nuclear Hamiltonian:\n\\bel{hamil}\n {\\hat H}={\\hat H}_0 + V\n= {\\hat H}_0 + Q {\\hat F} + \\frac{1 }{2} Q^2 \\left\\langle \\left(\\frac{\\partial^2\n{\\hat H}}{\\partial Q^2}\\right)_{Q=0}\\right\\rangle_0+...,\n\\end{equation}\n where\n$H_0$ is an unperturbed Hamiltonian. Introducing the\n collective variable $Q$ ($Q=0$ in equilibrium), one may write\n \\bel{foper}\n{\\hat F}=\\left(\\frac{\\partial {\\hat H}}{\\partial Q}\\right)_{Q=0}=\n\\left(\\frac{\\partial V }{\\partial Q}\\right)_{Q=0}.\n\\end{equation}\n The total Hamiltonian ${\\hat H}_{\\rm tot}$ is given by\n\\bel{hamiltot}\n{\\hat H}_{\\rm tot}={\\hat H}+q_{\\rm ext}(t){\\hat F}.\n\\end{equation}\n\nAs shown in \\cite{hofmann,hofbook}, a\nconservation of the nuclear energy\n$\\langle {\\hat H} \\rangle$ for the Hamiltonian ${\\hat H}$\n(\\ref{hamil}) leads to the equation of motion which is the\nsecular equation in the Fourier representation,\n\\bel{seculareq}\nk^{-1} +\\chi(\\omega )=0 .\n\\end{equation}\n The coupling constant $k$ is given by\n\\bel{kstiffC0chi0}\n -k^{-1} = C(0) + \\chi_{{}_{\\! FF}}(0),\n\\end{equation}\n$C(0)=\\left(\\partial^2 E(Q,S)\/ \\partial Q^2\\right)_{Q=0}$ is the stiffness\ncoefficient of the internal energy\n$E(Q,S)$ for the constant nuclear entropy, $S_0$,\nand $\\chi_{FF}(0)$ is the static (isolated) susceptibility.\n$\\langle {\\hat F} \\rangle_\\omega $ and $Q_\\omega $ are related then each\nother by the self-consistency condition\n \\bel{selfconsist}\nk \\langle {\\hat F} \\rangle_\\omega = Q_\\omega \n\\end{equation}\n with $Q_\\omega $ being the\nFourier component of the collective variable $Q(t)$.\nThe ergodicity condition,\n\\bel{ergodicity1}\n\\chi_{{}_{\\! FF}}(0)=\\chi_{FF}^{\\rm ad},\n\\end{equation}\n with $\\chi_{FF}^{\\rm ad}$ being the\nadiabatic susceptibility was not used really in the derivation of\nthe self-consistency condition (\\ref{selfconsist}) with the\ncoupling constant $k$ from (\\ref{kstiffC0chi0}) in\n\\cite{hofmann} if the definition of slow variation of the\ntime-dependent $\\langle {\\hat F} \\rangle$ is employed under the\ncertain physical conditions, see (3.7-14),(3.3-15) in\n\\cite{hofmann} and discussion there. The isolated susceptibility\n$\\chi_{{}_{\\! FF}}(0)$ is the static limit $\\omega \\rightarrow 0$ of the\nintrinsic response function $\\chi_{{}_{\\! FF}}(\\omega )$ defined by\n\\bel{defrespintr}\n\\langle {\\hat F}\\rangle_\\omega = -\\left(Q_\\omega +\nq_{\\rm ext}^\\omega \\right) \\chi_{{}_{\\! FF}}(\\omega ) .\n\\end{equation}\n Thus, the intrinsic response\nfunction $\\chi_{{}_{\\! FF}}(\\omega )$ is related to the collective response\nfunction $\\chi_{FF}^{\\rm coll}(\\omega )$ through the relation\n(\\ref{ddrespchiin}) \\cite{bohrmot,hofmann}. Within the FLDM\nformulated below, it is simpler to derive first the collective\nresponse function $\\chi_{FF}^{\\rm coll}(\\omega )$ by making directly\nuse of the\ndefinition (\\ref{chicollrho}). For comparison with\nthe microscopic quantum theory \\cite{hofmann} and for study of\nthe susceptibilities and of the ergodicity property, it is helpful\nto present the intrinsic response function $\\chi(\\omega )$ in terms\nof the collective response function $\\chi_{FF}^{\\rm coll}(\\omega )$ found\nfrom (\\ref{ddrespchiin}) as\n\\bel{respintr}\n\\chi_{{}_{\\! FF}}(\\omega )=\n\\frac{\\chi_{FF}^{\\rm coll}(\\omega ) }{1 - k \\chi_{FF}^{\\rm coll}(\\omega )}.\n\\end{equation}\n\n\n\\subsection{Fermi-Liquid Droplet Model}\n\\label{fldm}\n\nIn this section we follow \\cite{magkohofsh} for the basic grounds\nof the FLDM \\cite{magstrut,kolmagpl} for heavy nuclei taking into account\nthe quasiparticle Landau--Vlasov theory for the collective\ndynamics of the {\\it heated} Fermi liquids described in\n\\cite{heipethrev} and developed in the previous sections in more\ndetails for nuclear matter. The main idea is to apply this\nsemiclassical theory for the distribution function {\\it inside the\nnucleus} with the macroscopic {\\it boundary conditions}\n\\cite{strutmagbr,magstrut} like for normal liquids {\\it at its\nmoving surface}. These boundary conditions are used for the\nsolutions of the dynamical collisional Landau--Vlasov equation\n(\\ref{landvlas}) {\\it coupled with the thermodynamic relations}\nfor motion in the Fermi-liquid-drop interior. Our derivations are\nbased on the conception of the linearized dynamics near the {\\it\nlocal} equilibrium instead of the global one considered earlier in\n\\cite{kolmagpl,magkohofsh}. This is important for a {\\it low} frequency\nregion of the nuclear excitations which we are interested in this\nreview.\n\nWe shall consider below small isoscalar vibrations of the nuclear surface\nnear a spherical shape, which are induced by the external field\n$V_{\\rm ext}(t)$ (\\ref{extfield}). To this end, we define\na collective variable $Q(t)$ in the usual way:\n\\bel{surface}\nR = R_0\n\\left[1 + Q(t)Y_{L0}({\\hat r})\\right],\n\\end{equation}\nwhere $R_0$ is the\nequilibrium radius of nucleus, and $Y_{L0}({\\hat r})$ is the spherical\nharmonics which represent the axially symmetric shapes as\nfunctions of the radius vector angles ${\\hat r}$. For $Q(t)$\nwe expect the form\n\\bel{collvarq}\nQ(t) = Q_\\omega \\hbox{exp}\\left(-i\\omega \nt\\right)\n\\end{equation}\n with the same frequency $\\omega $ as for the external\nfield (\\ref{extfield}).\n\n\n\\subsubsection{EQUATIONS OF MOTION INSIDE THE NUCLEUS}\n\n Quasiparticle conceptions\nof the Landau Fermi-liquid theory can be justified\nin the nuclear volume, where variations of the density\n$\\rho({\\bf r},t)$ (\\ref{densit}) are small. Therefore, in\nthe interior of sufficiently heavy nuclei, one may describe the\nsemiclassical phase-space dynamics\nin terms of the distribution function $\\delta f({\\bf r},{\\bf p},t)$\n(\\ref{dfgeqdfleq}) which satisfies the\ncollisional Landau--Vlasov equation (\\ref{landvlas}). We recall now\nthe equations of Sec.\\ \\ref{eqmotion} which present the\ncollective dynamics linearized with respect to the local\nequilibrium (\\ref{intcoll}). Our interior nuclear collective\ndynamics is then described by 6 equations, see (\\ref{landvlas}) and\n(\\ref{consereq}), for the 6 local quantities $\\delta \\rho({\\bf r},t)$,\n$\\delta \\mu({\\bf r},t)$, ${\\bf u}({\\bf r},t)$ and $\\delta T({\\bf r},t)$ defined inside\nof the nucleus as for the nuclear matter. The conserving\nequations (\\ref{conteq}), (\\ref{momenteq}) [or (\\ref{navstokeq})\nfor a potential flow], (\\ref{enerconserv}) and (\\ref{entropyeq})\nare helpful to find them in the semiclassical approximation.\n\nFor the isoscalar multipole vibrations of the Fermi-liquid drop\nsurface (\\ref{surface}), we shall look for the solutions of\n(\\ref{landvlas}), (\\ref{consereq})\nin terms of a superposition\nof the plane sound waves (\\ref{planewave}) over all angles\n${\\hat q}$ of the unit wave vector ${\\bf q}$ with the amplitude\n$\\mathcal{A}_L({\\hat q})$,\n\\begin{eqnarray}\\label{superpos}\n \\delta f({\\bf r},{\\bf p},t) &=& \\int {\\rm d} \\Omega_{\\bf q}\n\\mathcal{A}_L({\\hat q})\n~\\delta {\\tilde f} ({\\bf q},{\\bf p},\\omega )\\; \\hbox{exp} [i({\\bf q}{\\bf r}-\\omega t)]\n\\nonumber\\\\\n&&{\\rm with} \\qquad \\mathcal{A}_L({\\hat q})=\nY_{L0} ({\\hat q}_z).\n\\end{eqnarray}\n Here $L$ is the multipolarity of collective vibrations,\n${\\hat q}_z$ is the projection of the unit\nvector ${\\hat q}={\\bf q}\/q$ on the symmetry $z$-axis.\nThe Fourier amplitudes\n$\\delta f({\\bf q},{\\bf p},\\omega )$ are presented as a spherical harmonic\nexpansion in momentum space,\n\\bel{tildef}\n\\delta {\\tilde\nf}({\\bf q},{\\bf p},\\omega ) = \\left(\\frac{\\partial f_{\\rm g.e.} (\\varepsilon_{\\bf p})}{\\partial\n\\varepsilon_{\\bf p}} \\right)_{\\rm g.e.} \\sum_{l^\\prime} \\mathcal{A}_{l^\\prime}(\\omega ,q)\nY_{l^\\prime 0} ({\\hat p} \\cdot {\\hat q}),\n\\end{equation}\n where $\\mathcal{A}_{l^\\prime}$ are small vibration amplitudes.\nFor such\nsolutions, the velocity field ${\\bf u}$ corresponds to the\npotential flow (\\ref{velpoten}).\n\nThe relaxation time $\\tau$ in (\\ref{intcoll}) is assumed to be\nfrequency and temperature dependent as in (\\ref{tautom}).\nFollowing \\cite{hofmann,kolmagpl,magkohofsh}, we take the form:\n\\bel{relaxtime}\n \\tau(\\omega ,T)=\\frac{\\hbar}{{\\it \\Gamma}(\\omega ,T)},\n\\end{equation}\nwhere\n\\bel{widthG}\n {\\it \\Gamma}(\\omega ,T)=\\frac{\\pi^2 }{{\\it\n\\Gamma}_0}~ \\frac{T^2 + c_o (\\hbar \\omega )^2 }{\n1+\\frac{\\pi^2}{c^2}\\left[T^2 + c_o (\\hbar \\omega )^2\\right]}.\n\\end{equation}\n For $c_o$ one has several values. For instance, $c_o = 1 \/ 4\\pi^2$,\naccording to \\cite{landau,ayik}, $c_o = 1 \/\\pi^2$ follows\nfrom \\cite{pinenoz,hofmann}, $3 \/\n4\\pi^2$ from \\cite{kolmagpl} and several numbers near these\nconstants were suggested in \\cite{sykbrook,brooksyk}. Formula\n(\\ref{widthG}) with the $c_o = 1\/\\pi^2$ and finite cut-off\nconstant $c$ which weakens the dependence on both\nfrequency $\\omega $ and\ntemperature $T$ at large values of these quantities may in some\nsense be compared with the expressions suggested in \\cite{hofmann} for\nthe imaginary part of the self-energy to be used in microscopic\ncomputations \\cite{hofmann,hofbook}\n[$c$ in (\\ref{widthG}) should not be\nconfused with the sound velocity $c$ used for the description of\nnormal liquids [see, g.e., (\\ref{liqparam}) and (\\ref{forstereq})].\nIn line\nof these computations, we shall use ${\\it \\Gamma}_o=33.3$ MeV and\n$c=20$ MeV in our FLDM calculations.\nThe value of the parameter $c_o=3\/4 \\pi^2$ is taken as in\n\\cite{kolmagpl,magkohofsh}. The specific value of this\nparameter is not important for the following derivations and\nresults in this section because we shall apply the temperature-dependent\nFermi-liquid theory for low frequencies and large temperatures.\nNote that for $c \\to \\infty$ the expression (\\ref{widthG}) was derived in\n\\cite{landau,pinenoz,sykbrook,brooksyk,kolmagpl}.\n\n\n\n\\subsubsection{BOUNDARY CONDITIONS AND COUPLING CONSTANT}\n\\label{boundary}\n\nThe dynamics in the surface layer of nucleus can be described by\nmeans of the macroscopic boundary conditions as in\n\\cite{magstrut} by using the effective surface approximation\n\\cite{strutmagbr,strutmagden,magboundcond}. For small vibration\namplitudes, they read:\n\\bel{bound1}\nu_r \\Big\\vert_{r=R_0} = {\\dot\nR}(t) \\equiv R_0 {\\dot Q}(t) Y_{L0}({\\hat r}),\n\\end{equation}\n \\bel{bound2}\n\\Pi_{rr} \\Big\\vert_{r=R_0} = P_{S} + P_{\\rm ext},\n\\end{equation}\n where $u_r$\nand $\\Pi_{rr}$ are the radial components of the velocity field\n${\\bf u}$ (\\ref{veloc}) and the momentum flux tensor\n$\\Pi_{\\alpha\\beta}$ (\\ref{momentflux}) which are determined in\nthe nuclear volume, see\n\\cite{bekhal,ivanov,magboundcond,abrditstrut,komagstrvv,abrdavkolsh}\nfor other (mirror and diffused) boundary conditions used directly\nfor the distribution function as a solution of the Landau--Vlasov equation.\nIn the case of the potential flow (\\ref{velpoten}),\n we shall use the specific expression for the\nmomentum flux tensor (\\ref{momentfluxpot}) with the shear modulus\n($\\lambda$) and viscosity ($\\nu$) coefficients given by\n(\\ref{shearmod}) and (\\ref{viscos}), respectively. The\nsurface pressure $P_{S}$, which is due to the tension forces\nfor the isoscalar motion in symmetric nuclei, is\ngiven by\n\\bel{surfpress}\nP_{S} = \\frac{\\alpha }{R_0} (L-1)\n(L+2)\\; Q(t) Y_{L0} ({\\hat r}),\n\\end{equation}\n where $\\alpha$ is the surface\ntension coefficient, see Sec.\\ \\ref{npcorivgdr} and\nAppendix D\nfor the isovector asymmetric modes. For the\ntension coefficient $\\alpha$, we used an expression found in\n\\cite{strutmagden} within the ESA.\nThis approximation is based on expansion of the nuclear\ncharacteristics, such as the particle density and the total energy in\nsmall parameter $a\/R_{0} \\sim A^{-1\/3}$, where $a$ is the diffuseness\nparameter and $R_{0}$ is the mean curvature radius of the nuclear surface\n\\cite{strutmagbr,strutmagden}, see also \\cite{magsangzh,BMRV} and\nAppendix D.\nIn this way, one derives the\nnuclear energy expansion [Wiezs\\\"acker formula (\\ref{EvEs}),\n(\\ref{Espm})], $E=E_{\\mathcal{V}} +\nE_{S} +...$,\nwith the volume part of the energy $E_{\\mathcal{V}}$\nproportional to the particle number $A$,\nand the surface energy $E_{S}$, $E_{S}=b_{{}_{\\! S}}A^{2\/3}$\n($b_{{}_{\\! S}}=4\\pi r_0^2 \\alpha$ corresponds to the surface tension\nconstant $\\alpha$, $b_{{}_{\\! S}}\\approx 20~\\mbox{MeV}$,\n$r_{{}_{\\! 0}}=R_0\/A^{1\/3} \\approx 1.1-1.2$ fm) and so on, see\n\\cite{strutmagbr,strutmagden,magsangzh,BMRV} and Appendices A.4\n(symmetrical nuclei) and D (asymmetrical ones) for more details\n(the suffix ``$+$'' is omitted here).\nAccording to (\\ref{sigma}) of \\cite{strutmagden,BMRV},\n\\bel{tensionconst}\n\\alpha \\approx 2\n\\mathcal{C}\n\\int\\limits^\\infty_0 {\\rm d} r \\left(\\frac{\n\\partial \\rho_{\\rm qs}}{\\partial r}\\right)^2.\n\\end{equation}\nHere and below we neglect the relatively small\ncorrections of the order of $A^{-1\/3}$ of the\nESA, which are in\nparticular related to the semiclassical $\\hbar$ corrections and\nexternal field. The coefficient $\\mathcal C$\nappears earlier in\nfront of the term which is proportional to $\\left({{\\bf \\nabla}}\n\\rho_{\\rm qs}(r)\\right)^2$\nin the nuclear energy-density formula [see (\\ref{enerden})],\n$\\mathcal{C}=40-60~ {\\rm MeV} \\cdot {\\rm fm}^5$ \\cite{BMRV}.\n\n\nAn external pressure\n$P_{\\rm ext}$ appears in (\\ref{bound2}),\nwhere we make connection to the external\npotential $V_{\\rm ext}$ (\\ref{extfield})\n\\cite{komagstrvv,magkohofsh},\n\\bel{extpress}\nP_{\\rm ext} =\n-\\int\\limits_0^\\infty dr \\> \\> \\rho_{\\rm qs} (r)\n\\frac{\\partial V_{\\rm ext}}{\\partial r}.\n\\end{equation}\n For the density in equilibrium,\none has\n\\bel{denseq}\n\\rho_{\\rm qs} (r) =\\rho_{{}_{\\! 0}} w(\\xi),\\quad \\xi=\\frac{r-R}{a},\\quad\na=\\sqrt{\\frac{\\mathcal{C}_{+}\\; \\rho_\\infty\\; K}{30\\;\nb_{\\mathcal{V}}^2}}.\n\\end{equation}\nThis density is expressed in terms of the\nprofile function $w(\\xi)$ with a sharp decrease from one to zero in\nthe narrow region of the order of the diffuseness parameter\n$a$\nnear $\\xi=0$ as in a step function ($w(\\xi) \\rightarrow \\theta(R-r)$\nfor $a \\rightarrow 0$), $b_{\\mathcal{V}} \\approx 16$ MeV\nis the separation energy per\nnucleon\n\\cite{strutmagbr,magstrut,strutmagden,BMRV}. The value\nof equilibrium density inside the nucleus $\\rho_{{}_{\\! 0}}$\n\\cite{strutmagbr} is given by\n\\bel{rho0}\n\\rho_{{}_{\\! 0}}=\\rho_\\infty\\left(1+\\frac{6\nb_{{}_{\\! S}} r_0}{K R_0}\\right),\n\\end{equation}\n where\n$\\rho_\\infty$ is the particle density of the infinite nuclear\nmatter, $\\rho_\\infty=3\/(4\\pi r_0^3)$. The surface energy constant,\n$b_{{}_{\\! S}}$, and in-compressibility modulus, $K$, in\n(\\ref{rho0}) depend on the condition of the constant temperature,\nentropy and of the static limit, as shown in Appendix C.\nIn (\\ref{rho0}) and below, we omit the index ${\\tt X}$ of these\nquantities which specifies one of these conditions, see Appendix C.\nFor instance, the in-compressibility in\n(\\ref{rho0}) is denoted simply as\n$K=K_{\\rm tot}(\\omega =0)=K^{\\varsigma}$, as shown above through\n(\\ref{incomprtot}), (\\ref{dtdrhoexp}) and (\\ref{incompradexp}).\nThe surface energy constant $b_{{}_{\\! S}}$ in (\\ref{rho0}) is also\nidentical to the adiabatic one as the in-compressibility\n(see Appendix C). The second term in the circle brackets of\n(\\ref{rho0}) is a small correction proportional to $A^{-1\/3}$,\ndue to the surface tension.\n\n\nBoundary conditions (\\ref{bound1}) and (\\ref{bound2}) were\nre-derived here from (\\ref{conteq}) and\n(\\ref{momenteq}) where all quantities are now extended to the\nsurface region with a sharp coordinate dependence of the particle\ndensity as in the approach \\cite{magstrut}.\nHowever, we used the specific properties of the {\\it\nheated} Fermi-liquid drop following the same ESA\n\\cite{strutmagbr,magstrut,strutmagden}. For the derivation of (\\ref{bound2}),\ng.e.,\nthe key equation\n(\\ref{keybound}) for the Gibbs thermodynamic potential per\nparticle $g$, which satisfies the thermodynamic relations\n(\\ref{thermrelmu}), was applied instead of the energy per particle\n$\\varepsilon$ of \\cite{magstrut}. The result\n(\\ref{bound2}) has the same form as in\n\\cite{magstrut,magkohofsh} because in its derivation we have\nsimultaneously to use (\\ref{gradrelener}) of the\ntemperature-dependent Fermi-liquid theory (with the entropy term\n$T d \\varsigma$), in contrast to the adiabatic equation (17) of\n\\cite{magstrut}, see Appendix A.4 for details.\n\n\nThe external field $V_{\\rm ext}$ (\\ref{extfield}) in\n(\\ref{extpress}) is determined by the operator ${\\hat F}({\\bf r})$\n(\\ref{foper}), and hence, $V_{\\rm ext}$ is concentrated in the surface\nregion of the nucleus. Indeed, for the operator ${\\hat F}({{\\bf r}})$\n(\\ref{foper}) in the FLDM, one gets the form\n\\bel{foperl}\n{\\hat F}({{\\bf r}}) = \\left(\\frac{\\delta V }{\\delta \\rho}~ \\frac{\\partial\n\\rho}{\\partial Q}\\right)_{Q=0}^{\\rm qs}\n=-R_0\\left(\\frac{\\partial V }{\\partial r}\\right)_{R=R_0}^{\\rm qs}\nY_{L0}({\\hat r}),\n\\end{equation}\n see (\\ref{dhdq}). After substitution of\n(\\ref{foperl}) into (\\ref{extpress}) we have\n\\bel{extpressl}\nP_{\\rm ext} = - \\frac{1 }{kR_0^3}\\; q_{\\rm ext}(t) Y_{L0}\n({\\hat r}),\n\\end{equation}\nwhere\n\\bel{kfld}\nk^{-1} = \\frac{K \\alpha R_0^4 }{18\n\\mathcal{C} \\rho_\\infty}\n \\left[1 + \\mathcal{O}\\left(A^{-1\/3}\\right)\\right]\n \\approx \\frac{K b_{{}_{\\! S}} r_0^5 }{54 \\mathcal{C}} A^{4\/3}.\n\\end{equation}\n The integration by parts in (\\ref{extpress}) and the\nequation (\\ref{couplfld}) for the quasistatic coupling constant $k^{-1}$\nwere used in the derivation of (\\ref{extpressl}), (\\ref{kfld}), see\nthe second equation of (\\ref{couplxchix}), and also applications to\ncalculations of the collective vibration modes in \\cite{yaf,gzhmagfed}.\n\n\n\n\\subsubsection{COLLECTIVE RESPONSE FUNCTION}\n\\label{collresponse}\n\nAs shown in \\cite{strutmagbr,strutmagden,galiqmod}, the \nlinearized dynamic part of the\nnucleonic density $\\delta \\rho ({\\bf r},t)$ for the isoscalar modes\ncan be represented as a sum\nof the \"volume\" and the \"surface\" term,\n\\bel{densvolsurf}\n\\delta\n\\rho({{\\bf r}},t) = \\delta \\rho^{\\rm vol}({{\\bf r}},t) w(\\xi)\n- \\frac{\\partial\nw}{\\partial r} \\rho_{{}_{\\! 0}} \\delta R,\n\\end{equation}\n where $\\delta R$ is the\nvariation of nuclear radius (\\ref{surface}), $\\delta\nR = R_0 Q(t) Y_{L0} ({\\hat r})$, $w$ is defined around\n(\\ref{denseq})\nand in Appendix D. For\nisovector vibration modes of the odd multipolarity (dipole), one has to\naccount for the mass center conservation \\cite{kolmagsh,BMR}\n[see (\\ref{trandenscl})].\nThe upper index \"vol\" in $\\delta\n\\rho^{\\rm vol}({{\\bf r}},t)$ of (\\ref{densvolsurf}) denotes that\nthe dynamical particle-density variation\nis determined by the equations of motion in the nuclear\nvolume and is given in terms of the local part $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\rm l.e.})$ (\\ref{dfleq}) of the distribution function\n$\\delta f({{\\bf r}}, {{\\bf p}},t)$ (\\ref{dfgeqdfleq}) through\n(\\ref{densit}).\n\n\nSolving (\\ref{navstokeq}) with the first\nboundary condition (\\ref{bound1}), one gets the potential\n$\\varphi$ in the form\n\\bel{potensolut}\n\\varphi({{\\bf r}},t) = \\frac{1 }{q^2}\n\\frac{qR_0}{ j^\\prime_L (qR_0)} {\\dot Q}(t) j_{{}_{\\! L}}(qr) Y_{L0}\n({\\hat r}),\n\\end{equation}\n where $j_{{}_{\\! L}}(x)$ is the spherical Bessel function\nand $j^\\prime_L(x) = dj_{{}_{\\! L}}(x)\/dx$. From the continuity equation\n(\\ref{conteq}) with (\\ref{potensolut}), one has\n\\bel{densolut}\n\\delta \\rho^{\\rm vol}({{\\bf r}},t) = \\rho_{{}_{\\! 0}} \\frac{qR_0 }{ j^\\prime_L(qR_0)}\nQ(t) j_{{}_{\\! L}}(qr) Y_{L0} ({\\hat r}).\n\\end{equation}\n Therefore, according to\n(\\ref{densvolsurf}) and (\\ref{densolut}), one finds\n\\bel{trandensolut}\n\\delta \\rho({{\\bf r}},t) = \\rho_{{}_{\\! 0}} Q(t) Y_{L0}({\\hat\nr}) \\left[\\frac{qR_0 }{ j^\\prime_L(qR_0)} j_{{}_{\\! L}}(qr) w(\\xi)\n- \\frac{\\partial w}{\\partial r} R_0\\right].\n\\end{equation}\n\nWith this solution, we may now proceed to calculate the response\nfunction $\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollrho})\nby expressing the integral over the coordinates ${\\bf r}$ for the average\n$\\langle {\\hat F}\\rangle_\\omega $ (\\ref{defresp}) in\nthe numerator of (\\ref{chicollrho})\nin terms of our collective variable $Q_\\omega $ given by (\\ref{collvarq}).\nIndeed, substituting the Fourier transform of\n(\\ref{trandensolut}) together with ${\\hat F}$ from\n(\\ref{foperl}) into (\\ref{chicollrho}), we obtain\n\\bel{chicollQq}\n \\chi_{FF}^{\\rm coll}(\\omega ) = - \\frac{Q_\\omega }{k q^\\omega _{\\rm ext}}.\n\\end{equation}\n\n\nUsing (\\ref{momentfluxpot}),\n(\\ref{surfpress}), (\\ref{extpressl}) and (\\ref{potensolut}), one may\nwrite the second boundary condition (\\ref{bound2}) in terms of the\ncollective variable $Q(t)$ and periodic time dependence of the\nexternal field $V_{\\rm ext}$ in the form of the equation of motion\n\\bel{eqmotionQ}\n\\mathcal{B}_L(x) {\\ddot Q} + \\mathcal{C}_L(x)Q +\n\\mathcal{Z}_L(x) {\\dot Q} = - q_{\\rm ext}.\n\\end{equation}\n We have introduced various new\nquantities,\n\\bel{x}\nx = \\frac{\\omega }{\\Omega}=\\frac{\\omega R_0 }{ v_{{}_{\\! {\\rm F}}} s}, \\quad \\mbox{with}\n\\quad\n\\Omega=\\frac{v_{{}_{\\! {\\rm F}}}}{R_0}\\sim \\frac{\\varepsilon_{{}_{\\! {\\rm F}}} }{A^{1\/3} \\hbar},\n\\end{equation}\n which is a complex function of $\\omega $ by\nmeans of (\\ref{despeq}) for the sound velocity $s$ with\n(\\ref{despfunc})-(\\ref{chinumer}),\n(\\ref{chitemzero})-(\\ref{chitemtwo}).\nIn (\\ref{x}), $\\Omega$ is the characteristic frequency of the classical\nparticle rotation\nin a mean potential well of the radius $R_0$ with the\nenergy near $\\varepsilon_{{}_{\\! {\\rm F}}}$, as a convenient frequency unit.\nOther quantities are defined as\n\\bel{blx}\n\\mathcal{B}_L(x) = m \\rho_{{}_{\\! 0}} R_0^5 \\frac{j_{{}_{\\! L}}(x)}{x\nj^\\prime_L(x)},\n\\end{equation}\n \\bel{clx} \\mathcal{C}_L(x) = C_L^{(S)} +\n\\mathcal{C}_L^{(\\lambda)} (x),\n\\end{equation}\n \\bel{LDstiff} C_L^{(S)} =\n{\\alpha} R^2_0 (L-1) (L+2)= \\frac{b_{{}_{\\! S}} }{4\\pi} A^{2\/3} (L-1)\n(L+2),\n\\end{equation}\n \\bel{clmu} \\mathcal{C}_L^{(\\lambda)}(x) = 2 \\lambda R^3_0\n\\frac{x }{j^\\prime_L(x)} (j^{\\prime\\prime}_L (x) + j_{{}_{\\! L}}(x))\\;,\n \\end{equation}\nand\n \\bel{zlx} \\mathcal{Z}_L(x) = 2 \\nu R^3_0 \\frac{x }{ j^\\prime_L(x)}\n(j^{\\prime \\prime}_L (x)+ j_{{}_{\\! L}}(x)).\n\\end{equation}\n\nFrom (\\ref{eqmotionQ}), one has\n\\bel{Qqext}\n\\frac{Q_\\omega }{q^\\omega _{\\rm ext}} = - \\frac{1 }{k D_L(\\omega )}\n\\end{equation}\n with\n\\begin{eqnarray}\\label{glx}\n&&\\mathcal{D}_L(\\omega ) \\equiv - \\left[\\mathcal{B}_L(x) \\omega ^2 -\n\\mathcal{C}_L(x) + i \\omega \n\\mathcal{Z}_L(x)\\right] ~~~\\nonumber\\\\\n&=& \\frac{C_L^{(S)} }{j^\\prime_{{}_{\\! L}} (x)}\n\\left\\{ j^\\prime_L(x) - \\frac{6 A^{1\/3}\\lambda\\;\nx }{b_{{}_{\\! S}}(L-1) (L+2) \\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}}} \\left[\n\\left(i \\nu \\omega -\\lambda\\right) \\right.\\right.~~~\\nonumber\\\\\n&\\times& \\left.\\left. j^{\\prime \\prime}_L (x) +\n\\left(\\frac{s^2\n\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{\\mathcal{G}_1} - \\lambda + i \\nu \\omega \\right)\nj_{{}_{\\! L}}(x)\\right] \\right\\}.~~~\n\\end{eqnarray}\nIn (\\ref{Qqext}),\n$k$ is the coupling constant (\\ref{kfld}), see (\\ref{blx})-(\\ref{zlx}).\nThe kinetic\ncoefficients $\\lambda$ and $\\nu$ are the shear modulus $\\lambda$\nand viscosity $\\nu$ given by (\\ref{shearmod}) and\n(\\ref{viscos}), respectively. The two latter quantities enter\n(\\ref{glx}) in the following combination\n\\bel{lambvischixz}\n\\lambda -i \\nu \\omega = s \\chi _{xz} \\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}}\n\\end{equation}\nthrough a function $\\chi _{xz}$ defined by\n(\\ref{chixz}). Finally, the\nresponse function $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollQq}) with\n(\\ref{Qqext}) writes\n\\bel{chicollfldm}\n\\chi_{FF}^{\\rm coll}(\\omega ) = \\frac{1 }{k^2 \\mathcal{D}_L(\\omega )}.\n\\end{equation}\nThe poles of this collective response function\nare determined by the following equation,\nsee (\\ref{glx}) for $\\mathcal{D}_L(\\omega )$,\n\\bel{poleseq}\n- \\mathcal{B}_L(x) \\omega ^2 + \\mathcal{C}_L(x) -\ni \\omega \\mathcal{Z}_L(x) = 0,\n\\end{equation}\n with $\\,x\\,$ defined by (\\ref{x}).\nThe complex\nsolution of the dispersion equation (\\ref{despeq}) for $s$\nhas two branches of the solutions. They are related\nasymptotically to the Landau--Placzek heat $s^{(0)}$ and the sound\n$s^{(1)}$ solutions considered all in the hydrodynamic limit for\nthe infinite nuclear matter in Sec.\\ \\ref{longwavlim}, see\n(\\ref{shp})-(\\ref{gambarone}) for the corresponding\nfrequencies $\\omega ^{(0)}$ and $\\omega ^{(1)}$. For each branch denoted\nbelow by the same upper index\n$n=0,1$ as well in\nSec.\\ \\ref{longwavlim}, we have the roots of the secular equation\n(\\ref{poleseq}) written as\n\\bel{rootomega}\n \\omega ^{(n)} =\n\\omega _i^{(n)} - i \\Gamma_i^{(n)}\/2, \\> \\> \\> i = 0,1,...,\n\\end{equation}\n where\n$i$ numbers $\\omega _i^{(n)}$ in order of their increasing magnitude.\nWe shall consider these roots with $\\omega _i^{(n)}$ in the frequency\nregion of about $\\hbar \\omega \\siml \\hbar \\Omega$ which overlaps the\nlow frequency energy region discussed below. We shall consider\nenough large\ntemperatures $T \\simg \\sqrt{c_o} \\hbar \\Omega\\; \\simg \\sqrt{c_o}\n\\hbar \\omega $ but smaller than the Fermi energy. This approximately\nmeans $2 MeV \\siml T \\siml 10 MeV$ for $c_o=3\/4\\pi^2$ of\n\\cite{kolmagpl} ($A \\sim 200$). (Low temperature limit is about\n$1$ MeV for $c_o=1\/4\\pi^2$ of\n\\cite{landau,ayik}.)\nFor above mentioned frequencies $\\omega $ and temperatures $T$, for which\nthe quasiparticle and local-equilibrium conceptions of the theory\nfor the heated Fermi liquids can be applied, the only lowest\nsolutions have been found in the infinite sequence\n(\\ref{rootomega}). They are\nassociated with $i=0$, $1$ and $2$ for the \"first sound\"\nbranch $n=1$, and\nthat with $i=0$ for the \"Landau--Placzek\"\nbranch ($n=0$). (Quote marks show that the corresponding names\nare realized in fact only asymptotically in the hydrodynamical\nlimit.) The total response function is the sum of the two\nbranches mentioned above. The response function\n(\\ref{chicollfldm}) contains all important information concerning\nthe excitation modes of the Fermi-liquid drop. One of the ways of\nthe receipt of this information is to analyze the response\nfunction poles (\\ref{rootomega}) and their residua. However, this\nway is often not so convenient and too complicate in the case when\na few poles are close to each other or they belong (or are close)\nto the imaginary axis of the complex plane $\\omega $. More transparent\nway which is free from such disadvantages is to describe the\nresponse function in terms of the transport coefficients\n\\cite{hofmann}.\n\n\n\\subsection{Transport properties for a slow collective motion}\n\\label{transprop}\n\nThe macroscopic response of nucleus to an external field\nis a good tool for calculations of\nthe transport coefficients. To achieve this goal we follow\nthe lines of \\cite{hofmann,hofbook}. For instance, in cranking model type\napproximations, one assumes the collective motion to be\nsufficiently slow such that the transport coefficients can be\nevaluated simply in the \"zero-frequency\" limit. For a such slow\ncollective motion we shall study here the transport coefficients\nwithin the FLDM having a look at excitation energies smaller than the\ndistance between gross shells \\cite{strutmag},\n\\begin{equation}\\label{hom}\n\\hbar \\Omega = \\frac{\\hbar v_{{}_{\\! F}}}{R} \\approx\n\\frac{\\varepsilon_{{}_{\\! F}}}{A^{1\/3}} = 7-10\\, {\\rm MeV}\n\\end{equation}\n in heavy nuclei [$\\Omega$ is the particle rotation frequency\n(\\ref{x})] $\\hbar\\omega \n\\siml \\hbar\\Omega$, i.e., less than or of the order of the giant\nmultipole resonance\nenergies. Within the low collective motion ($\\omega \\siml \\Omega$),\nwe shall deal with first more simple\ncase of the hydrodynamic approximation which can be applied for\nfrequencies much smaller than the characteristic \"collisional\nfrequency\" $1\/\\tau$ related to the relaxation time $\\tau$\n(\\ref{relaxtime}), $\\omega \\tau \\ll 1$. Using this hydrodynamic\nexpansion of the macroscopic response function (\\ref{chicollfldm})\nin small parameter $\\omega \\tau$, we shall look for in\nSec.\\ \\ref{zerofreqlim} the relation to the \"zero frequency limit\"\ndiscussed in \\cite{hofmann}. Another problem of our interest in\nthis section is related to the correlation functions, \"heat pole\nfriction\" and ergodicity property, see \\cite{hofmann,hofbook}. We shall\nconsider in the next Sec.\\ \\ref{heatcorrfun} a smaller\nfrequency region where the nuclear heat pole like the Landau--Placzek mode\nfor the infinite matter appears within the hydrodynamic\napproximation. This subsection will be ended by a more general\ntreatment of the transport coefficients in terms of the parameters\nof the oscillator response function.\nThe method of \\cite{hofmann,hofbook} can be applied for the low frequency\nexcitations, $\\omega \\siml \\Omega$, but also in the case when the\nhydrodynamic approach fails, i.e. for $\\omega \\tau \\simg 1$.\n\n\n\nFollowing \\cite{hofmann,hofbook}, we shall study the \"intrinsic\"\nresponse function $\\chi_{{}_{\\! FF}}(\\omega )$ related to the collective\none $\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollrho}) by\nthe relation (\\ref{respintr}). The collective response function\n$\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollfldm}) in the FLDM was\nderived straightly from (\\ref{chicollrho}) in terms of the\nsolution for the transition density $\\delta\\rho$\n(\\ref{densvolsurf}). The \"intrinsic\" response function can be\nthen got with help of (\\ref{respintr}). This way is more\nconvenient in the FLDM with respect to the opposite one used\nusually in the microscopic quantum calculations based on the shell\nmodel \\cite{hofmann}.\n\n\nBy making use of expansion of the denominator $\\mathcal{D}_L(\\omega )$\n(\\ref{glx}) of the response function $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollfldm}) up to fourth order terms in small parameter\n$\\omega \\tau$ in the low frequency region $\\omega \/\\Omega \\ll 1$, and then,\nof (\\ref{respintr}), one gets the response function in the $F$ mode\nin the form:\n\\begin{eqnarray}\\label{respfunas}\n\\chi(\\omega )&=& k^{-2}\\left(-M\\omega ^2- i\n\\gamma \\omega + C_{\\rm in} - \\frac{i \\Upsilon C } {2 \\omega }\\right)^{-1},\\nonumber\\\\\nC_{\\rm in}&=&C - k^{-1},\n\\end{eqnarray}\n $M$, $C$ and $\\gamma$ can be\ndefined as the $Q$-~mode mass, the stiffness and the friction coefficients\nwhich are the values of $B_L(x)$ (\\ref{blx}), $C_L(x)$\n(\\ref{clx}) and $\\mathcal{Z}_L(x)$ (\\ref{zlx}) for $x=0$ ($\\omega =0$).\nHere and below we omit the low index \"FF\" in the\n\"FF\"-response functions everywhere when it will not lead to\nmisunderstanding. Note that the formulas which we derive here and\nbelow for the \"intrinsic\" response function $\\chi(\\omega )$ can be\napplied also to the collective response function\n$\\chi^{\\rm coll}(\\omega )$ if we only omit the index \"in\" in $C_{\\rm in}$\nand in functions of $C_{\\rm in}$ denoted by the same index (except\nfor some approximations based on the specific properties of\n$C_{\\rm in}$ compared to $C$ and noted below if necessary).\nAnother argument of the presentation of our results in terms of\nthe \"intrinsic\" response functions is to compare them more\nstraightly with the discussed ones in \\cite{hofmann} in connection\nto correlation functions and ergodicity. For the\ninertia $M$ and stiffness $C$, we obtain the parameters of the\nclassic hydrodynamic model, namely:\n\\bel{mass0}\nM=\\mathcal{B}_L(0) =\n\\frac{1 }{L} m \\rho_{{}_{\\! 0}} R^5_0 \\equiv M_{\\rm LD},\n\\end{equation}\n the inertia of\nirrotational flow, and\n\\bel{stiffness0}\n C=\\mathcal{C}_L(0) =\nC_L^{(S)} \\equiv C_{\\rm LD}\n\\end{equation}\n with $C_L^{(S)}$ being the\nstiffness coefficient of the surface energy (\\ref{LDstiff}). (We\nintroduced here more traditional notations labeled by index \"LD\"\nwhich means the relation to the usual liquid-drop model of\nirrotational flow). For friction $\\gamma$, we arrive at\nthe temperature dependence typical for hydrodynamics,\n\\begin{eqnarray}\\label{friction0}\n\\gamma &=&\\mathcal{Z}_L(0)=2 \\nu_{{}_{\\! {\\rm LD}}} R_0^3 (L-1) \\nonumber\\\\\n&=& \\frac{3 A (L-1)\\varepsilon_{{}_{\\! {\\rm F}}}\n}{5 \\pi}~ \\frac{\\nu_{{}_{\\! {\\rm LD}}}(T) }{\\nu_{{}_{\\! {\\rm LD}}}(0)}~ \\tau \\equiv\n\\gamma_{{}_{\\! {\\rm LD}}}.\n\\end{eqnarray}\n Here, $\\nu_{{}_{\\! {\\rm LD}}}$ is the classical hydrodynamic\nlimit $\\nu^{(1)}$ (\\ref{shearvisone}) for the viscosity\ncoefficient, $\\tau$ the relaxation time\n(\\ref{relaxtime}), (\\ref{widthG}) for $\\omega =0$,\n\\bel{wdthGT}\n\\tau\\equiv\\tau(0,T)=\\frac{\\hbar }{{\\it \\Gamma}(0,T)},\\quad\n{\\it \\Gamma}(0,T)= \\frac{\\pi^2 T^2 }{{\\it \\Gamma}_0 \\left(1+\n\\pi^2 T^2 \/ c^2\\right)}.\n\\end{equation}\n However, our result\n(\\ref{friction0}) for the\nclassical liquid-drop model of irrotational flow, if only extended\nto include the two-body viscosity, differs from the one found in\n\\cite{nixsierk,davinixsierk}\nby an additional factor of\n$(2L+1)\/L$, see \\cite{magkohofsh,ikmPRC}. We neglected the fourth order\nterms in ${\\bar T}$ (Sec.\\ \\ref{lowtemlim}) in\n(\\ref{mass0}), (\\ref{stiffness0})\nand (\\ref{friction0}) because of the presence of more important\nlower order terms there. For the coefficient $\\Upsilon$ in the term\nproportional to $1\/\\omega $ in (\\ref{respfunas}), one obtains\n\\bel{gamma0hp}\n\\Upsilon=\\frac{13A^{1\/3}\\varepsilon_{{}_{\\! {\\rm F}}} \\pi^4 {\\bar T}^4 s_0^2\n}{60 b_{{}_{\\! S}}(L-1)(L+2) \\tau} =\n\\frac{24 \\mathcal{G}_0 \\varepsilon_{{}_{\\! {\\rm F}}} A^{1\/3}\n}{b_{{}_{\\! S}}(L-1)(L+2)}~ \\nu^{(2)}.\n\\end{equation}\n The expression in the middle\nof these equations turns into zero for the Landau--Placzek kind\n($n=0$) of the solutions (\\ref{shp}) of dispersion equation\n(\\ref{despeq}) for the velocity $s$. It is, however, finite for the first\nsound mode $n=1$ presented in (\\ref{sfirst}) for\n$s_0=s_0^{(1)}$. The second equation being true only for the first\nsound mode was obtained by making use of (\\ref{sfirst}) for\nthe first sound velocity $s_0^{(1)}$ and (\\ref{shearvishp}) for\nthe viscosity component $\\nu^{(2)}$ up to small temperature\ncorrections of the next order. The both equations (\\ref{gamma0hp}) show\nthe main term in the temperature expansion of the coefficient\n$\\Upsilon$ in front of $1\/\\omega $ in (\\ref{respfunas}). Note that it\nappears in the order ${\\bar T}^4$ and can not be neglected for enough\nsmall frequencies $\\omega $. As seen from (\\ref{respfunas})\nconsidered for the case of the collective response, i.e., with\nomitted index \"in\" in $C_{\\rm in}$ of (\\ref{respfunas}), for\nenough small frequencies $\\omega $, there is the pole which equals\napproximately $i\\Upsilon\/2$. Therefore, the physical meaning of the parameter\n$\\Upsilon$ (\\ref{gamma0hp}) is a \"width\" of the overdamped pole in\nthe asymptotic collective response function $\\chi^{\\rm coll}(\\omega )$ for\nenough low frequencies. As shown below, this pole and\ncorresponding pole of the intrinsic response function\n(\\ref{respfunas}) is overdamped. It is similar to the Landau--Placzek\npole in the infinite nuclear matter and to the nuclear heat pole found in\n\\cite{hofmann}, see more detailed discussion in\nSec.\\ \\ref{heatcorrfun}. The \"width\" $\\Upsilon$ (\\ref{gamma0hp}) of such\n\"heat pole\" is inversely proportional to the relaxation time\n$\\tau$ and increases with temperature and particle number.\nNote also that this \"width\" is proportional to the component\n$\\nu^{(2)}$ (\\ref{shearvishp}) of the viscosity discussed in\nSec.\\ \\ref{viscosthermcond}. It is somewhat similar to the\nviscose part of the standard expression for the first sound\n\"width\" $\\Gamma$ (\\ref{gammaland}) in terms of the first component\n$\\nu^{(1)}$ of the viscosity coefficient (\\ref{shearvisonehp}),\n$\\Gamma \\propto \\nu^{(1)}$. However, there is in\n(\\ref{gamma0hp}) the surface energy constant $b_{{}_{\\! S}}$ and\nparticle number factor $A^{1\/3}$ which are both the specific\nparameters of a {\\it finite} Fermi-liquid drop.\n\nThus, the denominator of the hydrodynamical response function\n(\\ref{respfunas}) contains the two friction terms. One of them is\nproportional to the friction coefficient $\\gamma$, $\\gamma\n\\propto \\nu^{(1)}$, and another one which is proportional to\n$\\Upsilon$ ($\\Upsilon \\propto \\nu^{(2)}$). We shall consider in\nthe next two Secs. \\ref{zerofreqlim} and \\ref{heatcorrfun}\nthe two limit cases neglecting first the heat pole $\\Upsilon$\nterm for enough large frequencies $\\omega $ within the hydrodynamic\napproximation $\\omega \\tau \\ll 1$, and then, the $\\gamma$ friction one\nfor smaller frequencies with the dominating heat pole, respectively.\n\n\n\\subsubsection{HYDRODYNAMIC SOUND RESPONSE}\n\\label{zerofreqlim}\n\nFor enough large frequencies $\\omega $ within the frequent\ncollisional (hydrodynamic) regime,\n\\begin{eqnarray}\\label{hydpolcond}\n&&\\omega _{\\rm crit} \\tau \\ll \\omega \\tau \\ll 1 \\;, \\qquad \\omega _{\\rm crit}\\tau\n=\\tau~\\sqrt{C\\Upsilon \/2\\gamma} \\nonumber\\\\\n&=&\\frac{\\pi^2\\sqrt{13 \\mathcal{G}_0 \\mathcal{G}_1}\n~s_0~\\nu_{{}_{\\! {\\rm LD}}}(0) }{36(L-1)~\\nu_{{}_{\\! {\\rm LD}}}(T)}~{\\bar T}^2 ,\n\\end{eqnarray}\n one finds the\n first sound ($i=1$) solution $s$ (\\ref{sfirst0}).\nIn this case, one can neglect\nthe last term proportional to $1\/\\omega $ compared to the friction\nterm in the denominator of the asymptotic expression\n(\\ref{respfunas}). The critical frequency $\\omega _{\\rm crit}$ is defined\nin the second equation of (\\ref{hydpolcond}) as a frequency\nfor which these two compared terms coincide,\n$\\omega _{\\rm crit}=\\sqrt{C\\Upsilon\/2\\gamma}=\\omega _{{}_{\\! {\\rm LD}}}\\sqrt{M\\Upsilon\/2\\gamma}~$\n($\\omega _{{}_{\\! {\\rm LD}}}=\\sqrt{C\/M}$ is the frequency of the surface\nliquid-drop\nvibrations). The critical value $\\omega _{\\rm crit}\\tau$ increases with\nincreasing temperature as ${\\bar T}^2$ and does not depend on\nparticle number for the first sound mode $n=1$. It equals zero\nfor the Landau--Placzek branch $n=0$, according to\n(\\ref{gamma0hp}) for $\\Upsilon$. For the $n=1$ mode,\n$\\omega _{\\rm crit}\\tau$ is small for all temperatures $T ~\\siml~\n10~\\mbox{MeV}$, $\\omega _{\\rm crit}\\tau \\approx 0.6~ {\\bar T}^2 ~\\ll~ 1$ at\ntypical values of the parameters, $\\varepsilon_{{}_{\\! {\\rm F}}}=40~\\mbox{MeV}$ and\n$r_0=1.2~\\mbox{fm}$, and\nfor a value $\\mathcal{C}$ of the Skyrme forces considered in\n\\cite{strutmagden},\n$\\mathcal{C} =80~\\mbox{MeV} \\cdot \\mbox{fm}^5$,\nwhich is somewhat larger than those of \\cite{BMRV}\n(Sec.\\ \\ref{npcorivgdr} and Appendix D)\nin the ESA, where $A^{-1\/3}$ is assumed to be\nsmall. We took here and below $L=2$ for the quadrupole\nvibrations, $\\mathcal{F}_0=-0.2$, $\\mathcal{F}_1=-0.6$ for the Landau\nconstants which are close to the values common used for the\ncalculations of the nuclear giant multipole resonances\n\\cite{spethwoud,hasse}, a little more \"realistic\" than in\n\\cite{kolmagpl,magkohofsh}. For frequencies $\\omega $ within the\ncondition (\\ref{hydpolcond}), we arrive at the\noscillator-like response function,\n\\bel{oscresponse}\n\\chi(\\omega )\\equiv k^{-2}\\chi_{\\rm osc}(\\omega ) =k^{-2}\\left(-M \\omega ^2 -i\n\\gamma \\omega +C_{\\rm in}\\right)^{-1},\n\\end{equation}\n with all hydrodynamic\ntransport coefficients presented in (\\ref{mass0}),\n(\\ref{stiffness0}) and (\\ref{friction0}). In the middle of (\\ref{hydpolcond}),\n$\\chi_{\\rm osc}(\\omega )$ is\nthe \"intrinsic\" oscillator response function which describes the\ndynamics in terms of the $Q(t)$ variable for the collective\nharmonic oscillator potential. As seen now, the constants $M$, $C$\nand $\\gamma$ were naturally called above as the transport\ncoefficients: The collective response function $\\chi^{\\rm coll}(\\omega )$\nwithin the approximation (\\ref{hydpolcond}) is the same\n(\\ref{oscresponse}) but with omitted index \"in\" in the\nstiffness coefficient, as noted above. This remark is related also\nto the oscillator $QQ$- response function $\\chi_{\\rm osc}^{\\rm coll}(\\omega )$\nuseful for the following analysis of the response functions in\nterms of the transport coefficients,\n\\bel{oscrespcoll}\n\\chi_{\\rm osc}^{\\rm coll}(\\omega ) =\\left(-M \\omega ^2 -i \\gamma \\omega \n+C\\right)^{-1}.\n\\end{equation}\n We obtain the $QQ$- response functions from\nthe $FF$-ones, for instance, from $\\chi(\\omega )$ (\\ref{oscresponse}),\nsimply multiplying by the constant $k^2$ because of the\nself-consistency condition (\\ref{selfconsist}). Note also that\nthe condition (\\ref{hydpolcond}) for the Landau--Placzek branch of\nthe solutions for the sound velocity $s$ [see (\\ref{shp})], is\nalways fulfilled for $\\omega \\tau \\ll 1$.\n\n\nIn order to compare our results with those of previous\ncalculations \\cite{hofmann},\nwe introduce the dimensionless quantity\n\\begin{eqnarray}\\label{eta}\n\\eta &=& \\gamma \/ \\left(2 \\sqrt{M |C|}\\right) \\nonumber\\\\\n&=& \\frac{2 \\varepsilon_{{}_{\\! {\\rm F}}}}{5 p_{{}_{\\! {\\rm F}}} r_0 A^{1\/6}} ~\n\\sqrt{\\frac{6 L (L-1)\n\\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_1 }{(L+2) b_{{}_{\\! S}}}}\n~\\frac{~\\nu_{{}_{\\! {\\rm LD}}}(T)~\\tau\n}{~\\nu_{{}_{\\! {\\rm LD}}}(0)},\n\\end{eqnarray}\n see\n(\\ref{friction0}), (\\ref{mass0}) and (\\ref{stiffness0}) for\n$\\gamma$, $M$ and $C$, respectively. The quantity $\\eta$ in\n(\\ref{eta}) characterizes the effective damping rate of the\ncollective motion. Neglecting small temperature corrections of the\nviscosity coefficient $\\nu_{{}_{\\! {\\rm LD}}}=\\nu^{(1)}$ in (\\ref{eta}), see\n(\\ref{shearvisone}), and substituting\n(\\ref{relaxtime}), (\\ref{widthG}) for the relaxation time\n$\\tau$ at $\\omega =0$, one writes\n\\bel{eta0}\n\\eta \\approx \\frac{2 \\hbar\n\\varepsilon_{{}_{\\! {\\rm F}}} {\\it \\Gamma}_0 }{5 \\pi^2 p_{{}_{\\! {\\rm F}}} r_0 A^{1\/6}} ~\\sqrt{{6 L\n(L-1) \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_1 \\over {(L+2)b_{{}_{\\! S}}}}} ~\n\\frac{1 + \\pi^2 T^2 \/\nc^2}{T^2}.\n\\end{equation}\n This hydrodynamic effective friction $\\eta$\nmainly decreases with temperature as $1\/T^2$. For large\ntemperatures and finite cut-off parameter $c$,\nthe dimensionless friction parameter\n$~\\eta$ (\\ref{eta0}) approaches the\nconstant.\n\nWe have the two kind of poles of the response function\n(\\ref{oscresponse}) as roots of the quadratic polynomial in the\ndenominator, the overdamped poles, see\n\\cite{hofmann},\n\\begin{eqnarray}\\label{overdamp}\n\\omega _{\\pm}^{\\rm over}&=&-i \\Gamma_{\\pm}^{\\rm in}\/2,\\nonumber\\\\\n\\Gamma_{\\pm}^{\\rm in}&=&2\\varpi_{\\rm in}\\left(\\eta_{\\rm in}\\pm\n\\sqrt{\\eta_{\\rm in}^2-1}\\right),\\quad \\eta_{\\rm in}>1,\n\\end{eqnarray}\n and the\nunderdamped ones,\n\\bel{underdamp}\n\\omega _{\\pm}^{\\rm under}=\n\\varpi_{\\rm in}\\left(\\pm \\sqrt{1-\\eta_{\\rm in}^2} - i\n{\\eta_{\\rm in}}~\\right), \\quad \\eta_{\\rm in}<1\\;.\n\\end{equation}\n These solutions\ndepend on the two parameters,\n\\bel{varpietain}\n\\varpi_{\\rm in}=\\sqrt{\\frac{|C_{\\rm in}|}{M}}~ \\qquad \\mbox{and} \\qquad\n\\eta_{\\rm in}= \\frac{\\gamma }{2 \\sqrt{M|C_{\\rm in}|}} .\n\\end{equation}\n Note also that\nthe two hydrodynamic poles in (\\ref{oscresponse}) coincide\napproximately for the both branches $n=0$ and $1$ of solutions to\nthe dispersion equation (\\ref{despeq}) for the velocity $s$. The\ndifference between these two modes is related only to the last\nterm proportional to $\\Upsilon$ in the brackets of r.h.s. of\n(\\ref{respfunas}), and it was neglected under the condition\n(\\ref{hydpolcond}).\n\nFor the real and imaginary parts of the response function\n$\\chi(\\omega )$ (\\ref{oscresponse}) with the help of\n(\\ref{varpietain}) for the overdamped case (\\ref{overdamp}),\nfor instance, one gets for completeness\n[see (\\ref{overdamp})-(\\ref{varpietain})]\n\\begin{eqnarray}\\label{oscrespre}\n\\chi^{\\prime}(\\omega )&=& \\frac{1 }{4 M k^2\n\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}}\n\\left(\\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. {\\Gamma_{+}^{\\rm in} \\over\n{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}}\\right),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{oscrespim}\n\\chi^{\\prime\\prime}(\\omega )&=& \\frac{\\omega }{4 M\nk^2\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}} \\left(\\frac{1\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. \\frac{1\n}{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right).\n\\end{eqnarray}\n\n\nFor a more simple case of the collective response in the FLDM, we\nomit index \"in\" in formulas of this section [see the comment\nafter (\\ref{respfunas})]. From (\\ref{eta}) for $\\eta$ with\nthe parameters used above for the estimate of $\\omega _{\\rm crit} \\tau$\nof (\\ref{hydpolcond}), and the \"standard\"\n${\\it \\Gamma}_0=33.3$ MeV \\cite{hofmann}; one has an overdamped motion, $\\eta > 1$,\nfor all temperatures $T \\siml 10~ {\\rm MeV}$ and particle numbers\n$A \\siml 230$, as seen from Fig.\\ \\ref{fig1}. Moreover, for such\ntemperatures and particle numbers, one can expand the \"widths\"\n$\\Gamma_{\\pm}$ in small parameter $MC\/\\gamma^2=(4\\eta^2)^{-1}$,\nsee (\\ref{overdamp}) omitting index \"in\". From\n(\\ref{overdamp}) (without index \"in\") one gets approximately\n\\bel{overdamp1}\n\\Gamma_{\\pm}=4\\varpi \\eta \\left\\{{1-\\left(4\n\\eta^{2}\\right)^{-1} \\atop {\\left(4 \\eta^2\\right)^{-1}}}\\right\\}\n= 2\\left\\{{\\gamma\/M \\atop {C\/\\gamma}}\\right\\},\\,\\,\n\\eta^2~ \\gg~ 1.\n\\end{equation}\n Fig.\\ \\ref{fig1} shows that the above\nmentioned parameter $1\/(4 \\eta^2)$ for the expansion in\n(\\ref{overdamp1}) is really small for all considered\ntemperatures. Using (\\ref{mass0}),\n(\\ref{stiffness0}), (\\ref{friction0}) for the transport\ncoefficients and the definition of $\\tau$ (\\ref{wdthGT}) as in\nthe derivation of (\\ref{eta}), (\\ref{eta0}), one obtains from\n(\\ref{overdamp1})\n\\begin{eqnarray}\\label{Gammaplus}\n\\Gamma_{+}&=& \\frac{16 \\mathcal{G}_1\nL(L-1)\\varepsilon_{\\rm F}^2 }{5 \\left(p_{{}_{\\! {\\rm F}}} r_0\\right)^2~A^{2\/3}}~\n\\frac{~\\nu_{{}_{\\! {\\rm LD}}}(T) ~\\tau }{ ~\\nu_{{}_{\\! {\\rm LD}}}(0)} \\nonumber\\\\\n&\\approx& \\frac{16 \\hbar^2\n\\mathcal{G}_1 L(L-1)\\Gamma_0 \\varepsilon_{\\rm F}^2 }{5\n\\pi^2\\left(p_{{}_{\\! {\\rm F}}}r_0\\right)^2~A^{2\/3}}~ ~\n\\frac{1 + \\pi^2 T^2 \/ c^2\n}{T^2},\n\\end{eqnarray}\n \\begin{eqnarray}\\label{Gammaminus}\n\\Gamma_{-}&=&\\frac{5 b_{{}_{\\! S}} (L+2)\n}{6 \\varepsilon_{{}_{\\! {\\rm F}}}~A^{1\/3}}~ \\frac{~\\nu_{{}_{\\! {\\rm LD}}}(0)\n}{~\\nu_{{}_{\\! {\\rm LD}}}(T) ~\\tau} \\nonumber\\\\\n&\\approx& \\frac{5 \\pi^2 b_{{}_{\\! S}} (L+2) }{6 \\hbar {\\it \\Gamma}_0\n\\varepsilon_{{}_{\\!{\\rm F}}}~A^{1\/3}}~ ~\\frac{T^2 }{1 + \\pi^2 T^2\/c^2 }.\n\\end{eqnarray}\n One of\nthe \"widths\" specified by $\\Gamma_{+}$ (\\ref{Gammaplus}) is\nmainly the decreasing function of temperature, $\\Gamma_{+}\n\\propto \\tau\\propto 1\/T^2$ at low temperatures. It is typical for\nthe hydrodynamic modes as the first sound vibrations in normal\nliquids; in contrast to another \"width\" $\\Gamma_{-}$\n(\\ref{Gammaminus}), $\\Gamma_{-} \\propto 1\/\\tau \\propto T^2$,\nsimilar to the zero sound damping in relation to the $\\tau $-\ndependence. They both become about a constant for high\ntemperatures, due to the cut-off factor $c$.\n\nNote, $\\Gamma_{+}$ (\\ref{Gammaplus}) decreases with particle\nnumber as $A^{-2\/3}$ while $\\Gamma_{-} \\propto A^{-1\/3}$, see\n(\\ref{Gammaminus}). The different $A$-dependence of the \"widths\"\n$\\Gamma_{-}$ (\\ref{Gammaminus}) and $\\Gamma_{+}$\n(\\ref{Gammaplus}) can not be nevertheless\nreferred even formally to the\nso-called \"one- and two-body\ndissipation\", respectively. (Collisions with potential walls\nwithout the integral collision term in the Landau--Vlasov equation\nbut with the mirror or diffused boundary conditions might lead to\nthe \"widths\" proportional to $\\Omega$\nin (\\ref{x}), $\\Omega\n\\propto A^{-1\/3}$, as in equation (49) of \\cite{komagstrvv} or\nthrough the wall formula \\cite{wall,blocki,BMY}.)\nThey both depend on the collisional\nrelaxation time $\\tau$ and correspond to the \"two-body\"\ndissipation. The latter means here the collisional damping of the\nviscose Fermi liquid as in \\cite{kolmagpl,magkohofsh}. The\nphysical source of the damping in the both cases is the same\ncollisions of particles in the nuclear volume, due to the integral\ncollision term (\\ref{intcoll}) with the relaxation time $\\tau$.\nWe would like to emphasize, however, that the collisional\n$\\Gamma_{-}$ (\\ref{Gammaminus}) depends on the surface energy\nconstant $b_{{}_{\\! S}}$ and disappears proportionally to $A^{-1\/3}$\nwith increasing particle number $A$ like $\\Omega$\nof (\\ref{x}) because we took into account a {\\it finite} size of\nthe system through the boundary conditions\n(\\ref{bound1}), (\\ref{bound2}). An additional overdamped pole with\nthe \"width\" $\\Gamma_{-}$ (\\ref{Gammaminus}) appears because of the\n{\\it finiteness of the system and collisions inside the nucleus.}\nThis looks rather in contrast to the wall friction \\cite{blocki,BMY}\ncoming from\nthe collisions with the only walls of the potential well.\n\n\nWe shall come back now to the {\\it intrinsic}\nresponse function $\\chi(\\omega )$ (\\ref{oscresponse}).\nFor the \"intrinsic stiffness\" $C_{\\rm in}$, one has\n\\bel{Cintr}\nC_{\\rm in}=-\\left(1-kC\\right)\/k\\approx -1\/k.\n\\end{equation}\n In the last equation,\nwe neglected a small parameter $kC$,\n\\begin{eqnarray}\\label{smallpar}\nkC&=& \\frac{54\n(L-1)(L+2) \\mathcal{C} }{4 \\pi K r_0^5 A^{2\/3}} \\nonumber\\\\\n&\\approx&\n\\frac{9(L-1)(L+2)\n\\mathcal{C} }{4 \\pi \\mathcal{G}_0 \\varepsilon_{{}_{\\! {\\rm F}}} r_0^5 A^{2\/3}}\n \\approx \\frac{3 }{A^{2\/3}},\n\\end{eqnarray}\n for the typical values of the\nparameters mentioned above before (\\ref{oscresponse}). We\nneglected also small temperature corrections of\n(\\ref{incompradexp}) for the in-compressibility modulus\n$K$, $K=K^{\\varsigma}$, in the second equation of (\\ref{smallpar}).\n\n\nUsing a smallness of the parameter $kC$ (\\ref{smallpar}), we shall\nget now the relation of the coupling constant $k^{-1}$ with the\nisolated susceptibility $\\chi(0)$ and stiffness $C$ as in\nequation (3.1.26) of \\cite{hofmann}. For this purpose, we take the limit\n$\\omega \\rightarrow 0$ in (\\ref{oscresponse}) for the \"intrinsic\"\nresponse function $\\chi(\\omega )$ and expand then the obtained\nexpression for $\\chi(0)$,\n$\\chi(0)=k^{-2}C_{\\rm in}^{-1}=-k^{-2}(1-kC)^{-1}$, in powers of the\nsmall parameter $kC$ (\\ref{smallpar}) up to {\\it second} order\nterms. As result, we arrive at the relation\n\\bel{kstiffCchi0}\n-k^{-1}=\\chi(0) + C.\n\\end{equation}\n\nThe liquid-drop transport coefficients $M$ (\\ref{mass0}), $C$\n(\\ref{stiffness0}) and $\\gamma$ (\\ref{friction0}) can be now\ncompared with the ones in the \"zero frequency limit\" $M(0)$,\n$C(0)$ and $\\gamma(0)$, respectively, defined by\nequations (3.1.84)-(3.1.86) in \\cite{hofmann}:\n\\bel{C0}\nC(0)=-\\left(1\/k\n+ \\chi(0)\\right)=C,\n\\end{equation}\n\\bel{gamma0} \\gamma(0)=-i\\left(\\partial\n\\chi(\\omega )\/\\partial \\omega \\right)_{\\omega =0}= \\gamma,\n\\end{equation}\n \\begin{eqnarray}\\label{M0}\nM(0)&=&\\left(\\frac{1}{2}\\partial^2\n\\chi(\\omega )\/\\partial\\omega ^2\\right)_{\\omega =0}= M\\left(1+\\gamma^2\nk\/M\\right) \\nonumber\\\\\n&=&M\\left(1+4\\eta_{\\rm in}^2\\right).\n\\end{eqnarray}\n Expanding $\\chi(\\omega )$\nnear the zero frequency $\\omega =0$ in the secular equation\n(\\ref{seculareq}), see \\cite{hofmann}, we assumed here and will\nshow below that the \"intrinsic\" response function $\\chi(\\omega )$ is\na smooth function of $\\omega $ for small frequencies $\\omega $ within the\nhydrodynamic condition (\\ref{hydpolcond}). The second and third\nequations in (\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0}) were\ngot approximately in the ESA from (\\ref{oscresponse}) up to\nsmall corrections in the parameter $kC$ with help of\n(\\ref{kstiffCchi0}), second equation in\n(\\ref{varpietain}) and (\\ref{Cintr}). As the liquid drop\nstiffness $C$ equals approximately the stiffness in the \"zero\nfrequency limit\" $C(0)$, according to (\\ref{C0}), the\nequation (\\ref{kstiffCchi0}) is identical to the relation\n(\\ref{kstiffC0chi0}) of the general response-function theory\n\\cite{hofmann} within the same ESA. As seen from\n(\\ref{C0})-(\\ref{M0}), the stiffness $C(0)$ and friction\n$\\gamma(0)$ equal to the liquid drop parameters, but the inertia\n$M(0)$ differs from the liquid-drop mass value $M$ by a positive\ncorrection.\n\nFor the definition of transport coefficients in \"the zero\nfrequency limit\" (\\ref{C0})-(\\ref{M0}), we needed to know also\nthe properties of the \"intrinsic\" response function in the secular\nequation (\\ref{seculareq}),\nconcerning its pole\nstructure. For the \"intrinsic\" case the quantity\n$\\eta_{\\rm in}$, see (\\ref{varpietain}), plays a\nrole similar to the effective damping $\\eta$ (\\ref{eta}) for the\ncollective motion.\nMoreover, $\\eta_{\\rm in}$ determines the correction to the liquid\ndrop mass parameter $M$ in (\\ref{M0}) for the inertia $M(0)$\nin \"the zero frequency limit\". Due to a smallness of the parameter\n$kC$ (\\ref{smallpar}), $\\eta_{\\rm in}$ is much smaller than $\\eta$\n(\\ref{eta0}) for large particle numbers $A \\approx 200-230$, as\nseen from Fig.\\ \\ref{fig1},\n\\begin{eqnarray}\\label{etaintr}\n &&\\eta_{\\rm in} = \\frac{\\gamma\n}{ 2 \\sqrt{M |C_{\\rm in}|}} \\approx \\eta^2 kC ~~~ \\nonumber\\\\\n&\\approx& \\frac{3 (L-1) \\hbar\n{\\it \\Gamma}_0 \\varepsilon_{{}_{\\! {\\rm F}}} }{5 \\pi^2 p_{{}_{\\! {\\rm F}}} r_0}~ \\sqrt{\n\\frac{6 l \\mathcal{C}\n\\mathcal{G}_1 }{\\pi \\mathcal{G}_0 b_{{}_{\\! S}} r_0^5~A}} ~\\frac{1 + \\pi^2\nT^2 \/ c^2 }{T^2},~~~\n\\end{eqnarray}\n see (\\ref{friction0}), (\\ref{mass0}), (\\ref{stiffness0}),\n(\\ref{Cintr}), and (\\ref{smallpar}).\n\n\nFor such heavy nuclei ($A \\approx 200-230$) and enough large\ntemperatures, $T \\simg 5 ~\\mbox{MeV} $, one has formally the\n\"underdamped\" pole structure (\\ref{underdamp}) ($\\eta_{\\rm in} < 1$)\nfor the parameters selected above. Using the expansion of the\npoles $\\omega _{\\pm}^{\\rm in}$ (\\ref{underdamp}) of the intrinsic\nresponse function in powers of small $\\eta_{\\rm in}^2$\n(\\ref{etaintr}) up to terms of the order of $\\eta_{\\rm in}^4$, one\nwrites\n\\begin{eqnarray}\\label{underdamp1}\n&&\\omega _{\\rm in}^{\\pm} = \\omega _{\\rm in}\\left[\\pm\n\\left(1-\\frac{1}{2} \\eta_{\\rm in}^2\\right) -i \\eta_{\\rm in}\\right] \\nonumber\\\\\n&\\approx& \\pm \\varpi_{{}_{\\! {\\rm LD}}}\/\\sqrt{kC}-i\\Gamma_{+}\/4 \\quad{\\rm\nfor}\\quad \\eta_{\\rm in}^2 \\ll 1;\n\\end{eqnarray}\n see (\\ref{varpietain}), (\\ref{overdamp1}) (for $\\Gamma_{+}$ on the\nvery r.h.s.), and (\\ref{Cintr}) (for $kC$ there) in the\nderivation of the second equation. The \"underdamped\" poles\n$\\omega _{\\rm in}^{\\pm}$ approach the real axis on a large distance from\nthe imaginary one as compared to the liquid drop frequency\n$\\varpi_{{}_{\\! {\\rm LD}}}=\\sqrt{C\/M}$, $|\\omega _{\\rm in}^{\\pm}| \\gg\n\\omega _{{}_{\\! {\\rm LD}}}$. They\nhave a small \"width\" $2\\omega _{\\rm in}\\eta_{\\rm in}=\\gamma\/M\\propto 1\/T^2$\nfor our choice of large temperatures ($T \\simg 5 {\\rm MeV}$); see\n(\\ref{varpietain}), (\\ref{friction0}), (\\ref{mass0}), and\n(\\ref{wdthGT}). By this reason, for the \"underdamped\" case of\nsmall $\\eta_{\\rm in}^2$ and low frequencies\n$\\omega \\siml \\omega _{{}_{\\! {\\rm LD}}}$, the\nintrinsic response function $\\chi(\\omega )$ is a smooth function of\n$\\omega $.\n\nFor smaller temperatures $T ~\\siml~ 4~ \\mbox{MeV}$ and for our\nparameters used in (\\ref{etaintr}), one has the \"overdamped\" poles\n(\\ref{overdamp}) of the intrinsic response function $\\chi(\\omega )$,\n$\\eta_{\\rm in} > 1$. For such temperatures, $\\eta_{\\rm in}$\n(\\ref{etaintr}) is enough large. We can use therefore the\nexpansion of the \"widths\" $\\Gamma_{\\pm}^{\\rm in}$ of\n(\\ref{overdamp}) in a small parameter $(M\n\\omega _{\\rm in}\/\\gamma)^2=(4 \\eta_{\\rm in}^2)^{-1}$ (see Fig.\\ \\ref{fig1}),\n\\begin{eqnarray}\\label{overdampintr}\n&&\\Gamma_{\\pm}^{\\rm in}= 4\\omega _{\\rm in}\\eta_{\\rm in}\n\\left\\{{1-\\left(4 \\eta_{\\rm in}^{2}\\right)^{-1} \\atop {\\left(4\n\\eta_{\\rm in}^2\\right)^{-1}}}\\right\\} \\nonumber\\\\\n&\\approx& 2~\\left\\{{\\gamma\/M\n\\atop {1\/(k\\gamma)}}\\right\\} \\quad {\\rm for} \\quad\n\\left(4\\eta_{\\rm in}^2\\right)^{-1} \\ll 1,\n\\end{eqnarray}\nsee\n(\\ref{varpietain}), (\\ref{Cintr}) and (\\ref{smallpar}). The\n\"intrinsic width\" $\\Gamma_{+}^{\\rm in}$ in the upper row of\n(\\ref{overdampintr}) and the \"collective width\" $\\Gamma_{+}$\n(\\ref{Gammaplus}) [see (\\ref{overdamp1}] are the same.\n$\\Gamma_{-}^{\\rm in}$ in the low row has the temperature dependence\nas for $\\Gamma_{-}$ in (\\ref{Gammaminus}) but a different\n$A$-dependence, $\\Gamma_{-}^{\\rm in} \\propto A^{1\/3}$ [see\n(\\ref{kfld}) and (\\ref{friction0})]. Moreover,\n$\\Gamma_{-}^{\\rm in} \\gg \\Gamma_{-}$ because of smallness of the\nparameter $kC$ (\\ref{smallpar}). It becomes clear after dividing\nand multiplying the last expression for the $\\Gamma_{-}^{\\rm in}$ in\n(\\ref{overdampintr}) by the factor $C$ and using\n(\\ref{stiffness0}) and (\\ref{friction0}).\n\n\nThe \"intrinsic width\" $\\Gamma_{+}^{\\rm in}$, see\n(\\ref{overdampintr}), is mainly larger than $\\Gamma_{-}^{\\rm in}$.\nThey become comparable when increasing temperature, i.e.,\n$\\Gamma_{+}^{\\rm in}\\simg \\Gamma_{-}^{\\rm in}$. As $\\Gamma_{-}^{\\rm in}$\nfrom (\\ref{overdampintr}),\n\\bel{Gammaminintr}\n\\Gamma_{-}^{\\rm in} =\\frac{10 \\pi \\mathcal{G}_0 b_{{}_{\\! S}}\nr_0^5 ~A^{1\/3}}{27 (L-1) \\mathcal{C} ~\\tau},\n\\end{equation}\n is large compared to the\ncharacteristic collisional frequency $1\/\\tau$\n(for the same choice of the parameters) the both poles are far\naway from the zero, see more discussions of the \"intrinsic widths\"\nbelow in connection with the heat pole in the next section\n\\ref{heatcorrfun}. Therefore, the intrinsic response function\n$\\chi(\\omega )$ (\\ref{oscresponse}) is a smooth function of $\\omega $ for\nthe \"overdamped\" case of enough large $\\eta_{\\rm in}$ used in\nthe derivations of (\\ref{overdampintr}) as for the\n\"underdamped\" one discussed above. Thus, we expect that the \"zero\nfrequency limit\" based on the expansion of the intrinsic response\nfunction $\\chi(\\omega )$ is a good approximation for low frequencies\nlarger the critical value $\\omega _{\\rm crit}$ within the\nhydrodynamic condition (\\ref{hydpolcond}) for all considered\ntemperatures.\nIt means that the definition of\nthe transport coefficients in this limit\n(\\ref{C0})-(\\ref{M0}) is justified within the hydrodynamic\napproximation (\\ref{hydpolcond}).\n\nThe correction to the liquid drop mass parameter in the inertia\n$M(0)$ (\\ref{M0}) is always positive. This correction is the\ndecreasing function of the temperature and particle number which\ncan be presented approximately as\n\\begin{eqnarray}\\label{masscorr}\n&&(M(0)-M)\/M =\nk\\gamma^2\/M =4\\eta_{\\rm in}^2 \\nonumber\\\\\n&\\propto& ~\\left(1 + \\pi^2 T^2 \/\nc^2\\right)^2 \/ \\left(A~ T^4\\right),\n\\end{eqnarray}\n see (\\ref{etaintr}).\nFor smaller temperatures when the expansion in\n(\\ref{overdampintr}) is justified, this correction is equal\napproximately to the ratio of the \"intrinsic widths\"\n$\\Gamma_{+}^{\\rm in}\/\\Gamma_{-}^{\\rm in}$ taken from (\\ref{overdampintr}).\nThe relative mass correction\n(\\ref{masscorr}) and the \"intrinsic width\" ratio\n$\\Gamma_{+}^{\\rm in}\/\\Gamma_{-}^{\\rm in}$, see (\\ref{overdampintr}),\ndecreases with temperature $T$\nmainly as\n$1\/T^4$ if $T$ is not too big, as shown in Fig.\\ \\ref{fig1}.\nThe dimensionless inertia correction (\\ref{masscorr}) is proportional\napproximately to $1\/A$. The zero frequency mass\n$M(0)$ exceeds much the liquid drop inertia and turns asymptotically to\nthe latter for high temperatures, see Fig.\\ \\ref{fig1}.\nNote, for enough large temperatures $T \\simg 5 {\\rm MeV}$ and\nparticle numbers $A \\sim 200$ when $\\eta_{\\rm in}^4$ terms can be\nneglected in accordance with (\\ref{etaintr}), all zero\nfrequency transport coefficients $C(0)$, $\\gamma(0)$ and $M(0)$\n[see (\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0})] approach the\ncorresponding liquid drop parameters.\n\n\n\nIt would be interesting now to get the \"overdamped\" correlation\nfunction $\\psi^{\\prime\\prime}(\\omega )$ determined by the imaginary\npart of the corresponding response function\n$\\chi^{\\prime\\prime}(\\omega )$ (\\ref{oscrespim}) through the\nfluctuation-dissipation theorem, see (\\ref{fludiptheor}) and\n (\\ref{corrfun}). In the semiclassical approximation\n(\\ref{corrfun}) and (\\ref{oscrespim})\nfor the first sound mode, one writes\n\\begin{eqnarray}\\label{corroscim}\n &&\\frac{1}{T}\\psi^{s \\; \\prime\\prime}(\\omega )=\\frac{2}{\\omega }\\chi^{\\prime\\prime}(\\omega )\n= \\frac{2 }{4 M\nk^2\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}} \\nonumber\\\\\n&\\times&\\left(\\frac{1\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4}- \\frac{1\n}{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right).\n\\end{eqnarray}\n Using the approximations\nas in (\\ref{overdampintr}) and (\\ref{Cintr}), one gets from\n(\\ref{corroscim})\n\\begin{eqnarray}\\label{corroscim1}\n&&\\frac{1}{T}\\psi^{s\n\\;\\prime\\prime}(\\omega )= \\frac{1 }{k} \\left(\\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. \\frac{1 }{4 \\eta_{\\rm in}^2}\\;\n\\frac{\\Gamma_{+}^{\\rm in} }{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right)\n\\approx \\frac{1}{k}\\; \\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4}.\n\\end{eqnarray}\n The second Lorentzian in the\nmiddle is negligibly small compared to the first one because\n\\bel{gammapmincond}\n\\Gamma_{+}^{\\rm in}\\simg \\Gamma_{-}^{\\rm in} \\gg\n1\/\\tau \\gg \\omega ,\n\\end{equation}\n and $4\\eta_{\\rm in}^2$ is large in these\nderivations, see the discussions in between\n(\\ref{overdampintr}) and (\\ref{Gammaminintr}). It seems that\nwe are left with the Lorentzian term of this correlation function\non very right of (\\ref{corroscim1}) which looks as the\nLandau--Placzek heat-pole correlation function\n(\\ref{corrfunhphof}) and equation (4.3.30) of \\cite{hofmann} with\nobvious constants $\\psi^{(0)}$ and ${\\it \\Gamma}_T$. However, we can\nnot refer the found correlation function\n$\\psi^{s\\;\\prime\\prime}(\\omega )$ (\\ref{corroscim1}) to the heat pole\none. The \"width\" $\\Gamma_{-}^{\\rm in}$ of (\\ref{overdampintr}) in\n(\\ref{corroscim1}) is finite and large compared to the\ncharacteristic collision frequency $1\/\\tau$ which, in turn, is\nmuch larger considered frequencies $\\omega $, as shown above, see\n(\\ref{gammapmincond}). The limit $\\Gamma_{-}^{\\rm in} \\to 0$ for a\nfixed finite $\\omega$ and the corresponding\n$\\delta(\\omega )$-function which would show the relation to the heat\npole correlation function do not make sense within the\napproximation (\\ref{hydpolcond}) used in (\\ref{corroscim1}).\nIn particular, the response (\\ref{oscresponse}) and the correlation\n(\\ref{corroscim1}) functions were derived for\nenough large frequencies $\\omega \\gg \\omega _{\\rm crit}$ due to the\ncondition (\\ref{hydpolcond}). Note also that the inertia\nparameter $M$ (\\ref{M0}) is not zero, as it should be for the heat\npole.\n\n\n\\subsubsection{HYDRODYNAMIC CORRELATIONS AND HEAT POLE}\n\\label{heatcorrfun}\n\nFor lower frequencies $\\omega $, which are smaller the critical value\n$\\omega _{\\rm crit}$, we should take into account the last additional term\nin the denominator of (\\ref{respfunas}) for the response\nfunction. For such small frequencies, this friction term being\nproportional to $\\Upsilon$ (\\ref{gamma0hp}) becomes dominating as\ncompared to the liquid-drop one $\\gamma=\\gamma_{{}_{\\! {\\rm LD}}}$. Within this\napproximation, we shall derive the heat-pole response and\ncorrelation functions, and relate $\\Upsilon$\n(\\ref{gamma0hp}) of (\\ref{respfunas}) with the corresponding heat\npole friction. This subsection will be ended by\ndiscussions of the nuclear ergodicity.\n\n\nFor smaller frequencies,\n\\bel{heatpolcond}\n\\omega \\tau \\ll\n\\omega _{\\rm crit}\\tau \\ll 1,\n\\end{equation}\n (see the second equation in\n(\\ref{hydpolcond}) for the critical frequency $\\omega _{\\rm crit}$),\none can neglect the friction $i\\gamma \\omega $ term in the\ndenominator of the asymptotic response function (\\ref{respfunas})\nas compared to the last one, $\\gamma \\omega \\ll \\Upsilon C\/ 2\\omega $. The\nmass term there is even smaller than the friction one for\nfrequencies $\\omega \\siml \\omega _{\\rm crit}$ for the considered parameters\nand will be neglected too, $M \\omega ^2 \\ll \\gamma\\omega $. In this\napproximation, from (\\ref{respfunas}) one\nobtains the heat pole response function $\\chi(\\omega ) \\approx\n\\chi^{\\rm hp}(\\omega )$, which is similar to\n(\\ref{chicollhp}), (\\ref{chichpqom}) for the infinite nuclear\nmatter,\n\\bel{hprespfunas}\n\\chi ^{\\rm hp}(\\omega )= \\frac{\\omega }{k^2 C_{\\rm in}\n\\left(\\omega +i \\Gamma^{\\rm hp}\/2\\right)} \\approx - \\frac{\\omega }{k\n\\left(\\omega +i \\Gamma^{\\rm hp}\/2\\right)},\n\\end{equation}\n where\n\\bel{heatpole}\n\\Gamma^{\\rm hp}= -C \\Upsilon\/C_{\\rm in} \\approx kC\\Upsilon.\n\\end{equation}\n In these derivations, we used the\nspecific properties of the {\\it intrinsic} response functions\nwhich we now are interested in for analysis of the correlation\nfunctions and ergodicity conditions \\cite{hofmann}. In\n(\\ref{hprespfunas}) and in all approximate equations below in this subsection,\nwe\napplied also the expansion in small parameter $kC$\n(\\ref{smallpar}) as in (\\ref{Cintr}).\n\n\nThe real and imaginary parts of the response function\n$\\chi ^{\\rm hp}(\\omega )$ (\\ref{hprespfunas}) are, respectively,\n\\begin{eqnarray}\\label{rehprespas}\n\\chi ^{\\rm hp\\;\\prime}(\\omega )&=& \\frac{\\omega ^2 }{k^2 C_{\\rm in} \\left[\\omega ^2\n+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx& - \\frac{\\omega ^2 }{k \\left[\\omega ^2\n+(\\Gamma^{\\rm hp})^2\/4\\right]},\n\\end{eqnarray}\n\\begin{eqnarray}\\label{imhprespas}\n\\chi^{\\rm hp\\;\\prime\\prime}(\\omega )&=& - \\frac{\\omega \\Gamma^{\\rm hp} }{2 k^2 C_{\\rm in}\n\\left[\\omega ^2+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx& \\frac{\\omega \n\\Gamma^{\\rm hp} }{2 k \\left[\\omega ^2+(\\Gamma^{\\rm hp})^2\/4\\right]}\n\\end{eqnarray}\nup to small $kC$ corrections, see (\\ref{smallpar}).\n\nWe shall derive now the correlation function\n$\\psi^{{\\rm hp}\\,\\prime\\prime}(\\omega )$ applying the\nfluctuation-dissipation theorem (\\ref{corrfun}) to the \"intrinsic\" response\nfunction $\\chi ^{{\\rm hp}\\;\\prime\\prime}(\\omega )$ (\\ref{imhprespas})\nobtained in the asymptotic limit (\\ref{heatpolcond}). From\n(\\ref{corrfun}) and (\\ref{imhprespas}) one gets\n\\begin{eqnarray}\\label{corrfunashp}\n\\frac{1}{T}\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )&=& \\frac{2}{\\omega }\n\\chi ^{{\\rm hp} \\;\\prime\\prime}(\\omega ) = -\n\\frac{\\Gamma^{\\rm hp}}{k^2 C_{\\rm in}\\;\n\\left[\\omega ^2+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx&\n\\frac{\\Gamma^{\\rm hp}}{k \\left[\\omega ^2+(\\Gamma^{\\rm hp})^2\/4\\right]}.\n\\end{eqnarray}\n This correlation\nfunction looks as the Landau--Placzek peak for the infinite Fermi\nliquid, see (\\ref{corrfunhphof}),\n\\bel{psifunhp}\n\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )= \\psi_{\\rm hp}^{(0)} \\frac{\\hbar\n{\\it \\Gamma}_T^{\\rm hp} }{(\\hbar \\omega )^2 + ({\\it\n\\Gamma}_T^{\\rm hp})^2\/4}.\n\\end{equation}\n It is identical to the r.h.s. of equation\n(4.3.30) in \\cite{hofmann}, but with the specific parameters\n$\\psi^{(0)}=\\psi_{\\rm hp}^{(0)}$ and\n${\\it \\Gamma}_T={\\it \\Gamma}_T^{\\rm hp}$,\n\\bel{psi0GammaT}\n\\frac{1}{T} \\psi_{\\rm hp}^{(0)}=-\\frac{1}{k^2 C_{\\rm in}}~\n\\approx \\frac{1}{k}, \\quad {\\it \\Gamma}_T^{\\rm hp} =\\hbar\n\\Gamma^{\\rm hp}\\approx \\hbar kC\\Upsilon.\n\\end{equation}\n The \"width\" $\\Gamma^{\\rm hp}$ in\n(\\ref{psi0GammaT}) is much smaller than the characteristic\ncollision frequency $1\/\\tau$,\n\\bel{Gammahptau}\n\\Gamma^{\\rm hp} = \\frac{13\n\\pi^4 \\mathcal{G}_1 \\mathcal{C}~ T^4 }{20\\varepsilon_{\\rm F}^4b_{{}_{\\! S}}r_0^5 {\\it\n\\Gamma}_0~A^{1\/3}~\\tau} \\ll \\frac{1}{\\tau},\n\\end{equation}\nsee (\\ref{gamma0hp})\nand (\\ref{smallpar}). The relationship (\\ref{Gammahptau}) for\n$\\Gamma^{\\rm hp}$ is in contrast to the one (\\ref{gammapmincond}) for\nthe \"intrinsic overdamped widths\" $\\Gamma_{\\pm}^{\\rm in}$\n(\\ref{overdampintr}) which are much larger the collision\nfrequency $1\/\\tau$ for the same selected parameters at all\ntemperatures $T \\siml 10$ MeV and particle numbers\n$A=200-230$.\n\nFor the following discussion of the friction coefficients,\nwe compare now the\n\"width\" $\\Gamma^{\\rm hp}$ (\\ref{heatpole}), (\\ref{Gammahptau}) with\n$\\Gamma_{-}^{\\rm in}$ in (\\ref{overdampintr})\n[see (\\ref{friction0}), (\\ref{stiffness0}), (\\ref{gamma0hp}),\n(\\ref{sfirst}), (\\ref{Cintr}) and (\\ref{kfld})],\n\\bel{zetaintr}\n\\frac{\\Gamma^{\\rm hp}}{\\Gamma_{-}^{\\rm in}} =\n\\frac{\\gamma C \\Upsilon }{2C_{\\rm in}^2} \\approx\n\\frac{1}{2}\\gamma C k^2 \\Upsilon.\n\\end{equation}\n For all temperatures and particle numbers\nwhich we discuss here, this ratio is small,\n\\bel{zetapar}\n\\frac{\\Gamma^{\\rm hp}}{\\Gamma_{-}^{\\rm in}}\n = \\frac{351 \\pi^2 (L-1) \\mathcal{C}^2 }{800\nb_{{}_{\\! S}}^2 r_0^{10}~A^{2\/3}} ~{\\bar T}^4 \\approx\n\\frac{15{\\bar T}^4 }{A^{2\/3}}.\n\\end{equation}\nIn the second\nequation of (\\ref{zetapar}) we used the same values of the\nparameters as in (\\ref{smallpar}). Note that the \"width\" of the\nLandau--Placzek peak $\\Gamma^{(0)}$, $\\Gamma^{(0)} \\sim\n\\tau_q^2\/\\tau \\ll 1\/\\tau$ for $\\tau_q \\ll 1$, is similar to\n$\\Gamma^{\\rm hp}$ and is unlike $\\Gamma_{\\pm}^{\\rm in}$\n(\\ref{overdampintr}) in (\\ref{corroscim1}) for the\nhydrodynamical sound correlation function. In contrast to the\nhydrodynamical sound case, see (\\ref{corroscim1}), we can consider\n(\\ref{corrfunashp}) for the correlation function approximation\nin the zero width limit $\\Gamma^{\\rm hp} \\to 0$ (or in the zero\ntemperature limit $T \\to 0$) taking any small but finite frequency\n$\\omega $ under the condition (\\ref{heatpolcond}). Therefore, for such\nfrequencies $\\omega $, the correlation function (\\ref{corrfunashp}) can\nbe approximated by $\\delta(\\omega )$-like function as in\n(\\ref{corrdeltafunlim}) for the correlation function of the\ninfinite Fermi liquid (\\ref{corrfunhphof}). Because of a very\nclose analogy of equation for the correlation function\n$\\psi^{{\\rm hp}\\,\\prime\\prime}(\\omega )$ (\\ref{psifunhp}) to the\nLandau--Placzek peak for the infinite Fermi-liquids in the\nhydrodynamic limit, see (\\ref{corrfunhphof}), and to\nequation (4.3.30) of \\cite{hofmann}, we associate the pole\n(\\ref{heatpole}) and corresponding asymptotics of the response\n (\\ref{hprespfunas})\nand correlation (\\ref{corrfunashp}) functions with the \"heat pole\". As in the case of the\ninfinite nuclear matter, this pole for the finite Fermi-liquid\ndrop is situated at zero frequency $\\omega =0$. Moreover, they are\nboth called as the \"heat poles\" because they disappear in the\nzero temperature limit $T \\to 0$ in line of the discussions near\nequation (4.3.30) of \\cite{hofmann} and after. In the case of the\ninfinite matter, we can see this property from\n(\\ref{corrfunhp}) because\n${\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\mathcal{P}}\n\\to 1 $) [or due to (\\ref{psi0}) for $\\psi^{(0)}$ in\n(\\ref{corrfunhphof})]. For the finite Fermi-liquid drop, the\nreason is that $\\Upsilon \\to 0$ in the zero temperature limit $T \\to 0\n$, see (\\ref{gamma0hp}), and the only hydrodynamical sound\ncondition (\\ref{hydpolcond}) is then satisfied with the response\nfunction (\\ref{oscresponse}) and correlation function\n(\\ref{corroscim1}) where the heat pole is absent, see the\ndiscussion after (\\ref{corroscim1}).\n\n\nTo get more explicit expressions for $\\psi^{(0)}$ and ${\\it\n\\Gamma}_T$ of (\\ref{psi0GammaT}) we use now (\\ref{kfld}),\n(\\ref{gamma0hp}), (\\ref{friction0}) and (\\ref{stiffness0}) for the\ncoupling constant $k^{-1}$, parameter $\\Upsilon$, friction $\\gamma$\nand stiffness $C(0)$, respectively. With these expressions, one\nobtains approximately from (\\ref{psi0GammaT})\n\\bel{psi0GammaT1}\n\\frac{1}{T} \\psi_{\\rm hp}^{(0)} = \\frac{\\mathcal{G}_0\\varepsilon_{{}_{\\! {\\rm F}}} b_{{}_{\\! S}}\nr_0^5 A^{4\/3} }{9 \\mathcal{C}} \\approx\n2 A^{4\/3},\n\\end{equation}\n \\begin{equation}\\label{GammaT2m}\n{\\it \\Gamma}_{T}^{\\rm hp} = \\hbar \\Gamma^{\\rm hp}\n=\\frac{13 \\pi^6 \\mathcal{G}_1\n\\mathcal{C} }{20 \\varepsilon_{\\rm F}^4 b_{{}_{\\! S}} r_0^5 {\\it \\Gamma}_0 ~\nA^{1\/3}}~ \\frac{T^6 }{\\left(1+\\pi^2T^2\/c^2\\right)}.\n\\end{equation}\n In the derivation of\n(\\ref{GammaT2m}), we used (\\ref{sfirst}) for the first\nsound solution $s_0=s_0^{(1)}$ ($n=1$) in (\\ref{gamma0hp})\nfor $\\Upsilon$ and (\\ref{wdthGT}) for the relaxation time $\\tau$.\nFor simplicity, we neglected small\ntemperature corrections in the viscosity coefficient $\\nu^{(1)}$\n(\\ref{shearvisone}) and in the first sound velocity $s_0^{(1)}$\n(\\ref{sfirst}). Other approximations are the same as well in the\nderivation of (\\ref{smallpar}) for $kC$ used in\n(\\ref{GammaT2m}) through (\\ref{heatpole}). The\ntemperature dependences of the \"intrinsic overdamped width\"\n$\\Gamma_{-}^{\\rm in}$ (\\ref{Gammaminintr}) and \"heat pole one\"\n$\\Gamma^{\\rm hp}$ (\\ref{GammaT2m}), (\\ref{Gammahptau}) are different,\nnamely $\\Gamma^{\\rm hp} \\propto {\\bar T}^4\/\\tau(0,T)$ and $\\Gamma_{-}^{\\rm in}\n\\propto 1\/\\tau(0,T)$ where the temperature dependence of the\nrelaxation time $\\tau(0,T)$ can be found in (\\ref{wdthGT}). The\nboth \"widths\" are the growing function of temperature as in\n\\cite{hofmann} but with a different power. The dependence on\nparticle number $A$ completely differs for these compared poles\nbeing the growing function of $A$ for the \"width\"\n$\\Gamma_{-}^{\\rm in}$, $\\Gamma_{-}^{\\rm in} \\propto A^{1\/3}$, and\ndecreasing function of $A$ for the $\\Gamma^{\\rm hp}$, $\\Gamma^{\\rm hp}\n\\propto A^{-1\/3}$. As noted above, like for the Landau--Placzek\npeak [see (\\ref{corrfunhphof}), (\\ref{GammaT}) and (\\ref{psi0})],\nthe heat pole with the \"width\" $\\Gamma^{\\rm hp}$ (\\ref{GammaT2m})\nexists only in heated systems with a temperature $T \\neq 0$.\nHowever, in contrast to the result (\\ref{GammaT}),\n(\\ref{gambarhpt}) for the ${\\it \\Gamma}_T$ of the heat pole in the\ninfinite Fermi liquid, the heat pole \"width\" ${\\it \\Gamma}_T$\n(\\ref{GammaT2m}) disappears with increasing particle number $A$,\ni.e., ${\\it \\Gamma}_T \\rightarrow 0$ for $A \\rightarrow \\infty$. It\nallows us to emphasize also that this kind of the heat pole\nappears only in a {\\it finite} Fermi system.\n\n\nThe correlation function $\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$\n(\\ref{psifunhp}) was obtained approximately\nnear the pole $-i\\Gamma^{\\rm hp}\/2$, see (\\ref{heatpole}). The\ncorresponding $QQ$- correlation function\n$\\psi_{QQ}^{{\\rm hp}\\;\\prime\\prime}(\\omega )=k^2\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$\nis identical to the oscillator correlation function\n$\\psi_{\\rm osc}^{{\\rm hp}\\;\\prime\\prime}(\\omega )$ defined through the imaginary part of\n$\\chi_{\\rm osc}(\\omega )$ from the second equation of\n(\\ref{oscresponse}) at the zero mass parameter $M$, $M=0$, see\n\\cite{hofmann},\n\\begin{eqnarray}\\label{oscresphp}\n\\frac{1}{T}\\psi_{\\rm osc}^{{\\rm hp}\\,\\prime\\prime}(\\omega )&=&\n\\frac{2}{\\omega }\\chi_{\\rm osc}^{{\\rm hp}\\;\\prime\\prime}(\\omega )=\n\\frac{2}{|C_{\\rm in}|}~ \\frac{\n|C_{\\rm in}|\/\\gamma ^{\\rm hp} }{\\omega ^2 + \\left(C_{\\rm in}\/\\gamma\n^{\\rm hp}\\right)^2} \\nonumber\\\\\n&\\approx& 2 k ~ \\frac{ 1\/\\left(k\\gamma ^{\\rm hp}\\right)\n}{\\omega ^2+ 1\/\\left(k\\gamma ^{\\rm hp}\\right)^2},\n\\end{eqnarray}\n see again\n(\\ref{Cintr}) for the last approximation. The\nresponse (\\ref{hprespfunas}) and\ncorrelation (\\ref{corrfunashp}) functions are\nidentical to the corresponding oscillator ones (\\ref{oscresphp})\nwith a friction coefficient $\\gamma ^{\\rm hp}$,\n\\bel{frictionhp}\n\\gamma^{\\rm hp} =2|C_{\\rm in}|\/\\Gamma^{\\rm hp} \\approx 2k^{-1}\n\\left(\\Gamma^{\\rm hp}\\right)^{-1} \\approx 2\/(k^2C\\Upsilon).\n\\end{equation}\n Here,\nthe same equation (\\ref{Cintr}) was used, $\\Gamma^{\\rm hp}$ is given by\n(\\ref{heatpole}), (\\ref{GammaT2m}), $k^{-1}$ is the coupling\nconstant (\\ref{kfld}). [For $C$ and $\\Upsilon$ in\n(\\ref{frictionhp}), one has (\\ref{stiffness0}) and\n(\\ref{gamma0hp}), respectively.] According to\n(\\ref{corroscim1}) and (\\ref{corrfunashp}) for the correlation\nfunctions $\\psi ^{s\\;\\prime\\prime}(\\omega )$ and\n$\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$, with the help of (\\ref{heatpole}),\n(\\ref{frictionhp}) and (\\ref{gamma0}), one gets\n\\begin{eqnarray}\\label{corrfriction}\n(1\/2T)\\psi^{s\\;\\prime\\prime}(0)&=&\\gamma=\\gamma(0),\\nonumber\\\\\n(1\/2T)\\psi^{{\\rm hp}\\,\\prime\\prime}(0)&=& 2\/(k^2C\\Upsilon)=\\gamma^{\\rm hp},\n\\end{eqnarray}\n in\nline of the last right equation in (3.1.85) of \\cite{hofmann}.\n\nFor the friction $\\gamma ^{\\rm hp}$ (\\ref{frictionhp}) related to the\n\"heat pole width\" $\\Gamma^{\\rm hp}$ (\\ref{heatpole}), one\napproximately writes\n\\begin{equation}\\label{frictionhpm}\n\\gamma ^{\\rm hp} =\\frac{40 \\mathcal{G}_0 {\\it \\Gamma}_0\n\\varepsilon_{\\rm F}^5b_{{}_{\\! S}}^2 r_0^{10} A}{117 \\pi^6\n\\mathcal{G}_1 \\mathcal{C}^2 T^6 }\\;\\left(1+\\frac{\\pi^2T^2}{c^2}\\right),\n\\end{equation}\n see\n(\\ref{GammaT2m}) for $\\Gamma^{\\rm hp}$ and (\\ref{kfld}) for\n$k^{-1}$ in (\\ref{frictionhp}). We neglected here small\ntemperature corrections in the adiabatic in-compressibility\nmodulus $K=K^{\\varsigma}$ (\\ref{incompradexp}). The heat pole\nfriction $\\gamma ^{\\rm hp}$ (\\ref{frictionhpm}) is proportional to\n$1\/T^6$ for smaller temperatures and $1\/T^4$ for larger ones (due\nto the cut-off parameter $c$). This decreasing temperature\ndependence is much more sharp compared to the liquid drop one\n$\\gamma$ (\\ref{friction0}), (\\ref{wdthGT}); $\\gamma \\propto\n1\/T^2$ for smaller temperatures, and $\\gamma$ is a constant for\nlarge ones.\nNotice, according to (\\ref{frictionhp}), the \"width\" ratio\n$\\Gamma^{\\rm hp}\/\\Gamma_{-}^{\\rm in}$ (\\ref{zetaintr}),(\\ref{zetapar})\nhas a clear physical meaning as the ratio of the hydrodynamic\nfriction coefficient $\\gamma$ (\\ref{friction0}) to the heat pole\none $\\gamma^{\\rm hp}$ (\\ref{frictionhpm}),\n\\bel{zintr1}\n\\Gamma^{\\rm hp}\/\\Gamma_{-}^{\\rm in} \\approx \\gamma\/\\gamma ^{\\rm hp} \\approx\n\\gamma(0)\/\\gamma ^{\\rm hp}.\n\\end{equation}\n A smallness of this ratio shown above\nclaims that the heat pole friction $\\gamma ^{\\rm hp}$ is much larger\nthan the typical hydrodynamic one $\\gamma$, see more discussions\nconcerning this comparison of different friction coefficients\nbelow.\n\n\nAs seen from the inequalities (\\ref{heatpolcond}) with the\ndefinition of $\\omega _{\\rm crit}$ from (\\ref{hydpolcond}), the heat\npole appears only in the \"sound\" branch $n=1$ and does not exist\nfor the Landau--Placzek branch of the solutions of\n(\\ref{despeq}) for $s$. We realize it immediately noting that\nthe width parameter $\\Upsilon$ (\\ref{gamma0hp}) is proportional to\n$s_0$ which is finite for $n=1$ and zero for $n=0$ case, see\n(\\ref{shp}) and (\\ref{sfirst}), respectively.\n\n\nAs shown in \\cite{hofmann}, for enough small ${\\it \\Gamma}_T$, the\ncoefficient $\\psi^{(0)}$ in front of the Lorentzian-like correlation\nfunction, see (\\ref{corrfunhphof}) and (\\ref{psifunhp}),\nis related to the difference of\nsusceptibilities,\n\\bel{psichi}\n(1\/T) \\psi^{(0)}\n= \\chi^T-\\chi(0)=\\chi^T-\\chi^{\\rm ad} +\\chi^{\\rm ad}-\\chi(0).\n\\end{equation}\nNeglecting a small difference $\\chi^T-\\chi^{\\rm ad}$ according to\n(\\ref{chiTchiad1}), (\\ref{chiTchiad}), see Appendix C\nand \\cite{hofmann} for details, one notes that the ergodicity\ncondition (\\ref{ergodicity1}) means smallness of the $(1\/T)\n\\psi^{(0)}$ compared to the stiffness $C$.\n\nHowever, from (\\ref{psi0GammaT}), (\\ref{psi0GammaT1}) one gets\na large quantity $(1\/T) \\psi_{\\rm hp}^{(0)}\/C \\approx 1\/(kC) \\gg 1$.\nNote, in the derivations of\n(\\ref{corrfunashp}), (\\ref{psi0GammaT}) we took first $\\omega \n\\rightarrow 0$ (small $\\omega \\tau$) for the finite ${\\it\n\\Gamma}_T^{\\rm hp}$, see also (\\ref{gamma0hp}) for $\\Upsilon$ in the\nsecond equation of (\\ref{psi0GammaT}), and then, considered\n${\\it \\Gamma}_T^{\\rm hp} \\rightarrow 0$ (small temperature limit $T\n\\to 0$ ).\nWe emphasize that the limits $\\omega \\rightarrow 0$ and ${\\it\n\\Gamma}_T^{\\rm hp} \\rightarrow 0$ are not commutative, i. e., the\nresult of the correlation function calculations depends on the\norder of executing of these two operations like for the infinite\nFermi-liquid matter \\cite{forster}. This\nis obvious if we take into account that the \"heat pole\" last term\nin the denominator of (\\ref{respfunas}) appears in the next\n(${\\bar T}^4$) order in ${\\bar T}$ and is proportional to $1\/(\\omega \n\\tau)$ in contrast to the other classical (sound) hydrodynamic\nterms, i.e., this $\\Upsilon$-term turns into zero for ${\\it \\Gamma}_T\n\\to 0 $ ($T \\to 0$).\n\nThe relation (\\ref{psichi}) was derived in \\cite{hofmann} using\nthe opposite sequence of the above mentioned limits, namely, first\n${\\it \\Gamma}_T \\rightarrow 0$ and then $\\omega \\rightarrow 0$ in\nline of the recommendations of Forster \\cite{forster} [first ${\\it\n\\Gamma}_T \\propto q^2 \\rightarrow 0$ [or $\\tau_q \\to 0$, see\n(\\ref{GammaT}), (\\ref{shp})], and then, $\\omega \\rightarrow 0$ ($s\n\\rightarrow 0$) for the infinite Fermi-liquid]. In this case\nthere is no contradiction with the ergodicity for the finite\nFermi-liquid drop. In the limit ${\\it \\Gamma}_T^{\\rm hp} \\rightarrow\n0$ ($T \\to 0$) for {\\it a finite} value of $\\omega $, the condition\n(\\ref{hydpolcond}) is fulfilled instead of (\\ref{heatpolcond}),\nand the \"heat pole\" term proportional to $1\/\\omega $ in the\ndenominator of the response function (\\ref{respfunas}) disappears\nwithin the ESA used in the FLDM, as noted above. It means\nformally that one can neglect $\\psi_{\\rm hp}^{(0)} $ in (\\ref{psifunhp}),\nand we have small quantities on the both sides of\n(\\ref{psichi}) taking into account the ergodicity condition\n(\\ref{ergodicity1}) derived in Appendix C.\nIt is not\nobvious that the relation (\\ref{psichi}) can be also derived for\nthe opposite consequence of the above mentioned limit transitions\nunlike the Forster recommendations, i.e., taking first limit $\\omega \n\\rightarrow 0$ for a finite ${\\it \\Gamma}_T$, and then, considering\nthe limit ${\\it \\Gamma}_T \\rightarrow 0$. In particular,\n(\\ref{corroscim1}) for the overdamped correlation function was\nobtained for the last choice of the limit sequences.\nEquation (\\ref{corroscim1}) does have also the Lorentzian-like shape\nbut it is not related to the \"heat pole\" because the coefficient\nin front of the Lorentzian is {\\it not} equal to\n$\\chi^T-\\chi(0)$.\nThis equation was derived only for\nlarge $\\Gamma_{-}^{\\rm in}$ compared to the $1\/\\tau$, see\n(\\ref{gammapmincond}),\nand is true {\\it only} under these conditions and within\ninequalities (\\ref{hydpolcond}). There is no a $\\delta(\\omega )$\nfunction-like peak in (\\ref{corroscim1}) for all possible\nvariations of the parameters for which this equation was derived.\nThe overdamped shape of the correlation function like\n(\\ref{corrfunhphof}) does not mean yet that this function is\nthe \"heat pole\" one though the opposite statement is true. We\npoint out again that\n $(1\/T)\\psi^{(0)}$ (\\ref{corroscim1}) is really\nlarge compared to the stiffness $C$, $(1\/T)\\psi^{(0)}=1\/k$, and the\nergodicity condition (\\ref{ergodicity1}) is fulfilled rather than\nthe relation (\\ref{psichi}) between $(1\/T)\\psi^{(0)}$ and\n$\\chi^T-\\chi(0)$ within the hydrodynamic conditions\n(\\ref{hydpolcond}).\n\n\n\nFollowing\nthe Forster's recommendations \\cite{forster},\ni.e., take first the limit of small ${\\it \\Gamma}_T$ (${\\it\n\\Gamma}_T \\rightarrow 0$) or small temperature ($T \\rightarrow 0$),\none gets the typical hydrodynamic response function\n(\\ref{oscresponse}) without \"heat pole\" terms. The next limit $\\omega \n\\rightarrow 0$ ($\\omega \\tau \\rightarrow 0$) in (\\ref{oscresponse})\nleads to the finite value,\n\\bel{isorespk}\n\\chi(0)=\\frac{1\n}{k^2C_{\\rm in}} \\approx - \\frac{1}{k} -C,\n\\end{equation}\n up to the\nrelatively small corrections of higher order in parameter $kC$\n(\\ref{smallpar}). This is in line of Appendix C,\nand the\nergodicity condition (\\ref{ergodicity1}) is fulfilled for the\nfinite Fermi-liquid drop within the ESA. Note that we accounted\nabove for the $kC$ correction at the second order\nin (\\ref{isorespk}). In this way, we got the relation\n(\\ref{kstiffCchi0}) between the coupling constant $k^{-1}$,\nisolated susceptibility $\\chi(0)$ and stiffness $C$ provided that\nthe condition (\\ref{hydpolcond}) is true, see also\n(\\ref{kstiffC0chi0}) with the stiffness $C(0)=C$ of the \"zero\nfrequency limit\". Note also that the \"heat pole\" response\nfunction $\\chi^{\\rm hp}(\\omega )$ (\\ref{hprespfunas}) has a sharp peak\nnear the zero frequency, and hence, is not smooth, i.e., \"the zero\nfrequency limit\" for the transport coefficients can not be\napplied in the case (\\ref{heatpolcond}).\n\nThus, all properties of the finite Fermi liquids within the ESA\nconcerning the ergodicity relation (\\ref{ergodicity1}), as applied\nto (\\ref{psichi}), are quite similar to the ones for the\ninfinite nuclear matter [besides the expressions (\\ref{Gammaminus}),\n$\\Gamma_{-} \\propto b_{{}_{\\! S}}\/A^{1\/3}$, and\n (\\ref{Gammahptau}), $\\Gamma^{\\rm hp} \\propto 1\/( b_{{}_{\\! S}} A^{1\/3})$,\nthemselves depending on $b_{{}_{\\! S}}$]. Our study of these properties is helpful\nfor understanding the microscopic shell-model approach\n\\cite{hofmann,hofivmag,hofbook}. We point out\nthat the strength function corresponding to the asymptotics\n(\\ref{respfunas}) is the curve with the two maxima which are\nrelated to the \"heat pole\" and standard (sound) hydrodynamic\nmodes. However, for intermediate frequencies $\\omega $ of the order\nof $\\omega _{\\rm crit}$ in the low frequency region, see\n(\\ref{hydpolcond}) and (\\ref{heatpolcond}), the asymptotic\nresponse function (\\ref{respfunas}) can not be presented exactly\nin terms of a sum of the two oscillator response functions like\n(\\ref{oscrespcoll}). For instance, in this case we have the\ntransition from the \"heat pole\" mode to the sound hydrodynamic\npeak, and the response function (\\ref{respfunas}) is more\ncomplex. We have a similar problem when the hydrodynamic\ncondition $\\omega \\tau \\ll 1$ becomes not valid. However, as shown in\nthe next subsection, such problems can be overcome approximately\nusing an alternative definition for the transport coefficients\nsuggested in \\cite{hofmann}.\n\n\nFor larger frequencies, i.e., for $\\omega \\tau$ larger or of the order\nof 1, but within the low frequencies $\\omega $ smaller than $\\Omega$,\nsee (\\ref{x}), the equation for the collective motion becomes more\ncomplicate. It is not reduced generally speaking to the second\norder differential equation with the constant coefficients as in\nthe zero frequency limit of the hydrodynamic approach\n(\\ref{hydpolcond}). As shown and applied in\n\\cite{hofivyam,hofmann} (see also \\cite{magkohofsh} in\nconnection to the FLDM), the problem of the definition of transport\ncoefficients can be nevertheless overcome by defining them\nthrough a procedure of fitting an oscillator response function\n(\\ref{oscrespcoll}) to selected peaks of the collective\nresponse function $\\chi_{QQ}^{\\rm coll}(\\omega )$ of\n(\\ref{chicollfldm}) with respect to the parameters $M$, $C$ and\n$\\gamma$. Here such a fitting procedure would also be adequate for\ntemperatures mentioned above, especially because our response\nfunction (\\ref{chicollfldm}) has several poles (\\ref{rootomega}),\nfor instance, with $i = 0,1,2; n=1$ and $i=0;n=0$. Some of them\nare the overdamped poles close to the imaginary axis in the\n$\\omega $-complex plane. This procedure can be done analytically in\nthe zero frequency limit provided that the response function\n(\\ref{chicollfldm}) can be approximated\nby the oscillator\nresponse functions as in (\\ref{oscrespcoll}) or by\n$\\chi_{\\rm osc}^{\\rm hp}(\\omega )$ in (\\ref{oscresphp}). In this case, we\nhave analytical fitting of the collective response function\n(\\ref{chicollfldm}) by these oscillator response functions and\nget the expressions for the transport coefficients\n(\\ref{C0}) -\n(\\ref{M0}) in the zero frequency limit\n(\\ref{hydpolcond}) [or (\\ref{frictionhp}) for the heat pole\nfriction in a smaller frequency region (\\ref{heatpolcond})]. For\nlarger frequencies, we need to carry out the fitting procedure\nnumerically.\n\nWe should also comment a little more the definition of the\ntransport coefficients in the zero frequency limit in connection\nto the one through the fitting procedure to avoid some possible\nmisunderstanding. The transport coefficients in the zero frequency\nlimit can be related to the \"intrinsic\" response function and its\nderivatives taken at $\\omega \\rightarrow 0$ \\cite{hofmann,hofbook};\nsee (\\ref{C0}),\n (\\ref{gamma0}), and (\\ref{M0}). For\napplication of this method of the transport coefficient\ncalculations, we should be carefully in the case when we have\nseveral peaks in the strength function but we need to get the\ntransport coefficient, for instance, for the second or more high\npeaks. In these cases the zero frequency limit might be applied\nalso, but we have first to remove all lower peaks in the collective\nresponse function and take then the corresponding \"intrinsic\"\nresponse function and its derivatives without these lower peaks.\nIn practical applications, this limit for the transport\ncoefficients obtained in a such way is close to the same limit for\nthe oscillator response function which fits the selected peak. The\nlatter could be also the second or more high one.\n\n\nWe shall consider now the hydrodynamical approximation\n$\\omega \\tau \\ll 1$ for the response function, see\n(\\ref{respfunas}), for the two cases: The sound response\nfunction (\\ref{oscresponse}) for the sound condition\n(\\ref{hydpolcond}) and the heat-pole response function\n(\\ref{hprespfunas}) for the heat pole condition\n(\\ref{heatpolcond}). The corresponding correlation functions are\nthe sound correlation function (\\ref{corroscim1}) and the\nheat-pole correlation one (\\ref{corrfunashp}). These two different\napproximations are realized for different consequence of the\nlimit transitions, i.e., the approximate result depends on the\nconsequence of their applying. The heat pole case\n(\\ref{hydpolcond}) is realized when we take first the limit $\\omega \n\\to 0$ for a finite width ${\\it \\Gamma}_T$, and then, ${\\it\n\\Gamma}_T \\to 0$ (or zero temperature limit $T \\to 0$). This leads\napproximately to the $\\delta(\\omega )$-like function for the\ncorrelation function. In contrast to this, the sound pole case\n(\\ref{heatpolcond}) is realized when we take first ${\\it\n\\Gamma}_T \\to 0$ (or $T \\to 0$) to remove the last heat pole term\nproportional to $\\Upsilon$ in the hydrodynamical response\n(\\ref{respfunas}), and then, $\\omega \\to 0$. We like to follow this\nlast consequence of the limit transition in line of the Forster\nrecommendations \\cite{forster} when we have the response\n(\\ref{oscresponse}) and correlation (\\ref{corroscim1}) functions\nwithout heat pole. In this case the transport coefficients for\n$\\omega \\tau \\ll 1$ are the standard hydrodynamical ones\n(\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0}) related to the parameters\nof the standard hydrodynamical model (\\ref{stiffness0}),\n(\\ref{friction0}) and (\\ref{mass0}), respectively. Exception\nshould be done for the\nmodified mass parameter in (\\ref{M0}) which turns into the\nirrotational flow inertia (\\ref{mass0}) for high temperatures.\n\n\n\\subsection{Discussion of the results}\n\\label{discuss}\n\nIn this subsection, we discuss the\nresults of the FLDM calculations for the collective response\nfunction and transport coefficients. We shall explain now in more details\nthe application of the general fitting procedure for the definition\nof the transport coefficients. We discuss also the\nstiffness and inertia parameters found within the FLDM.\nThis subsection will be ended by\nthe discussion of the friction versus temperature. One\nof the important points of this discussion is the \"heat pole\"\nfriction and comparison with the quantum shell-model calculations\n\\cite{hofivyam,hofmann,hofbook}.\n\n\n\nWe show first the imaginary part of the response function\n$\\chi_{QQ}^{\\rm coll}(\\omega )$ (\\ref{chicollfldm}) (its strength) for\ndifferent temperatures in Fig.\\ \\ref{fig2}. The total collective\nresponse function $\\chi_{QQ}^{\\rm coll}$ is presented in\nFig.\\ \\ref{fig2} as a sum of the two branches $n=0$ and $1$ of\neigen-frequencies $\\omega ^{(n)}$, see (\\ref{rootomega}), in the\nimaginary part (strength) of the response function\n(\\ref{chicollfldm}). They are related to the two different\nsolutions of the dispersion equation (\\ref{despeq}) for the sound\nvelocity $s^{(n)}$. These solutions are similar to the\nLandau--Placzek (Raleigh) and the sound (Brillouin) ones in normal\nliquids. The latter are approached exactly by $s^{(0)}$ and\n$s^{(1)}$ solutions for sound velocity $s$ in the\nhydrodynamic limit $\\omega \\tau \\rightarrow 0$, which are related to\nthe eigen-frequencies of the infinite-matter vibrations\n$\\omega ^{(0)}$ (\\ref{shp}) and $\\omega ^{(1)}$ (\\ref{sfirst}),\nrespectively. The integral collision term is parametrized in\nterms of the relaxation time $\\tau(\\omega ,T)$ (\\ref{relaxtime}),\n(\\ref{widthG}) with $c=20~\\mbox{MeV}$. We took the nucleus Pu-230\nwith particle numbers $A=230$ as an example of enough heavy\nnucleus.\n\nFor the intermediate temperatures $4~\\mbox{MeV} ~\\siml~ T~ \\siml~\n6~\\mbox{MeV}$ we have the three peak structure. More detailed\nplots for smaller frequencies are shown in Fig.\\ \\ref{fig3} for the\ntemperature $T=6~\\mbox{MeV}$ for which the first two peaks (\"heat\npole\" and usual hydrodynamic ones) are seen better in a normal\nscale. In Fig.\\ \\ref{fig3}, we show also the separate contributions\nof the two branches $n=0$ (dotted line) and $n=1$\n(dashed one) for the eigen-frequencies $\\omega ^{(n)}$\n(\\ref{rootomega}) calculated from the secular equation\n(\\ref{poleseq}) at each $s^{(n)}$ ($n=0,1$) as in\nFig.\\ \\ref{fig2}. We present also the imaginary part of the\nasymptotic response function (\\ref{respfunas}) obtained\nanalytically above in the hydrodynamic frequent-collision limit.\nAs seen from Fig.\\ \\ref{fig3}, we found from (\\ref{chicollfldm})\nthe $n=1$ mode with the two ($i=0, 1$) peaks and the $n=0$ mode\nwith one peak ($i=0$) for small frequencies $\\omega $ and small\nparameter $\\omega \\tau$ in agreement with asymptotics\n(\\ref{respfunas}). The heat pole contribution is shown separately\nby the dotted curve.\nNote that the two curves for $i=0 $ and $1$ at $n=1$\nin Fig.\\ \\ref{fig3} coincide because they both\nwere calculated without the last $\\Upsilon$ term in\n(\\ref{respfunas}). For the dotted curve,\none has $\\Upsilon \\propto\ns_0^{(0)}=0$, and for the dashed one,\nthe last $\\Upsilon$ term in\n(\\ref{respfunas}) is omitted under the asymptotical sound\ncondition (\\ref{hydpolcond}). Therefore, the upper asymptotical\ndata (thin solid) marked also by the condition (\\ref{hydpolcond})\nare in factor about two larger than the dotted,\nor dashed, or asymptotical (\\ref{respfunas}) ones.\n\n\nThe third peak in Fig.\\ \\ref{fig2} appears for intermediate\ntemperatures and larger frequencies. This peak is coming from the\nthird pole $i=2$ which belongs to the branch $n=1$ in\n(\\ref{rootomega}). This is the essentially Fermi-liquid\nunderdamped mode due to the Fermi-surface distortions related to\nthe shear modulus $\\lambda$ given by (\\ref{shearmod}). Such a peak is\nmoving from a large zero-sound-frequency region of the giant\nresonances to smaller frequencies with increasing temperature.\nThe second ($i=1$) peak in the $n=1$ branch and first ($i=0$)\npeak in the $n=0$ one in the low frequency region ($\\omega \\tau \\ll\n1$) are related to the overdamped motion described approximately\nby the overdamped oscillator response function like\n(\\ref{oscrespcoll}) for the same cut-off parameter $c=20~\\mbox{MeV}$.\nFor $c=\\infty$ the overdamped motion turns into the underdamped\none for large temperatures $T \\simg 7~\\mbox{MeV}$. The next\n(third) peak in a more high frequency region ($\\omega \\tau ~\\simg~ 1$)\ncorresponds to the underdamped mode for the both $c$ values. The\nfirst lowest peak in Figs.\\ \\ref{fig2} and \\ref{fig3}, which is not seen\nin Fig.\\ \\ref{fig2} being too close to the ordinate axis and\nstudied separately in Fig.\\ \\ref{fig3}, is due to the overdamped\n\"heat pole\" $i\\Upsilon\/2$ in the collective response function, see\n(\\ref{gamma0hp}) for $\\Upsilon$. The most remarkable property of\nthis \"heat pole\" peak for smaller temperatures is that it has\nmainly a very narrow width (\\ref{gamma0hp}) which increases with\nthe temperature as $T^6$, see the comments concerning the heat\npole \"width\" after (\\ref{gamma0hp}) and (\\ref{GammaT2m}).\nThis is in\ncontrast to the temperature behavior of the width $\\Gamma_{-}$\n(\\ref{Gammaminus}) like $T^2$ for the hydrodynamic sound peak at\nlarge temperatures.\nFig.\\ \\ref{fig2} shows the three peaks only for the intermediate\ntemperatures $4~ \\siml~ T ~\\siml~ 6~\\mbox{MeV}$ because for\nsmaller temperatures the third peak moves to the high frequency\nregion larger $\\Omega$ corresponding to the giant resonances and\nfirst peak is very close to the ordinate axis.\n\n\nThe transport coefficients for such two- or three resonance\nstructure were calculated by a fitting procedure of the oscillator\nresponse functions to the selected peaks. We subtract first the\n\"heat pole\" peak known analytically, see (\\ref{hprespfunas}), from\nthe total response function (\\ref{chicollfldm}). We are left then\nwith the two-humped curve and fit then it by the sum of the two\noscillator response functions as (\\ref{oscrespcoll}). One of\nthem which fits the first (hydrodynamic) peak in the curve with\nthe remaining two maxima is the overdamped oscillator response\nfunction ($\\eta ~>~ 1$) and other one (more high in the low energy\nregion) corresponds to the underdamped motion ($\\eta ~<~ 1$). In\nthis way, we get the two consequences of the transport coefficients\npresented in Figs.\\ \\ref{fig4}-\\ref{fig7}. In these Figures,\nthe heavy squares\nare related to the second, hydrodynamic-sound peak of\nFigs.\\ \\ref{fig2},\\ref{fig3} for the mostly overdamped modes with\nthe effective friction $\\eta~\n>~1$. The open squares show the third Fermi-liquid peak (see\nFig.\\ \\ref{fig2}) related to the underdamped motion ($\\eta~ < ~1$)\nand Fermi-surface distortions, very specific for the\nFermi liquids, in contrast to the normal liquids.\n\n\nFor the temperatures smaller about $6~\\mbox{MeV}$ the second peak\n$i=1$ in the total response function is {\\it overdamped} and is\ncoming from the two poles $(i=1,n=1)$ and $(i=0,n=0)$ which are\nclose to the standard hydrodynamic approach. The third peak, due\nto the Fermi-surface distortions as noted above, can not be found\nin principle in the hydrodynamic limit. The main difference\nbetween the second and third peaks can be found in the comparison\nof the stiffness coefficient $C$ with the liquid-drop value\n$C_{\\rm LD}$ obtained both from the fitting procedure mentioned above.\nFor the third (\"Fermi liquid\" in sense of the relation to the\nFermi surface distortions specific for the Fermi liquids, in\ncontrast to normal ones) peak the stiffness $C$ is much high\nthan the liquid drop value $C_{\\rm LD}$ in contrast to the second\n(typical hydrodynamical) one for which the stiffness $C$ is very\nclose to $C_{\\rm LD}$ almost for all temperatures, see\nFig.\\ \\ref{fig4}. It means that the third peak is essentially of\ndifferent nature than the second one because exists only due to\nthe Fermi-surface distortions. A measure of these distortions is\nthe anisotropy (or shear modulus) coefficient $\\lambda$, see\n(\\ref{shearmod}), which disappears in the hydrodynamic limit.\n\nFor enough large temperature (larger than or of the order of $7~\n\\mbox{MeV}$~) all three peaks are not distinguished in\nFig.\\ \\ref{fig2}. For such large temperatures the fitting procedure\nis a little modified to select these three peaks which are close\nto each other. For the finite $c=20~\\mbox{MeV}$ and all large\ntemperatures presented in Fig.\\ \\ref{fig2} nearly $7-10~\\mbox{MeV}$,\nwe have one wide peak which can be analyzed as the superposition\nof the three peaks, namely the \"heat-pole\", usual overdamped\nhydrodynamic and underdamped \"Fermi-liquid\" ones. Subtracting the\nfirst \"heat pole\" peak [see (\\ref{hprespfunas})] as for lower\ntemperatures, we fit then the remaining curve by the only one\noverdamped oscillator function like (\\ref{oscrespcoll}) for $\\eta\n> 1$. We subtract then again this overdamped fitted oscillator function\nfrom the response function (\\ref{chicollfldm}) without the heat\npole one (\\ref{hprespfunas}) and fit the rest by the single\nunderdamped oscillator. The found parameters of the two last\noscillator response functions are used as initial values for the\niteration fitting procedure of the sum of the two oscillator\nresponse functions of the same types to the response function\n(\\ref{chicollfldm}) (without the heat pole). The found transport\ncoefficients are presented in Figs.\\ \\ref{fig4}-\\ref{fig7}. For\nenough large temperature nearly $10~\\mbox{MeV}$ in the case\n$c=\\infty$ the only one underdamped oscillator can be used for\nfitting procedure of one peak [after an exclusion of the heat\npole from (\\ref{chicollfldm})].\n\nWe show also the mass parameters found from the above described\nfitting procedure for several selected peaks in Fig.\\ \\ref{fig5}.\nFor the third \"Fermi-liquid\" peaks the mass parameter $M$ is\nclose to the liquid drop values $M_{\\rm LD}$ related to the\nirrotational flow. The mass parameter of the second\n\"hydrodynamic\" peak, due to the mixture of the identical\n($i=1,n=1$) and ($i=0, n=0$) poles, is significantly smaller than\nthe liquid drop value $M_{\\rm LD}$ but finite. For the first \"heat\npole\" ($i=0,n=1$) peak the mass parameter can be approximated\nonly by zero. As noted above,\nthe stiffness parameter for the\nthird peak is much larger than the one for other (hydrodynamic)\npoles which is mainly close to the liquid drop value (see\nFig.\\ \\ref{fig4}).\nAs shown in Figures \\ref{fig4} and \\ref{fig5},\nfor enough large temperatures the temperature\ndependences of the stiffness ($C$) and mass ($M$) parameters are\nclose to their zero frequency limit, see (\\ref{C0}) for $C(0)$\nand (\\ref{M0}) for $M(0)$. For\nsmaller temperatures, the inertia $M(0)$ [Fig.\\ \\ref{fig5}]\nbecomes essentially larger than that found from the response function\n(\\ref{chicollfldm}). It is in contrast to the stiffness $C(0)$\nwhich is identical to the liquid-drop quantity in the\nsemiclassical limit $\\hbar \\rightarrow 0$ when $C(0)$ does not contain\nquantum shell corrections.\n\n\nFigs. \\ref{fig6} and \\ref{fig7} show the results for the friction\ncoefficient $\\gamma\/\\hbar$ versus the temperature for the\ncollective response function $\\chi_{QQ}^{\\rm coll}(\\omega ) =k^2(T)\n\\chi_{FF}^{\\rm coll}(\\omega )$ related to the $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollfldm}). We used here the same parameters as well\nin Figs.\\ \\ref{fig2} and \\ref{fig3} for the response function.\nThe solid line for the friction\n$\\gamma$ (\\ref{friction0}) corresponds to the response function\n(\\ref{oscrespcoll}) in the hydrodynamic limit (\\ref{hydpolcond}),\nthe same as for the zero frequency approach (\\ref{gamma0}). The\nheavy squares show the result of the fit of (\\ref{chicollfldm})\nto the oscillator response function (\\ref{oscrespcoll}).\nWe presented also the \"heat\npole\" contribution to the friction obtained from the fitting\nprocedure by one \"heat pole\" (overdamped) oscillator response\nfunction (\\ref{hprespfunas}), see circles in Fig. \\ref{fig7}. We\nmight compare the results of this fit to the friction\nanalytically found in terms of the heat pole asymptotics\n(\\ref{frictionhp}) valid for smaller temperatures and shown by\nsolid thin lines in Figs.\\ \\ref{fig6} and \\ref{fig7}. They are in a\ngood agreement for smaller temperatures where the overdamped\n\"heat pole\" with the \"width\" $\\Upsilon$ (\\ref{gamma0hp}) is more\nimportant. This \"heat pole\" friction is too big as compared to\nother friction components related to the hydrodynamical-sound\n(full squares) and \"Fermi-liquid\" poles in the usual scale of\nFig.\\ \\ref{fig6}. Therefore, we use the logarithmic scale in\nFig.\\ \\ref{fig7}.\n\n\nOur FLDM friction, except for the \"heat pole\" one, is similar to\nthe corresponding result of SM calculations\n\\cite{hofivyam,hofmann}, see Fig.\\ \\ref{fig8}. A large SM\nfriction coming from the diagonal matrix elements in\nFig.\\ \\ref{fig8} and standard hydrodynamic friction\n(\\ref{friction0}) as well as heavy squares shown in\nFigs.\\ \\ref{fig6} and \\ref{fig7} are obviously similar. All these curves\nfor temperatures $T \\simg 2~\\mbox{MeV}$ show the mainly\ndiminishing friction, $\\gamma \\propto \\tau \\propto 1\/T^2$ roughly\nlike in hydrodynamics, see (\\ref{friction0}). Some deflection of\nthe friction temperature dependence in Fig.\\ \\ref{fig6} for large\ntemperatures $T$ from usual hydrodynamic one $1\/T^2$, i.e., a\nconstant asymptotics is related to a different temperature\nbehavior of the ${\\it \\Gamma}(0,T)$ (\\ref{widthG}) for a finite\nand infinite cut-off parameter $c$: This ${\\it \\Gamma}(0,T)$ goes\nto a constant for large temperatures if $c$ is finite and to zero\nfor $c=\\infty$, see the solid and dashed lines in Fig.\\ \\ref{fig6}.\n\n\nIt is noted also a similarity concerning the third\n(\"Fermi-liquid\") peak presented by the lower open squares with\nmainly increasing friction in Figs.\\ \\ref{fig6}, \\ref{fig7} and by\njoint full squares in Fig.\\ \\ref{fig8}. For $c=20~\\mbox{MeV}$ and\ntemperatures smaller about $10~\\mbox{MeV}$ the friction of this\nmode increases, see Figs.\\ \\ref{fig6}, \\ref{fig7}, in contrast to\nthe standard hydrodynamic behavior (for $c=\\infty$ this friction\nincreases first up to about $6-7~\\mbox{MeV}$, and then, decreases\nat larger temperatures). In Fig.\\ \\ref{fig8} the lower curve with\ngrowing dependence on the temperature for $c= 20~\\mbox{MeV}$ was\nobtained by excluding the contribution of the diagonal terms in\nthe response function within the quantum approach based on the\nSM, see \\cite{hofmann,hofbook} for the detailed explanations.\nWithin the conceptions of the FLDM and classical hydrodynamics of\nthe normal liquid drops the first \"heat pole\" friction obtained\nfor enough small frequencies (\\ref{heatpolcond}) within the\nhydrodynamic collision regime $\\omega \\tau \\ll 1$ at finite\ntemperature is the physical mode which can be excited when this\nregime might be realized like the Landau--Placzek pole for normal\nliquids. However, the hydrodynamic collision regime being still\nwithin a low frequency region (enough small collision frequency\n$1\/\\tau$) is expected to be not achieved in fission experiments.\nTherefore, the friction is related mainly to another Fermi-liquid mode\ncorresponding to the only third peak owing to the Fermi-surface\ndistortions. The friction of this mode is much smaller than the\nhydrodynamic one for small temperatures, and they become\ncomparable for high ones. The Fermi-surface distortion friction\ncan be characterized by completely other, mainly growing\ntemperature behaviour, see the lower curve marked by open squares\nin Figs.\\ \\ref{fig6} and \\ref{fig7}. Concerning the SM calculations, it\nseems that we should omit the diagonal matrix elements, see\n\\cite{hofmann}, because of similar arguments: The hydrodynamic\ncollision regime seems to be not realized for nuclear fission\nprocesses. (These diagonal matrix elements might correspond to the\nphysical hydrodynamic mode if it is excited, say in another\nsystems like a normal liquid drop). The quantum shell-model\nfriction without contributions of diagonal matrix elements is\nrelated probably to another non-hydrodynamic mode, such as the third peak\nfor a Fermi-liquid drop, and this might be the physical reason\nfor an exclusion of these matrix elements.\n\nNote that in the SM response-function derivations the diagonal\nmatrix elements mentioned above do not contribute in the Forster's\nsequence of the limit transitions discussed at the end of the\nprevious section, first ${\\it \\Gamma}_T \\to 0$, for exclusion of\nthe diagonal matrix elements at finite $\\omega$, and then, $\\omega \\to\n0$ limit. In this case, we have not contribution of the diagonal\nmatrix elements in the friction, and we are left with the low\nfriction curves with increasing temperature dependence shown in\nFigs.\\ \\ref{fig6}-\\ref{fig8}. For the opposite limit sequence if we\nconsider first the small frequency limit $\\omega \\to 0$ for the\nfinite (large) ${\\it \\Gamma}_T$ we have the contribution of\ndiagonal matrix elements to the friction shown by the curves\ndecreasing with temperature which correspond to the hydrodynamic\nlimit here. As noted above, the exclusion of diagonal matrix\nelements for this last case could be justified because the\nphysical condition of the hydrodynamic limit $\\omega \\tau \\ll 1$ is\nnot probably realized in fission processes. In that case, we\nexpect the increasing friction; which has essentially other,\nnon-hydrodynamic nature. We might interpret it within the FLDM as\nrelated to the third peak, due to the Fermi-surface distortions.\n\n\n\n\\section{NEUTRON-PROTON CORRELATIONS AND IVGDR}\n\\label{npcorivgdr}\n\n\\subsection{Extensions to the asymmetric nuclei}\n\\label{extensiontoas}\n\nThe FLDM was successfully applied for studying the global properties\nof the isoscalar multipole\ngiant resonances having nice agreement of their basic characteristics,\nsuch as the energies and sum rules, with experimental data\nfor collective excitations of heavy nuclei\n\\cite{strutmagden0,kolmagpl}. For the collective excitation modes in\nasymmetric neutron-proton nuclei, the FLDM was straightly extended\n in particular for calculations of the IVGDR structure\n\\cite{denisov,kolmagsh,BMV,BMR}.\nIn this case,\none has the two coupled (isoscalar and isovector) Landau--Vlasov equations\nfor the dynamical variations of distribution functions,\n$\\delta f^{}_{\\pm}({\\bf r},{\\bf p},t)$, in the nuclear phase-space volume \\cite{kolmagsh},\n\\begin{eqnarray}\\label{LVeq}\n\\frac{\\partial }{\\partial t}\n \\delta f^{}_{\\pm}({\\bf r},{\\bf p},t) &+&\n \\frac{{\\bf p} }{m_\\pm^*}\n {\\bf \\nabla}_r \\left[ \\delta f^{}_{\\pm}({\\bf r},{\\bf p},t) ~~\\right.\\nonumber\\\\\n &+&\\left.\n \\delta \\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right)\n \\delta \\varepsilon_{\\pm}\n +V_{\\rm ext}^{\\pm}\\right]\n =\\delta St^{}_{\\pm}.~~\n\\end{eqnarray}\nHere $m_\\pm^*$ are the isoscalar (+) and isovector (-) effective masses,\n$\\varepsilon=p^2\/(2 m_\\pm^*)$ , $\\varepsilon_{{}_{\\! {\\rm F}}}=\n(p_{\\rm F}^{\\pm})^2\/(2 m_\\pm^*)$ is the Fermi energy.\nThe splitting between\nthe Fermi momenta $p_{\\rm F}^{\\pm}$ is originated by the difference of the\nneutron and proton potential well depths, due to the Coulomb interaction\n\\cite{migdal,kolmagsh},\n\\bel{momentumdef}\np_{\\rm F}^\\pm =p_{{}_{\\! {\\rm F}}}(1\\mp \\Delta,\\quad \\Delta=2(1+\\mathcal{F}_0')\n\\mathcal{I}\/3,\n\\end{equation}\n where\n$\\mathcal{F}_0'=3J\/\\varepsilon_{{}_{\\! {\\rm F}}}-1$ is the isotropic isovector\nLandau constant of the quasiparticle interaction (\\ref{fasymint}),\n$J$ is the volume symmetry energy constant \\cite{myswann69}.\nThe asymmetry parameter\n$~\\mathcal{I}=(N-Z)\/A~$ is assumed to be small\nnear the nuclear stability line, $N$ and $Z$ are the neutron and proton\nnumbers in the nucleus ($A=N+Z$).\nIn (\\ref{LVeq}), for the dynamical variations of the\nself-consistent quasiparticle\n(mean-field) interaction\n$\\delta \\varepsilon_\\pm ({\\bf r},{\\bf p},t)$, one has\n\\bel{interaction}\n\\delta \\varepsilon_{\\sigma}=\\pi^2\\hbar^3\n\\sum_{\\sigma'}\\left[\\frac{F_{0,\\sigma \\sigma'}\n}{p_{\\rm F}^{\\sigma'} m_{\\sigma'}^*}\n~\\delta \\rho_{\\sigma'} +\n\\frac{m F_{1,\\sigma \\sigma'} }{m_{\\sigma'}^* p_{\\rm F}^\\sigma\\left(\np_{\\rm F}^{\\sigma'}\\right)^2}\n~{\\bf p} \\cdot {\\bf j}_{\\sigma'}\\right].\n\\end{equation}\nThe sum is taken over the sign index $\\sigma=\\pm$.\nThe dynamical variations\nof the quasiparticle interaction\n$\\delta \\varepsilon_\\pm$ at the first order with respect to the equilibrium energy\n$p^2\/(2 m_\\pm^*)$ is defined through those of the\nparticle density,\n\\bel{densitydef}\n\\delta \\rho_\\pm({\\bf r},t)=\n\\int \\frac{2{\\rm d}{\\bf p} }{(2 \\pi\\hbar)^3}\\;\\delta f_\\pm({\\bf r},{\\bf p},t)\n\\end{equation}\n[zero ${\\bf p}$-moments\nof the dynamical distribution functions $\\delta f_{\\sigma}({\\bf r},{\\bf p},t)$\n(\\ref{planewave})], and the current density,\n\\bel{currentdef}\n{\\bf j}_\\pm({\\bf r},t)=\\int \\frac{2{\\rm d}{\\bf p} }{(2 \\pi\\hbar)^3}\n~\\frac{{\\bf p}}{m}~ \\delta f_\\pm({\\bf r},{\\bf p},t)\n\\end{equation}\n(their first ${\\bf p}$-moments).\nThe Landau interaction constants $F_{l,\\sigma \\sigma'}$ in\n(\\ref{interaction}) are defined by\nexpansion of the scattering quasiparticle's interaction amplitude\n$F_{\\sigma \\sigma'}({\\bf p},{\\bf p}')$\nin the Legendre polynomial series,\n\\bel{fasymint}\nF_{\\sigma \\sigma'}({\\bf p},{\\bf p}')=F_{0,\\sigma \\sigma'} + F_{1,\\sigma \\sigma'}\n{\\hat p} \\cdot {\\hat p}'+ ..., \\quad {\\hat p}={\\bf p}\/p.\n\\end{equation}\nFor the sake of simplicity,\nwe assume that $F_{l,\\sigma \\sigma'}$ is a symmetrical matrix\n($l \\leq 1)$ and\n$F_{l,pp}-F_{l,nn}$\nis of the second order in parameter $\\Delta$ [see below (\\ref{LVeq})], and\ncan be neglected in the linear approximation with respect to $\\Delta$,\n\\bel{symmetry}\nF_{l,pp}=F_{l,nn},\\quad F_{l,pn}=F_{l,np}.\n\\end{equation}\nThus, we arrive at usual simple definitions for the isoscalar\n$F_0$ and $F_1$ and isovector $F_0'$ and $F_1'$ Landau interaction\nconstants \\cite{migdal,kolmagsh},\n\\begin{eqnarray}\\label{landauconst}\nF_{l}&=&(F_{l,pp}+F_{l,pn})\/2,\n\\nonumber\\\\\n F_{l}^\\prime&=&(F_{l,pp}-F_{l,pn})\/2,\n\\qquad l=0,1.\n\\end{eqnarray}\nThese constants are related to the\nSkyrme interaction constants\nin the usual way \\cite{liu}.\nThe isoscalar ($\\mathcal{F}_0$)\nand isovector ($\\mathcal{F}_0'$) isotropic interaction constants are\nassociated with\nthe volume in-compressibility modulus $K$ and symmetry energy constant $J$,\nrespectively.\nThe anisotropic interaction constants $\\mathcal{F}_1$ and\n$\\mathcal{F}_1'$ correspond to\nthe effective masses by equations $m_{+}^*=m(1+\\mathcal{F}_1\/3)$ and\n$m_{-}^*=m(1+\\mathcal{F}_1'\/3)$.\nThe periodic time-dependent external\nfield in (\\ref{LVeq}) is given by $V_{\\rm ext} \\propto \\hbox{exp}(-i \\omega t)$\nas in (\\ref{extfield}). The collision term $\\delta St^{}_{\\pm}$ is\ntaken in the simplest $\\tau_\\pm$-relaxation time approximation\n(\\ref{intcoll}).\nFor simplicity, we consider in this section the low temperature\nlimit $T \\rightarrow 0$ neglecting the difference between the local and\nglobal equilibrium for the quasistatic distribution function.\n\n\nSolutions of these equations (\\ref{LVeq}) associated with the dynamic\nmultipole particle-density variations,\n$\\delta \\rho_\\pm({\\bf r},t) \\propto Y_{L0}({\\hat r})$ in\nthe spherical coordinates $r$, $\\theta$ , $\\varphi$, can be found\nin terms of a superposition of the plane waves (\\ref{planewave})\nover angles of the wave vector ${\\bf q}$ as\n\\begin{eqnarray}\\label{planewaves}\n&& \\delta f_{\\pm}=\\delta\\left(\\varepsilon-\n(p_{\\rm F}^{\\pm})^2 \/2m_{\\pm}^*\\right) ~~\\nonumber\\\\\n&\\times&\\int {\\hbox{d}}\\Omega_{\\bf q} \\mathcal{A}_{\\pm} Y_{L0}\\left({\\hat q}\\right)~\n\\hbox{exp}\\left[i\\left({\\bf q}{{\\bf r}}-\\omega t\\right)\\right] , \\quad {\\hat q}={\\bf q}\/q,~~\n\\end{eqnarray}\n$\\omega =p_{\\rm F}^{\\pm}s q \\sqrt{NZ\/A^2}\/m_\\pm^*$, $q=|{\\bf q}|$.\nThe factor $~\\sqrt{NZ\/A^2}~$ ensures the conservation of the center-of-mass\n position for the odd vibration multipolarities $~L~$\n\\cite{eisgrei}), in particular, for the dipole modes ($L=1$).\nThe amplitudes of the Fermi surface\ndistortions $\\mathcal{A}_\\pm~$ are determined by (\\ref{LVeq}).\nFor the simplest case of the zero anisotropic interaction ($F_1=F_1'=0$)\nin the collisionless limit $\\omega \\tau \\to \\infty$,\nthe dispersion equation for the sound velocity $s$\ntakes the form:\n\\begin{eqnarray}\\label{dispeq}\n&&4F_0F_0'\\left(F_0 Q_1(s)-1\\right) \\nonumber\\\\\n&-&\\frac14\\Delta^2 F_0^2{F_0'}^2\n\\left(\\frac{s^2}{s^2-1} + Q_1(s)\\right)^2=0,\n\\end{eqnarray}\n(We accounted for a small $\\Delta$\nand large $\\omega \\tau$ at the zero temperature.)\nThis equation has the two solutions $s=s_n$ related to the main peak $n=1$\nand $2$ for its satellite, see (26) of \\cite{kolmagsh}\nfor the finite $\\omega \\tau$ and nonzero $F_1$ and $F_1'$.\nIn the limit $\\Delta \\to 0$, the dispersion equations\ngiven by (25) of \\cite{kolmagsh}\nwith our definitions for $s_1$ and $s_2$ modes $n=1$ and $2$\nare resulted in the two (isovector and isoscalar) equations\nfor the equations for the separated zero sounds,\nrespectively,\n\\bel{splitdispeq}\nQ_1(s)= 1\/F_0', \\qquad \\mbox{and}\\qquad\nQ_1(s)= 1\/F_0.\n\\end{equation}\n\n For the finite Fermi-liquid drop with a sharp ES\n\\cite{strutmagden,magstrut,magboundcond},\nthe macroscopic boundary conditions for the pressures\nand those for the velocities were derived in\n\\cite{kolmagsh,magsangzh,BMRV}. For small isovector vibrations near\nspherical shape, the radial mean-velocity $u_{r}$ and\nmomentum-flux-tensor $\\Pi_{rr}$ components, defined through\nthe moments of the distribution function\n$\\delta f_{-}$ as solutions of the kinetic equation (\\ref{LVeq}) [see\n(\\ref{veloc}) and (\\ref{momentflux})]\nare given by (\\ref{bound1}) and (\\ref{bound2}) with\n$u_{r}=u_{r}^{+}-u_{r}^{-}$ and\n$\\Pi_{rr}=\\Pi_{rr}^{+}-\\Pi_{rr}^{-}$.\nThe r.h.s.s of these boundary conditions are the isovector\nES velocity $u_{{}_{\\! S}}=R \\dot{Q}_S Y_{L0}({\\hat r})$ and capillary\npressure exceed\n\\bel{boundcondiv}\n\\delta P_S=2 Q_S b_S^{(-)} \\rho_{{}_{\\! 0}} A^{1\/3}\nY_{10}(\\hat{r})\/3,\n\\end{equation}\ngiven through the isovector\nsurface energy constant $b_S^{(-)} \\propto \\alpha_{-}$ [see (\\ref{sigma}) and\n(\\ref{bsplusminus})], where $Q_S$ is\nthe dynamical isovector-dipole ($L=1$)\namplitude of the motion of the neutron drop\nES against the proton one (\\ref{surface}),\nkeeping also the volume and the position of the center of mass conserved).\nNote that another interpretation of the surface symmetry-energy\nconstant $b_{\\rm S}^{(-)}$ in (\\ref{boundcondiv}) is considered in\n\\cite{denisov,abrIVGDR}.\nThis constant essentially differs from the isovector stiffness introduced\nin \\cite{myswann69} for the description of the neutron skin as a\ncollective variable, see more detailed discussions in \\cite{BMV,BMRV}.\n\n The energy constant, $~D=\\hbar \\omega A^{1\/3}~$,\nand energy weighted sum\nrules (EWSR),\n\\begin{equation}\\label{strength}\n{\\tt S}_{1}=\\frac{\\hbar^2}{\\pi} \\int \\hbox{d} \\omega \\; \\omega \\; {\\mbox {\\rm Im}} \\chi^{\\rm coll}(\\omega ),\n\\end{equation}\nfor the IVGDR can be found from the collective response function\n$\\chi^{\\rm coll}(\\omega )$ .\nThe response function (\\ref{chicollrho}) is determined by the\ntransition density (\\ref{densvolsurf}) generalized\nto the dynamic isoscalar and isovector components \\cite{BMR}:\n\\begin{eqnarray}\\label{trandenscl}\n\\delta \\rho_{\\pm}({{\\bf r}},t) &=&\n\\delta \\rho_{\\pm}^{\\rm vol}({{\\bf r}},t)\\; w_{\\pm}(\\xi) \\nonumber\\\\\n&-&\n\\frac{1}{a}\\frac{\\hbox{d} w_{\\pm}(\\xi)}{ \\hbox{d} \\xi}\\; \\overline{\\rho}\\;\n\\left[\\delta R_{\\pm}- \\delta \\aleph_{L}^{\\pm} \\;\nY_{L0} ({\\hat r})\\right],\n\\end{eqnarray}\nwhere\n$\\delta \\aleph_{L}^{\\pm}$ is defined by the mass center conservation\n($\\int \\hbox{d} {\\bf r} \\;{\\bf r}\\; \\delta \\rho_{\\pm}=0$),\n$w_\\pm(\\xi)$ is given by (\\ref{ysolplus}) and (\\ref{ysolminus}).\nIn Fig.\\ \\ref{fig9etf}, a strong SO dependence of the isovector\ndensity $w_{-}(\\xi)$\nis compared with\nthat of the isoscalar one $w_{+}(\\xi)$\n(low index ``+'' is omitted here and below)\nfor the SLy7 force as a typical example\n\\cite{magsangzh,BMRV}.\nAs shown in \\cite{BMRV}, the isoscalar $w(\\xi)$, and therefore, the isovector\n$w_{-}(\\xi)$ densities depend rather strongly on the\nmost of the Skyrme forces \\cite{chaban,reinhard} near the ES.\nIn Fig.\\ \\ref{fig10etf} (in logarithmic scale),\none observes notable differences in the isovector densities $w_{-}$ derived\nfrom different Skyrme forces\nwithin the edge diffuseness. In particular,\nthis is important for the calculations of the neutron skins of nuclei\n\\cite{BMRV}.\n\nWe emphasize that the dimensionless densities, $w(x)$\n(\\ref{ysolplus}) and $w_{-}(x)$\n(\\ref{ysolminus}), shown in Figs. \\ref{fig9etf} and \\ref{fig10etf} were\nobtained in\nthe leading ES approximation ($a\/R \\ll 1$) as functions of the\nspecific combinations\nof the Skyrme force parameters, such as $\\beta$ and $ c_{sym}$ of\n(\\ref{defpar}). Therefore, they are the universal distributions\nindependent of the specific properties of the nucleus such as the neutron and\nproton numbers, and the deformation and curvature of the nuclear ES;\nsee also \\cite{strtyap,strutmagden,magsangzh}.\nThese distributions yield approximately the spatial coordinate dependence\nof local densities in the normal-to-ES direction $\\xi$.\nWith the correct asymptotical behavior outside of the ES layer for any\nES deformation, they satisfy the leptodermic\ncondition $a\/R \\ll 1$, in particular,\nfor the semi-infinite nuclear matter.\n\n\nThe universal functions $w(\\xi)$ (\\ref{ysolplus}) and $w_{-}(x)$\n(\\ref{ysolminus}) of the leading order in the ESA\ncan be used [explicitly analytically in the quadratic\napproximation for $\\epsilon(w)$] for the calculations of the surface\nenergy coefficients\n$b_S^{(\\pm)}$ (\\ref{sigma}), the neutron skin and isovector stiffness\n(see \\cite{BMRV}). As shown in Appendices B and C of \\cite{BMRV},\nonly these particle-density distributions $w_{\\pm}(\\xi)$\nwithin the surface layer\nare needed through their derivatives [the lower limit of the integration\nover $\\xi$ in (\\ref{sigma}) can be approximately extended to\n$-\\infty$ because of no contributions from the internal volume region\nin the evaluation of the main surface terms of the pressure and energy].\nTherefore, the surface symmetry-energy coefficient\n$k_{{}_{\\! S}}$ in (\\ref{bsplusminus}) and (\\ref{Jm}) (also the neutron skin\nand the isovector stiffness \\cite{BMRV})\ncan be approximated analytically in terms of the functions\nof the definite critical combinations of the Skyrme parameters such as $\\beta$,\n$c_{sym}$, $a$ [see (\\ref{defpar})],\nand the parameters of the infinite nuclear matter\n($b_{\\mathcal{V}}, \\rho_\\infty, K$). Thus, they are independent of the\nspecific properties\nof the nucleus (for instance, the neutron and proton numbers), and\nthe curvature and deformation of the nuclear surface in the\nconsidered ESA.\n\n\nSolving the Landau--Vlasov equations (\\ref{LVeq}) in terms of the\nzero sound plane waves\n(\\ref{planewaves}) with using the dispersion equations (26)\nin \\cite{kolmagsh} for the sound velocities $s_n$ and\nmacroscopic boundary conditions\n(\\ref{bound1}) and (\\ref{bound2}) with (\\ref{boundcondiv}) on the\nnuclear ES, from (\\ref{chicollrho}) and (\\ref{trandenscl}) one obtains\n\\begin{eqnarray}\\label{respfun}\n &&\\chi_{L}^{\\rm coll}(\\omega )=\\sum_n \\frac{\\mathcal{A}_{L}^{(n)}(q)\n}{\\mathcal{D}_{L}^{(n)}(\\omega -i\\Gamma\/2)},\\quad{\\rm with}\\quad\n\\mathcal{D}_{L}^{(n)}(\\omega ) \\nonumber\\\\\n&=&\nj_1'(qR)+\\frac{3 \\varepsilon_{{}_{\\! {\\rm F}}} qR}{2b_S^{(-)}A^{1\/3}}\\;\n\\left[c_n j_1''(qR)+d_nj_1(qR)\\right].\n\\end{eqnarray}\nHere,\n$~c_1\\approx 1-s_1^2+\\mathcal{F}_0'~$,\n$d_1\\approx 1-s_1^2+\\mathcal{F}_0'$ for the main ($n=1$) IVGDR\npeak. Small anisotropic $\\mathcal{F}_1$ and $\\mathcal{F}_1'$\ncorrections\n and more bulky expressions for $s_2$ of the satellite ($n=2$) peak\nof a smaller ($\\propto I$)\nstrength were omitted (see (D11) in \\cite{kolmagsh} for more precise\nexpressions). We present here also the simplest expressions for the amplitudes,\n$\\mathcal{A}_1(q)\\approx -\\rho_\\infty R^3 j_1(qR)\/(m \\omega ^2)$ and\n$\\mathcal{A}_1(q)\\propto \\Delta \\propto I$\nfor the $n=1$ and $2$ modes [see a more complete equation\n(60) in \\cite{kolmagsh}]. The Bessel functions $j_1(z)$\nand its derivative $j_1'$ were defined after (\\ref{potensolut}) ($L=1$).\nThe poles of the response function\n$\\chi^{\\rm coll}(\\omega )$\n(\\ref{respfun}) (roots $\\omega _n$ of\nthe equation $D^{(n)}(\\omega -i \\Gamma\/2)=0$ or $q_n$ )\ndetermine the IVGDR energies $\\hbar \\omega $ as their\nreal part (the IVGDR width $\\Gamma$\nis determined by their imaginary part).\nThe residue $\\mathcal{A}_n$ is important for the calculations\nof the EWSR (\\ref{strength})\nat a small width of the IVGDR $\\Gamma$.\nNote that the expression like (\\ref{respfun})\nfor the only one main peak (without the IVGDR structure)\nin symmetrical nuclei ($N=Z$)\nwith using the phenomenological boundary conditions\nwhich have the same form as (\\ref{bound1}) and (\\ref{bound2}),\nwhere however the isovector neutron-skin stiffness was applied\ninstead of the surface symmetry-energy constant $b_{{}_{\\! S}}^{(-)}$ in the\ncapillary pressure exceed (\\ref{boundcondiv})\nwas obtained earlier in \\cite{denisov}.\n\n\n\\subsection{ Discussions of the asymmetry effects}\n\\label{discaseff}\n\n The isovector surface energy constants $k_{{}_{\\! S}}$ (\\ref{bsplusminus})\nin the ESA using the simplest quadratic approximation\nfor $\\epsilon(w)$ of\nthe energy density (\\ref{enerden})\nare shown in Table 1 for several critical Skyrme forces\n\\cite{chaban,reinhard}. These constants are rather sensitive to the choice of\nthe Skyrme forces. The modulus of $k_{{}_{\\! S}}$\nfor the Lyon Skyrme forces SLy4-7\n\\cite{chaban} is significantly larger than for other forces,\nall of them much smaller than those related to\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85}.\nFor T6 \\cite{chaban}, one has $\\mathcal{C}_{-}=0$,\nand therefore, $k_S=0$,\nin contrast to all of other forces shown in Table 1. Notice that the\nisovector gradient terms which are important for the consistent\nderivations within the ESA\n\\cite{BMRV} are not also included ($\\mathcal{C}_{-}=0$) into the energy density in \\cite{danielewicz1,danielewicz2}. For RATP \\cite{chaban},\nthe isovector stiffness ($\\propto -1\/k_{{}_{\\! S}}$),\ncorresponding inversed $k_{{}_{\\! S}}$ but with the opposite sign\n\\cite{BMRV}, is even negative as $\\mathcal{C}_{-}>0$\n($k_{{}_{\\! S}}>0$). The reason of significant differences in these values\nmight be related to those of the critical isovector\nSkyrme parameter $\\mathcal{C}_{-}$\n in the gradient terms of the energy density (\\ref{enerden}).\nDifferent experiments used for fitting this parameter were found to be\nalmost insensitive in determining uniquely its value, and hence, $k_S$\n[or $b_S^{(-)}$, see (\\ref{bsplusminus})],\nas compared to the well-known isoscalar surface-energy constant $b_S^{(+)}$.\nThe isovector surface-energy constant $k_{{}_{\\! S}}$ (\\ref{bsplusminus})\nand the corresponding stiffness\ndepend much on the SO $\\beta$ parameter through the constant\n$\\mathcal{J}_{-}$ (\\ref{Jm}).\n\n\nThe IVGDR energy constants $D=\\hbar \\omega ^{(-)} A^{1\/3}$\nof the hydrodynamic model (HDM)\nare roughly in good agreement with the well-known experimental value\n$D_{\\rm exp}\\approx 80$ MeV for heavy nuclei within a precision better\nor of the order of 10\\%,\nas shown in \\cite{BMV,BMRV} (see also\n\\cite{denisov,kolmagsh,plujko}).\nMore precise $A^{-1\/3}$ dependence of $D$\nseems to be beyond the accuracy of these HDM calculations. This takes place\n even accounting more consistently for the ES motion because of several\nother reasons (the macroscopic Fermi-surface distortions\n\\cite{denisov}, also including structure of the IVGDR\n\\cite{kolmag,kolmagsh,BMR,abrIVGDR,abrdavpl,plujko},\ncurvature, Coulomb, quantum-shell,\nand pairing \\cite{belyaevzel} effects\ntowards the realistic\nself-consistent calculations\nbased on the Skyrme HF approach\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2}.\nLarger values 30-80 MeV of the isovector stiffness\n\\cite{myswann69}\n(smaller $k_{{}_{\\! S}}$ ) were found in\n\\cite{myswnp80pr96,myswiat85,vinas2,brguehak}.\nWith smaller $|k_{{}_{\\! S}}|$ (see Table 1, or larger the isovector\nstiffness) the fundamental parameter of the LDM expansion in\n\\cite{myswann69,myswnp80pr96}\nis really small for $A \\simg 40$, and therefore, the results\nobtained by using this expansion are justified \\cite{BMRV}.\n\n\n Table 1 shows also the mean IVGDR energies $D$\nobtained \\cite{BMV,BMRV} within a more precised FLDM \\cite{kolmagsh}.\nThe IVGDRs even for the spherical nuclei have a double-resonance structure,\nthe main peak $n=1$ which exhausts mainly the EWSR\nfor almost all Skyrme forces and the satellite one $n=2$ with\nthe significantly smaller EWSR contributions proportional to\nthe asymmetry parameter $I$, typical for heavy nuclei. The\nlast row shows the average\n$D(A)$ weighted by their EWSR\ndistribution in rather good agreement with the experimental data\nwithin the same accuracy about 10 \\%, and in agreement with the results\nof different other macroscopic IVGDR models\n\\cite{denisov,abrIVGDR,abrdavpl,plujko}.\nExclusion can be done (see Table 1) for the\nSkyrme forces SIII \\cite{chaban} and SkL3 \\cite{reinhard} where we\nobtained a little larger IVGDR energies.\nNote that the main characteristics of the\n IVGDR described by mean $D$\nare almost insensitive to the isovector surface-energy constant\n$k_{{}_{\\! S}}$ \\cite{BMRV,BMV}. Therefore, we suggested \\cite{BMRV,BMR}\nto study the IVGDR\ntwo-peak (main and satellite) structure in order to fix the ESA value of\n$k_{{}_{\\! S}}$ \\cite{BMRV} from comparison with the experimental data\n\\cite{adrich,wieland,kievPygmy} and theoretical results\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2,ditoro}.\n\n\n \\section{NUCLEAR COLLECTIVE ROTATIONS}\n\\label{semshellmi}\n\n\\subsection{General ingradients of the cranking model}\n\\label{cranmod}\n\nWithin the cranking model, the nuclear collective\nrotation of the Fermi independent-particle system\nassociated with a many-body Hamiltonian,\n$H^{\\boldsymbol\\omega }=H+H_{\\rm CF}^{\\boldsymbol\\omega }$,\ncan be described, to a good approximation \\cite{eisgrei},\nin the restricted subspace of Slater determinants, by\nthe eigenvalue problem for\n a s.p.\\ Hamiltonian, usually called the {\\it Routhian}.\nFor this Routhian, in the body-fixed rotating frame \\cite{bohrmot,mix,fraupash},\none has\n\\bel{raussian}\nh^{\\boldsymbol\\omega }=h + h_{\\rm CF}^{\\boldsymbol\\omega },\\qquad\nh_{\\rm CF}^{\\boldsymbol\\omega }=-\\boldsymbol\\omega \\cdot\n\\left(\\boldsymbol\\ell + {\\bf s}\\right),\n\\end{equation}\nwhere $h_{\\rm CF}^{\\boldsymbol\\omega }$ is the s.p.\\ cranking field which is\napproximately equal to the Coriolis interaction\n(neglecting a smaller centrifugal term, $\\propto \\omega ^2$).\nThe Lagrange multiplier $\\boldsymbol\\omega $\n(rotation frequency of the body-fixed coordinate system) is defined\nthrough the constraint on the nuclear angular momentum ${\\bf I}$,\nevaluated through the quantum average\n$\\langle \\boldsymbol\\ell +{\\bf s} \\rangle^{\\boldsymbol\\omega }={\\bf I} $, of the\ntotal s.p.\\ operator, $\\boldsymbol\\ell + {\\bf s}$,\nwhere $\\boldsymbol\\ell$\nis the orbital angular momentum and ${\\bf s}$ is the spin of the quasiparticle,\nthus defining a function\n$\\boldsymbol\\omega =\\boldsymbol\\omega ({\\bf I})$.\nThe quantum average of the total s.p.\\ operator $\\boldsymbol\\ell\n+ {\\bf s}$ is obtained by evaluating expectation values of the\nmany-body Routhian\n$H_{\\rm CF}^{\\boldsymbol\\omega }$ in the subspace of Slater determinants.\nFor the specific case of a rotation around the $x$ axis ($\\omega =\\omega _x$)\nwhich is perpendicular to the symmetry $z$ axis of the axially-symmetric mean\nfield $V$, one has (dismissing for simplicity spin (spin-isospin) variables),\n\\bel{constraint0}\n\\langle \\ell_x \\rangle^{\\omega } \\equiv\nd_s \\sum_i n_i^{\\omega } \\int \\hbox{d} {\\bf r}\n\\;\\psi_i^{\\omega }\\; \\left({\\bf r}\\right) \\; \\ell_x\\;\n\\overline{\\psi}_i^{\\omega }\\left({\\bf r}\\right)=I_x,\n\\end{equation}\nwhere $d_s$ as the spin (spin-isospin) degeneracy in the case of the\ncorresponding symmetry of the mean potential $V$.\nThe occupation numbers $n_i^{\\omega }$\nfor the Fermi system of independent nucleons are given by\n\\bel{ocupnumbi}\nn_i^{\\omega }\\equiv n\\left(\\varepsilon_i^{\\omega }\\right)\n=\\{1+ \\hbox{exp}\\left[(\\varepsilon_i^{\\omega } - \\mu^{\\omega })\/T\\right]\\}^{-1}.\n\\end{equation}\nIn (\\ref{constraint0}), $\\psi_i^{\\omega }({\\bf r})$ are the eigenfunctions\nand\n$\\overline{\\psi}_i^{\\omega }({\\bf r})$ their complex conjugate,\n$\\varepsilon_i^{\\omega }$\nthe eigenvalues of the Routhian $h^{\\omega }$ (\\ref{raussian}),\n$\\mu^{\\omega }$ is the chemical potential. For relatively small frequencies\n$\\omega $ and temperatures $T$, $\\mu^{\\omega }$ is to a good approximation\nequal to the Fermi energy,\n$\\mu^{\\omega } \\approx \\varepsilon_{{}_{\\! {\\rm F}}} =\\hbar^2 k_{\\rm F}^2\/2 m^*$,\nwhere $k_{{}_{\\! {\\rm F}}}$ is the Fermi momentum in units of $\\hbar$.\nFrom (\\ref{constraint0}), the rotation frequency $\\omega $ can be\nspecifically expressed in terms of a given angular momentum of nucleus\n$I_x$, $\\omega =\\omega \\left(I_x \\right)$.\nWithin the same approach,\none approximately has for the particle number\n\\bel{partconspert}\nA= d_s\\sum_i n_i^{\\omega }\n\\int \\hbox{d} {\\bf r}\\; \\psi_i^{\\omega }({\\bf r})\\;\\overline{\\psi}_i^{\\omega }({\\bf r})\n\\approx d_s \\int \\hbox{d} \\varepsilon \\; n(\\varepsilon),\n\\end{equation}\nwhich determines the chemical potential $\\mu^{\\omega }$ for a given number of\nnucleons $A$. As we introduce the continuous\nparameter $\\omega $ and ignore the uncertainty\nrelation between the angular\n momentum and angles of the body-fixed coordinate system,\nthe cranking model is semiclassical\nin nature \\cite{ringschuck}.\nThus, we may consider the collective MI $\\Theta_x$ (for a rotation\naround the $x$ axis, and omitting, to simplify the notation, spin and\nisospin variables) as a response of the quantum average\n$\\delta \\langle \\ell_x \\rangle^{\\omega }$ (\\ref{constraint0}), to the\nexternal cranking field $h_{\\rm CF}^{\\omega }$ in (\\ref{raussian}).\nSimilarly to the magnetic or isolated\nsusceptibilities\n\\cite{richter,fraukolmagsan,magvvhof,yaf},\none can write\n\\bel{response}\n\\delta \\langle \\ell_x \\rangle^{\\omega }=\n\\Theta_x(\\omega ) \\delta\\omega ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{thetaxdef}\n&&\\Theta_{x}(\\omega )=\n\\partial \\langle \\ell_x\\rangle^\\omega \/\\partial \\omega =\n\\partial^2 E(\\omega )\/\\partial \\omega ^2,\n\\quad\\;\\;\\;\n\\nonumber\\\\\n&& E(\\omega )=\\langle h \\rangle^\\omega \n\\equiv d_s \\sum_i n_i^{\\omega }\\int \\hbox{d} {\\bf r}\n\\;\\psi_i^{\\omega }\\; \\left({\\bf r}\\right) \\;h \\;\n\\overline{\\psi}_i^{\\;\\omega }\\left({\\bf r}\\right).\\;\\;\\;\n\\end{eqnarray}\nTraditionally \\cite{mix,dfcpuprc2004,mskbPRC2010},\nanother parallel (alignment)\nrotation with respect to the\nsymmetry $z$ axis can be also considered as presented\nin Appendix A of \\cite{mskbPRC2010}.\n\nAs was shown in\n\\cite{inglis,bohrmotpr,belyaevzel,valat,bohrmot,fraupash,mix}, one can treat\nthe term\n$-\\boldsymbol\\omega \\cdot \\boldsymbol\\ell = -\\omega \\, \\ell^{}_x$ as\na perturbation for a nuclear rotation around the $x$ axis.\nWith the constraint (\\ref{constraint0})\nand the MI (\\ref{thetaxdef}) treated in second order perturbation theory,\none obtains the well known Inglis cranking formula.\nInstead of carrying out the rather involved calculations\npresented above, one could, to obtain the yrast line energies $E(I_x)$\nfor small enough temperatures $T$ and frequencies $\\omega $, approximate\nthe angular frequency by $\\omega =I_x\/\\Theta_x$ and write the energy in the form\n\\bel{yrast}\nE(I_x)= E(0) + \\frac{I^{2}_x}{2 \\Theta_x}.\n\\end{equation}\nAs usually done, the rotation term above needs to be quantized\nthrough $I^{2}_x \\rightarrow I_x (I_x+1)$ in order to study the\nrotation bands.\n\n\\subsection{Self-consistent ETF description of nuclear rotations}\n\\label{etfmi}\n\nFollowing\nreference \\cite{bartelnp}, a microscopic description of rotating nuclei\nwas obtained in the Skyrme Hartree--Fock formalism, within the Extended\nThomas--Fermi density-functional theory up to order $\\hbar^2$.\nWithin a variational space restricted to Slater determinant,\n the minimization of the\nexpectation value of\nthe nuclear Hamiltonian\nlead to the s.p. Routhian $h^{\\boldsymbol\\omega}_{\\rm q}$ (\\ref{raussian})\nthat is determined by a one-body potential $V_{\\rm q}({\\bf r})$,\na spin-orbit field ${\\bf W}_{\\rm q}({\\bf r})$ and an effective mass\nform factor $f^{\\rm eff}_{\\rm q}({\\bf r})=m\/m^*_{\\rm q}$ (see\nalso \\cite{brguehak}). In addition,\nin the case when the time\nreversal symmetry is broken, a cranking field form factor\n$\\boldsymbol\\alpha_{\\rm q}({\\bf r})$ and a spin field form factor\n${\\bf S}_{\\rm q}({\\bf r})$\nalso appear. In this subsection the (roman) subscript $\\rm q$\nrefers to the nucleon isospin\n($\\rm q = \\{n,p\\}$) and should not be confused\nwith the wave number $q$ in other sections.\nAll these fields can be written\nas functions of local densities and their derivatives, like the\nneutron-proton particle\ndensities $\\rho_{\\rm q}({\\bf r})$, the kinetic energy densities\n$\\tau_{\\rm q}({\\bf r})$,\nthe spin densities (also referred to\nas spin-orbit densities)\n${\\bf J}_{\\rm q}({\\bf r})$, the current densities\n${\\bf j}_{\\rm q}({\\bf r})$, and\nthe spin-vector densities\n$\\boldsymbol\\rho_{\\rm q}({\\bf r})$.\nNote that in the present subsection, $\\tau_{\\rm q}({\\bf r})$\nstands for the kinetic energy\ndensity which should not be confused with the relaxation time in\nprevious sections (here, however, with a different subscript ${\\rm q}$\nas compared to $q$ in \nsections \\ref{kinapp},\\ref{respfuntheor},\\ref{npcorivgdr} and Appendices\nA,B). In principle, two\nadditional densities appear, a spin-vector kinetic energy density\n$\\boldsymbol\\tau_{\\rm q}({\\bf r})$ and a tensor coupling\n$J^{}_{\\alpha \\beta}({\\bf r})$\nbetween spin and gradient vectors, which have, however, been neglected\nsince their contribution should be small, as suggested by \\cite{BFH87}.\n\n\n\nThe cranking-field form factor\n$\\boldsymbol\\alpha_{\\rm q}({\\bf r})$ contains two\ncontributions.\nOne of them is coming from the orbital part of the constraint,\n$-\\boldsymbol\\omega \\, \\boldsymbol\\ell$,\nwhich has been shown in \\cite{BV78} to correspond to the Inglis cranking\nformula \\cite{inglis}. The other,\na Thouless--Valatin self-consistency\ncontribution \\cite{TV62} has its origin in the self-consistent response\nof the mean field to the time-odd part of the density matrix generated by\nthe cranking term of the Hamiltonian.\nThe aim is now to find functional relations for the local densities\n$\\tau_{\\rm q}({\\bf r})$, ${\\bf J}_{\\rm q}({\\bf r})$,\n${\\bf j}_{\\rm q}({\\bf r})$ and $\\boldsymbol\\rho_{\\rm q}({\\bf r})$ in\nterms of the particle densities $\\rho_{\\rm q}({\\bf r})$,\nin contrast to those given by\nGrammaticos and Voros \\cite{GV79} in terms of the form factors\n$V_{\\rm q}$,\n$f^{\\rm eff}_{\\rm q}$, ${\\bf W}_{\\rm q}$,\n$\\boldsymbol\\alpha_{\\rm q}$ and\n${\\bf S}_{\\rm q}$.\nTaking advantage of the fact that, at the leading Thomas--Fermi order,\nthe cranking field form factor is given by \\cite{bartelnp}\n\\bel{ETF02}\n \\boldsymbol\\alpha^{({\\rm TF})}_{\\rm q}\n = f^{\\rm eff}_{\\rm q}\\left( {\\bf r}\n \\times \\boldsymbol\\omega \\right),\n\\end{equation}\none simply obtains the rigid-body value for the Thomas--Fermi current density\n\\bel{ETF03}\n {\\bf j}^{({\\rm TF})}_{\\rm q} = \\displaystyle \\frac{m}{\\hbar}\n \\left( \\boldsymbol\\omega \\times {\\bf r} \\right) \\, \\rho_{\\rm q}.\n\\end{equation}\nThis result is not that trivial, since it is only through the effect\nof the Thouless--Valatin self-consistency terms that such a simple\nresult is obtained. Notice also that (\\ref{ETF03}) corresponds to a\ngeneralization to the case $f^{\\rm eff}_{\\rm q} \\neq 1$ of a result\nalready found by Bloch \\cite{Bl54}.\nEquation (\\ref{ETF03})\ncan be also considered as an extension of\nthe Landau quasiparticle (generalized TF) theory\n\\cite{landau,abrikha} presented in\nSecs. \\ref{etfmi}, \\ref{cranmod} to the case of rotating Fermi-liquid systems,\ncf. (\\ref{ETF03}) with (\\ref{currentdef})\nfor the current density as an average of the \\textit{particle}\nvelocity, ${\\bf p}_{\\rm rot}\/m=\\boldsymbol\\omega \\times {\\bf r}$,\nrotating with the frequency\n$\\boldsymbol\\omega $.\nIn particular, the re-normalization of the cranking field form factor\n$\\boldsymbol\\alpha^{({\\rm TF})}_{\\rm q}=\nf_{\\rm q}^{\\rm eff} \\boldsymbol\\alpha_{\\rm o}$ with\n$\\boldsymbol\\alpha_{\\rm o} =\n\\left( {\\bf r} \\times \\boldsymbol\\omega \\right)$,\nby (\\ref{ETF02}) can be also explained\nas related to the effective mass corrections,\n$f_{\\rm q}^{\\rm eff} \\neq 1$, obtained\nby Landau \\cite{landau} with using\nboth the Galileo principle and the Thouless--Valatin self-consistency\ncorrections to a particle mass $m$ due to the quasiparticles'\n(self-consistent) interaction\nthrough a mean field. They lead in \\cite{bartelnp}\nto the self-consistent TF angular momentum\nof the \\textit{quasiparticle}\n$\\boldsymbol\\ell_{\\rm q} =f_{\\rm q}^{\\rm eff} \\boldsymbol\\ell_{\\rm o}$ with the\nclassical angular momentum $\\boldsymbol\\ell_{\\rm o}= {\\bf r} \\times {\\bf p} $\nof the particle,\nso that $-\\boldsymbol\\omega \\cdot \\boldsymbol\\ell_{\\rm q}=\n\\boldsymbol\\alpha_{\\rm q} \\cdot {\\bf p}$. This effect is similar to that\nfor the kinetic energies of the \\textit{quasiparticles},\n$\\varepsilon_{\\rm q}=p^2\/(2 m^*_{\\rm q})=f_{\\rm q}^{\\rm eff}\\varepsilon_{\\rm o}$\nwhere $\\varepsilon_{\\rm o}=p^2\/(2m)$, see after (\\ref{LVeq}).\nWith this transparent connection to the Landau quasiparticle\ntheory, it is clear that\nthere is no contradictions with the TF limit\nof the current densities (\\ref{currentdef}), $\\hbar \\rightarrow 0$,\naccounting for the particle\ndensities (\\ref{densitydef}), as well as\nwith the definitions in subsections \\ref{extensiontoas} and\n\\ref{semshell}, because $\\hbar$ in (\\ref{ETF03})\n appears formally due to a traditional\n use of the dimensionless units for the angular momenta in the\nquantum-mechanical picture to compare with experimental\nnuclear data. Another reason is related to\na consistent treatment of the essentially quantum\nspin degrees of freedom, beyond the Landau quasiparticle approach\nto the description of Fermi liquids,\nwhich have no \\textit{straight} classical limit,\nin contrast to the orbital angular momentum\n$\\boldsymbol\\ell$.\nThe convergence in the TF limit $\\hbar \\rightarrow 0$ can be realized\nfor smooth already quantities after the statistical (macroscopic)\naveraging over many\ns.p.\\ (more generally speaking, many-body) quantum states\nto remove the fluctuating (shell) effects which\nappear in the denominators of the exponents within the POT (see \nSec.\\ \\ref{semshell} for\nmore detailed discussions). Finally, the spin paramagnetic effect\n can be considered as a macroscopic one in the MI\nlike the orbital diamagnetic contribution.\nFor instance, the spin-vector density does not\nhave a \\textit{straight} classical analogue,\nsuch as the orbital angular momentum,\nand is considered as the object of leading order $\\hbar$.\n\n\nStarting from these results and taking advantage of the fact that in\nthe functional ETF expressions up to the order\n$\\hbar^2$, it is sufficient to replace quantities, such as\nthe cranking field form factor $\\boldsymbol\\alpha_{\\rm q}$, by\ntheir Thomas--Fermi expressions\n(after the statistical averaging mentioned above).\nIn order to obtain a\nsemiclassical expression,\nthat is correct to that order in $\\hbar$, one obtains for the spin-vector\ndensities $\\boldsymbol\\rho_n$ and $\\boldsymbol\\rho_p$, which are of order\n$\\hbar$ in the considered ETF expansion, a system of linear\nequations. They can be easily resolved \\cite{bartelnp}.\nOne also notices from this system of\nequations that the spin-vector densities are proportional to the\nangular velocity $\\omega$.\nExploiting the well known analogy of the microscopic Routhian problem with\nelectromagnetism, one may then define {\\it spin susceptibilities}\n$\\chi_{\\rm q}$,\n\\bel{ETF04}\n \\boldsymbol\\rho_{\\rm q} = \\hbar \\,\n\\chi_{\\rm q} \\, \\boldsymbol\\omega \\; .\n\\end{equation}\n\n\nThe key question now is to assess the sign of\nthese susceptibilities and to\ndecide whether or not the corresponding alignment is of a\n``Pauli paramagnetic''\ncharacter. The study of \\cite{bartelnp} shows that this is the case, i.e., that\nthe spin polarization is, indeed, of paramagnetic character, thus confirming\nthe conclusions of the work performed\nby Dabrowski \\cite{Da75} in a simple model of non-interacting\nnucleons.\n\n\n\nSince the cranking field factor\n$\\boldsymbol\\alpha_{\\rm q}$ is, appart from that of the\nconstraining field\n$\\boldsymbol\\alpha_{\\rm o}$ determined\nonly by the current densities ${\\bf j}_{\\rm q}$\nand the spin-vector densities\n$\\boldsymbol\\rho_{\\rm q}$, one can then write down \\cite{bartelnp}\nthe contributions to the\ncurrent densities ${\\bf j}_{\\rm q}$\ngoing beyond the Thomas--Fermi approach.\nThe semiclassical corrections\nof order $\\hbar^2$ can be split into\ncontributions\n$(\\delta {\\bf j}_{\\rm q})_{\\ell}$ and\n$(\\delta {\\bf j}_{\\rm q})_{s}$ coming\nrespectively from the orbital motion and the spin degree of freedom. It is\nfound \\cite{bartelnp} that the orbital correction\n$(\\delta {\\bf j}_{\\rm q})_{\\ell}$\ncorresponds to a surface-peaked {\\it counter-rotation} with respect to the\nrigid-body current proportional\nto $\\left( \\boldsymbol\\omega \\times {\\bf r} \\right)$,\nthus recovering the Landau diamagnetism characteristic of a finite Fermi gas.\nWith the expressions of the current densities ${\\bf j}_{\\rm q}$\nand the spin-vector\ndensities $\\boldsymbol\\rho_{\\rm q}$ up to order $\\hbar^2$,\none can write down the\ncorresponding ETF expressions for the kinetic energy density\n$\\tau_{\\rm q}({\\bf r})$ and\nspin-orbit density ${\\bf J}_{\\rm q}({\\bf r})$.\n\n\nHaving now at hand the ETF functional expressions up to order $\\hbar^2$ of all\nthe densities entering our problem, one is able to write down the energy of the\nnucleus in the laboratory frame as a functional of these local densities,\n\\bel{endentau}\nE = \\int \\hbox{d} {\\bf r} \\; \\rho \\; \\mathcal{E}[\\rho_{\\rm q},\n\\tau_{\\rm q}, {\\bf J}_{\\rm q}, {\\bf j}_{\\rm q},\n\\boldsymbol\\rho_{\\rm q}],\n\\end{equation}\nwhere $\\rho=\\rho_n+\\rho_p$ as in Appendix D,\n$\\rho \\approx \\rho_\\infty w_{+}$.\nUpon some integration by parts, one finds that\n$\\mathcal{E}$ can be written as a sum of\n the energy density per particle of\nthe non-rotating system\n$\\mathcal{E}(0)$ and its rotational part,\nin line of (\\ref{yrast}).\nWithin the ETF approach, one has from (\\ref{endentau})\n\\bel{endentauSCM}\nE_{\\rm ETF}= \\int \\hbox{d}{\\bf r} \\rho \\mathcal{E}(0) +\n\\frac12 \\Theta_{\\rm ETF}^{({\\rm dyn})}\\,\\omega ^2,\n\\end{equation}\nwhere $\\Theta_{\\rm TF}^{({\\rm dyn})}$ is the ETF dynamical moment of\ninertia for the\nnuclear rotation with\nthe frequency $\\boldsymbol\\omega $. This MI is given in the form:\n\\begin{eqnarray}\\label{ETF05}\n \\Theta_{\\rm ETF}^{({\\rm dyn})} &=& m \\sum_{\\rm q} \n\\int \\hbox{d} {\\bf r} \\; \\left\\{ r_\\perp^2 \\,\n\\rho_{\\rm q}\n - \\left(3 \\pi^2 \\right)^{-2\/3} f^{\\rm eff}_{\\rm q} \\; \\rho^{1\/3}_{\\rm q} \n\\right.\\nonumber\\\\\n&+& \\left.\n \\left[\\frac{\\hbar^2}{2m} + W^{}_0 \\left(\\rho + \\rho_{\\rm q}\\right)\n\\right] \\, \\chi_{\\rm q} \\right\\},\n\\end{eqnarray}\nwhere\n$ r_\\perp$ is the distance of a given point to the rotation axis and\n$W^{}_0$ is the Skyrme-force strength parameter of the spin-orbit\ninteraction \\cite{brguehak}.\n\n\n\n\nOne notices that the Thomas--Fermi term which comes from the orbital motion\nturns out to be the rigid-body moment of inertia. Semiclassical corrections\nof order $\\hbar^2$ come from both the orbital motion\n($\\Theta_{\\rm orb.}^{({\\rm dyn})}$) and from the spin degrees of freedom\n($\\Theta_{\\rm spin}^{({\\rm dyn})}$). The contribution $\\Theta_{\\rm orb.}^{({\\rm dyn})}$\nis\nnegative corresponding to a surface-peaked counter rotation in the rotating\nframe. Such a behavior is to be expected\nfor a N-particle system bound by\nattractive short-range forces (see \\cite{BJ76}).\nThe spin contribution\n$\\Theta_{\\rm spin}^{({\\rm dyn})}$ turns out to be of the {\\it paramagnetic} type,\nthus leading to a positive contribution which corresponds to an alignment of\nthe nuclear spins along the rotation axis. It can\nalso be shown (see \\cite{BBQ93})\nthat the ETF kinematic moment of inertia,\n\\bel{mikin}\n \\Theta_{\\rm ETF}^{({\\rm kin})} = \\displaystyle\n\\frac{\\langle{\\boldsymbol\\ell}+{\\bf s}\\rangle^{\\omega }}{\\omega},\n\\end{equation}\nis identical\nto the ETF dynamical moment of\ninertia presented above.\n\n\n\n\n\nIt is now interesting to study the importance of the Thouless--Valatin\nself-consistency terms. This has accomplished by calculating\nthe moment of inertia\nin the Thomas--Fermi approximation but omitting, this time, the\nThouless--Valatin terms. One then finds \\cite{bartelnp} the following\nexpressions for the dynamical moment of inertia, in what is simply the\nInglis cranking (IC) limit\n\\begin{eqnarray}\\label{ETF06}\n \\Theta_{\\rm IC}^{({\\rm dyn})} &=& m \\sum_{\\rm q} \\int \\hbox{d} {\\bf r} \\;\n\\left[ \\frac{\\rho_{\\rm q}}{\\left(f^{\\rm eff}_{\\rm q}\\right)^2} \\right.\n\\nonumber\\\\\n &+& \\left.\\frac{m B_3}{\\hbar^2} \\, \\rho_{\\rm q} \\, \\rho_{\\bar {\\rm q}}\n \\left(\\frac{1}{f^{\\rm eff}_{\\rm q}} -\n\\frac{1}{f^{\\rm eff}_{\\bar {\\rm q}}} \\right)^{\\!\\!2} \\,\n \\right] r_\\perp^2,\n\\end{eqnarray}\nwhere ${\\bar \\rm q}$ is the {\\it other}\ncharge state (${\\bar \\rm q} \\!\\!=\\!\\! p$ when\n$\\rm q \\!\\!=\\!\\! n$ and vice-versa)\nand $B_3$ is defined through the Skyrme force parameters\n$t^{}_1, t^{}_2, x^{}_1$ and $x^{}_2$ (see \\cite{bartelnp}).\nApart from the corrective term in $\\rho_{\\rm q} \\, \\rho_{\\bar {\\rm q}}$,\none notices\nthat the first term in the expression above, which is the leading term,\n yields, at least for a standard HF-Skyrme\n force where $f^{\\rm eff}_{\\rm q} \\geq 1$, to a smaller moment of\n inertia than the corresponding term in (\\ref{ETF05})\ncontaining the Thouless--Valatin corrections.\nIt is also worth noting that in\nthis approximate case, the kinematic moment of inertia is given by\n\\bel{ETF07}\n \\Theta_{\\rm IC}^{({\\rm kin})}\n = m \\sum_{\\rm q} \\int \\hbox{d} {\\bf r} \\;\n\\frac{\\rho_{\\rm q}}{f^{\\rm eff}_{\\rm q}} \\;\nr_\\perp^2,\n\\end{equation}\nwhich turns out to be quite different from the above given\ndynamical moment\nof inertia, (\\ref{ETF06}),\nobtained in the same limit (Thomas--Fermi limit, omitting the\nThouless--Valatin self-consistency terms).\n\n\nTo investigate the importance of the different contributions to the total\nmoment of inertia, we have performed\nself-consistent ETF calculations up to\norder $\\hbar^4$ for 31 non-rotating nuclei,\nimposing a spherical symmetry,\nand\nusing\nthe SkM$^*$ Skyrme effective nucleon-nucleon interaction \\cite{BQB82}. Such\ncalculations yield variational semiclassical density profiles for neutrons\nand protons \\cite{brguehak} which are then used to calculate the above given\nmoments of inertia. The nuclei included in our calculations are $^{16}$O,\n$^{56}$Ni, $^{90}$Zr, $^{140}$Ce, $^{240}$Pu and three isotopic chains for\nCa ($A \\!=\\! 36-50$), Sn ($A \\!=\\! 100-132$) and Pb ($A \\!=\\! 186-216$).\nThe results of these calculations\nare displayed in figure \\ref{fig11} taken from\n\\cite{bartelnp}.\n\nOne immediately notices the absence of any significant isovector dependence.\nThe good reproduction of the total ETF moment of inertia obtained by the\nThomas--Fermi (rigid-body) value is also quite striking. One finds that the\norbital and spin semiclassical corrections are not small individually but\ncancel each other to a large extent. To illustrate this fact the ETF moments\nobtained by omitting only the spin contribution are also shown on the figure.\nOne thus obtains a reduction of the Thomas--Fermi result that is about 6\\% in\n$^{240}$Pu but as large as 43\\% in $^{16}$O.\n\n\nThe Inglis cranking approach performed at the Thomas--Fermi level underestimates\nthe kinematic moment of inertia by as much as 25\\% and the dynamical moment of\ninertia by about 50\\% in heavy nuclei, demonstrating in this way the importance\nof the Thouless--Valatin self-consistency terms.\n\n\n\nIn \\cite{bartelnp}, a crude estimate of the semiclassical corrections due to orbital and spin degrees of freedom has been made by considering the nucleus as a piece of\nsymmetric nuclear matter (no isovector dependence as already indicated by the\nself-consistent results shown in figure \\ref{fig11} above).\nIt turns out that these\nsemiclassical corrections have an identical $A$ dependence\n($A_{}^{-2\/3}$ relative to the leading order Thomas--Fermi,\ni.e.\\ rigid-body, term)\n\\bel{ETF08}\n \\Theta_{\\rm ETF} = \\Theta^{\\rm (RB)}\n \\left[ 1 + \\left( \\eta_{\\ell} +\n\\eta_s \\right) A^{-2\/3} \\right] \\; .\n\\end{equation}\nA fit of the parameters $\\eta_{\\ell}$ and $\\eta_s$ to\nthe numerical\nresults displayed\nin Fig.\\ \\ref{fig11} yields $\\eta_{\\ell} = -1.94$ and $\\eta_s = 2.63$\ngiving a total (orbital\n+ spin) corrective term of $0.69 \\, A^{-2\/3}$. For a typical rare-earth\nnucleus (A = 170) all this would correspond to a total corrective term equal to\n2.2\\% of the rigid-body value, resulting from a -6.3\\% correction\nfor the orbital\nmotion and a 8.5\\% correction for the spin degree of freedom.\n\n\n\nWhereas in the calculations that lead to figure \\ref{fig11} above,\nspherical symmetry\nwas imposed, fully variational calculations have been performed in\n\\cite{bartelpl}, imposing however the nuclear shapes to be of spheroidal form.\nIn this way, the\nnuclear rotation clearly impacts on the specific form of the\nmatter densities $\\rho^{}_n$ and $\\rho^{}_p$ which, in\nturn, in the framework\nof the ETF approach determine all the other local densities, as explained\nabove.\n\n\nTrying to keep contact with usual shape parametrizations,\nby the standard quadrupole\nparameters $\\beta$ and $\\gamma$ equating the semi-axis lengths of the spheroids\nwith the lengths of a standard quadrupole drop.\n\n\nAs a result,\nfigure \\ref{fig12} shows the evolution of the equilibrium\nsolutions (the ones that minimize the energy for given angular momentum I)\nas a function of I. One clearly observes that at low\nvalues of\nthe angular momentum\n(I in the range between 0 and 50 $\\hbar$) the nuclear drop takes on\nan oblate\nshape, corresponding to increasing values of the quadrupole parameter $\\beta$\nwith increasing I values, but keeping the non-axiality parameter fixed at\n$\\gamma = 60^{\\circ}$. For larger values of the total angular momentum\n(I beyond 55 $\\hbar$), one observes a transition into triaxial shapes, where\nthe nucleus evolves rapidly to more and more elongated shapes. For even\nhigher values of I (I beyond 70 $\\hbar$) the nucleus approaches\nthe fission\ninstability. These results are in excellent qualitative agreement with those\nobtained by Cohen, Plasil and Swiatecki \\cite{CPS74} in a rotating LDM.\n\n\nIt is\namusing to observe\nhere\na {\\it backbending} phenomena at the semiclassical\nlevel when one is plotting,\nas usual, the moment of inertia $\\Theta_{\\rm ETF}$\nvs the rotational angular momentum,\nsee Fig.\\ \\ref{fig13}.\nOne should, however,\ninsist on the fact that this {\\it backbending} has strictly nothing to do with\nthe breaking of a Cooper pair. The rapid increase\nof the moment of inertia\nat about I$ \\,= 60 \\hbar$\nwith a practically constant (or even slightly\ndecreasing)\nrotational frequency $\\omega $ comes simply\nfrom the fact that at such a value of I\n(between I $\\approx$ 60 and I $\\approx$ 70) the nucleus elongates\nsubstantially\nincreasing in this way its deformation and at the same time its\nmoment of inertia.\n\nIt is therefore interesting to notice that the semiclassical ETF\napproach leads to a moment of inertia that is very well approximated\nby its Thomas--Fermi, i.e. rigid-body value. Thouless--Valatin terms which\narise from the self-consistent response of the mean field to the time-odd\npart of the density matrix generated by the cranking piece of the\nHamiltonian are naturally taken care of in this approach. Semiclassical\ncorrections of order $\\hbar^2$ coming from the orbital motion and the spin\ndegree of freedom are not small individually, but compensate each other\nto a large extent. One has, however, to keep in mind that the\nshell and pairing effects, that go beyond the ETF approach, are not included\nin this description. These effects are not only both present, but influence\neach other to a large extent, especially for collective high-spin\nrotations of strongly deformed nuclei, as shown\nin \\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev}.\n\n\n\n\\subsection{MI shell structure and periodic orbits}\n\\label{semshell}\n\nWe shall outlook first the basic points of the POT for the semiclassical\nlevel-density and free-energy shell corrections\n\\cite{strut,fuhi,migdalrev}. We apply then the POT\nfor the derivation of the MI through the rigid-body MI\n(with the shell corrections, see Appendix E)\nin the NLLLA\nrelated to the equilibrium collective rotation\nwith a given frequency $\\omega$ \\cite{mskbPRC2010}. For simplicity,\nwe shall discard the spin and isospin degrees of freedom, in particular,\nthe spin-orbit and asymmetry interaction.\n\nNotice also that from the results presented in\nFigs. \\ref{fig11} and \\ref{fig13} (with the help of\nFig. \\ref{fig12}), one may conclude that the main contribution to\nthe moment of inertia\nof the strongly deformed heavy nuclei can be found within the\nETF approach to the rotational problems as a smooth\nrigid body MI.\n\n\\medskip\n\n\\subsubsection{\nGREEN'S FUNCTION TRAJECTORY EXPANSION}\n\\label{greenfun}\n\nFor the derivations of shell effects \\cite{strut} within the POT\n\\cite{gutz,strutmag,bt,creglitl,sclbook,migdalrev},\nit turns out to be helpful\nto use the coordinate representation of the MI\nthrough the Green's functions $G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon \\right)$\n\\cite{magvvhof,yaf,gzhmagsit,mskbg,mskbPRC2010},\n\\begin{eqnarray}\\label{thetaxdefG}\n\\Theta_{x}&=&\\frac{2 d_s}{\\pi}\\;\\int^{\\infty}_0 \\hbox{d}\\varepsilon\\;\nn({\\varepsilon}) \\int \\hbox{d} {\\bf r}_1 \\int \\hbox{d} {\\bf r}_2 \\; \\ell_{x}({\\bf r}_1)\\; \\ell_{x}({\\bf r}_2)\n\\nonumber\\\\\n&\\times&\n {\\mbox {\\rm Re}} \\left[ G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon \\right) \\right]\n{\\mbox {\\rm Im}} \\left[ G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right) \\right].\n\\end{eqnarray}\nThe Fermi occupation numbers $n(\\varepsilon)$ (\\ref{ocupnumbi})\nare approximately considered at $\\omega =0$ ($\\varepsilon=\\varepsilon_i$).\nIn (\\ref{thetaxdefG}), $\\ell_x({\\bf r}_1)$ and $\\ell_x({\\bf r}_2)$\nare the s.p.\\ angular-momentum projections onto the perpendicular rotation\n$x$ axis at the spatial points ${\\bf r}_1$ and ${\\bf r}_2$, respectively.\nWith the usual energy-spectral representation for the one-body\nGreen's function $G$ in the mean-field approximation, one finds\nthe standard cranking model expression, which however includes\nthe diagonal matrix elements of the operator $\\ell_x$.\nIn this sense, equation (\\ref{thetaxdefG}) looks more general beyond the\nstandard perturbation approximation,\nsee \\cite{mskbPRC2010}.\nMoreover, the quantum criterion of the application\nof this standard cranking model approximation, which is a smallness\nof the cranking field perturbation $h^\\omega _{\\rm CF}$ in\n(\\ref{raussian}) as\ncompared to the distance between\nthe neighboring states of the non-perturbative spectrum, becomes weaker\nin the semiclassical approach, see more comments below in relation to\n\\cite{belyaevfirst,belyaevbif}.\n\n\nFor the MI calculations by (\\ref{thetaxdefG}),\nthrough the Green's function $G$, one may use the\nsemiclassical Gutzwiller trajectory expansion \\cite{gutz} extended to\ncontinuous symmetry \\cite{strutmag,smod,magosc,creglitl,magkolstr,sclbook}\nand symmetry breaking \\cite{spheroidptp,maf,sclbook,migdalrev} problems,\n\\bel{GRdefsem}\nG\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)=\n\\sum_{\\rm CT} G_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right),\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{Gct}\n&&G_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)=\n\\mathcal{A}_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right) \\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar}{\\rm S}_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)\n - \\frac{i \\pi}{2}\\sigma_{{}_{\\! {\\rm CT}}}-\ni\\phi_{\\rm d}\\right].\n\\end{eqnarray}\nThe sum runs over all isolated\n classical\ntrajectories (CTs) or their families inside the\npotential well $V({\\bf r})$ which, for a given energy $\\varepsilon$, connect the two\nspatial points ${\\bf r}_1$ and ${\\bf r}_2$.\nHere ${\\rm S}_{\\rm CT}$ is the classical action along such a CT,\nand $\\sigma_{{}_{\\! {\\rm CT}}}$ denotes the phase associated with the Maslov index\nthrough the number of caustic and turning points along the path CT,\n$\\phi_{\\rm d}$\nis the constant phase depending on the dimension\nof the problem \\cite{sclbook,strutmag,maf,migdalrev}.\nThe amplitudes $\\mathcal{A}_{\\rm CT}$ of the Green's function depend on the\nclassical stability factors and trajectory degeneracy, due to the symmetries\nof that potential \\cite{sclbook,strutmag,smod,spheroidptp,maf}.\nFor the case of the isolated CTs\n\\cite{gutz,sclbook},\n one has the explicit semiclassical expression for the amplitudes\nthrough the stability characteristics of classical dynamics,\n\\bel{Agutz}\n\\mathcal{A}_{\\rm CT}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)\n=-\\frac{1}{2\\pi\\hbar^2}\\,\\sqrt{\\Big|\\mathcal{J}_{\\rm CT}({\\bf\np}_{1},t_{{}_{\\! {\\rm CT}}};{\\bf r}_{2},\\varepsilon)\\Big|}.\n\\end{equation}\nHere, $\\mathcal{J}_{\\rm CT}({\\bf p}_{1},t_{{}_{\\! {\\rm CT}}};{\\bf r}_{2},\\varepsilon)$\nis the Jacobian for the transformation between the two sets of variables\n${\\bf p}_{1},t_{{}_{\\! {\\rm CT}}}$ and ${\\bf r}_{2},\\varepsilon$; ${\\bf p}_1$ and $t_{{}_{\\! CT}}$ are the\ninitial momentum and time of motion of the particle along a\nCT, $t_{{}_{\\! {\\rm CT}}}=\\partial S_{\\rm CT}\/\\partial\\varepsilon$ ,\n${\\bf r}_2$ and $\\varepsilon$ are its final coordinate and energy.\nIn more general case, if the mean field Hamiltonian $h$ obeys a\nhigher symmetry like that of spherical or harmonic-oscillator\npotentials with rational ratios of frequencies,\none has to use other expressions for the amplitude\n$\\mathcal{A}_{\\rm CT}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)$ for close trajectories\nof a finite action (with reflection from the potential boundary),\ntaking into account such symmetries.\nThey account for an enhancement in $\\hbar$ owing to their classical\ndegeneracy (see \\cite{strutmag,sclbook,maf,migdalrev} and the discussion\nin subsection below). In the case of\nthe bifurcation of POs, generated by a symmetry-breaking, one\nmay use the ISPM\n\\cite{spheroidptp,maf}, especially\nfor superdeformed shapes of the potential. Some examples\nof the specific amplitudes for the degenerate families of\nclosed POs in the harmonic oscillator (HO) potential are given in \nAppendix E of\n\\cite{mskbPRC2010}.\nNote that (\\ref{Agutz}) can be applied for any potential wells for the\ncontributions of closed and\nnon-closed trajectories which can be considered as isolated\n(no PO families) ones for the given end points ${\\bf r}_1$ and ${\\bf r}_2$.\n\n\nAmong all of CTs in (\\ref{GRdefsem}),\nwe may single out ${\\rm CT}_{0} $\nwhich connects directly ${\\bf r}_1 $ and ${\\bf r}_2 $ without\nintermediate turning points, see Fig.\\ \\ref{fig14}.\nIt is associated with the component\n$G_{{\\rm CT}_0}$ of the sum (\\ref{GRdefsem}) for the semiclassical\nGreen's function.\nTherefore, for the Green's function $G({\\bf r}_1,{\\bf r}_2;\\varepsilon)$ (\\ref{GRdefsem}),\none has then a separation,\n\\bel{Ggsplit}\nG = G_{CT_{0}} + G_{1} \\approx G_0 + G_1.\n\\end{equation}\nIn the NLLLA \\cite{gzhmagsit,gzhmagsit,mskbPRC2010},\n \\begin{equation}\\label{nllla}\ns_{{}_{\\! 12}} \\ll \\hbar\/p_{{}_{\\! {\\rm F}}},\n\\end{equation}\nthe first term\n$G_{CT_{0}}$ of the splitting in the middle of (\\ref{Ggsplit}) is given\nby\n\\begin{equation}\\label{G0}\nG_{CT_{0}} \\approx G_0(s_{{}_{\\! 12}},p)=\n-\\frac{m}{2 \\pi \\hbar^2 s_{{}_{\\! 12}}} \\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} s_{{}_{\\! 12}} \\, p\\left({\\bf r}\\right) \\right],\n\\end{equation}\nwhere $p\\left({\\bf r}\\right) = \\sqrt{2 m [\\varepsilon - V({\\bf r})]\\,}~$, $~V({\\bf r})~$\nis a mean nuclear potential,\n\\bel{srvar}\ns_{{}_{\\! 12}}=\\left|{\\bf r}_2-{\\bf r}_1\\right|,\\qquad {\\bf r}=\\left({\\bf r}_1+{\\bf r}_2\\right)\/2,\n\\end{equation}\n$p=|{\\bf p}|$, ${\\bf p}=({\\bf p}_1+{\\bf p}_2)\/2$.\nThe second term $G_1$ in (\\ref{Ggsplit})\nis the fluctuating part of the Green's function\n(\\ref{GRdefsem}) determined by all other trajectories\n${\\rm CT}_1 \\neq {\\rm CT}_{0}$ in the sum\n(\\ref{GRdefsem}) with reflection points at the potential surface\n(see one of such trajectories ${\\rm CT}_1$ in Fig.\\ \\ref{fig14}),\n\\bel{Gosc}\nG_{1}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)=\n\\sum_{{\\rm CT}_1}\nG_{{\\rm CT}_1}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right),\n\\end{equation}\nwhere $G_{{\\rm CT}_1}$ is the Green's function component (\\ref{Gct}) taken at\nthe ${\\rm CT} \\neq {\\rm CT}_0$, i.e., ${\\rm CT}_1$.\n\n\n\\medskip\n\n\\subsubsection{\nLEVEL-DENSITY AND ENERGY SHELL CORRECTIONS}\n\\label{general}\n\nThe\nlevel density, $g(\\varepsilon)=\\sum_i\\delta(\\varepsilon-\\varepsilon_i)$, where\n$\\varepsilon_i$ is the quantum spectrum, is identically\nexpressed in terms of the Green's\nfunction $G$ as\n\\bel{totdensgen}\ng(\\varepsilon)=- \\frac{1}{\\pi}\\;{\\mbox {\\rm Im}}\n\\int \\hbox{d} {\\bf r}\\; \\left[G({{\\bf r}}_1,{{\\bf r}}_2;\\varepsilon)\n\\right]_{{\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}}.\n\\end{equation}\nAccording to (\\ref{Ggsplit}), this level density\ncan be presented semiclassically\nas a sum of the smooth and oscillating components\n\\cite{gutz,strutmag,sclbook,migdalrev},\n\\bel{totdensscl}\ng_{\\rm scl}(\\varepsilon)=g_{{}_{\\! {\\rm ETF}}}(\\varepsilon) + \\delta g_{\\rm scl}(\\varepsilon),\n\\end{equation}\nwhere\n$g_{{}_{\\! {\\rm ETF}}}(\\varepsilon)$ is given by the ETF approach related to the\ncomponent $G_0$ in (\\ref{Ggsplit}) in the NLLLA (\\ref{nllla})\n${\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}$\n\\cite{brguehak,sclbook,gzhmagfed,migdalrev}. The local part of\n$g_{{}_{\\! {\\rm ETF}}}(\\varepsilon)$ is the main simplest Thomas--Fermi (TF) level density\n$g_{{}_{\\! {\\rm TF}}}(\\varepsilon)$ \\cite{sclbook}.\nThe second oscillating term $\\delta g_{\\rm scl}(\\varepsilon)$\nof the level density (\\ref{totdensscl}) corresponds to the fluctuating\n$G_1$ in the sum\n(\\ref{Ggsplit}) for the Green's function $G$ near\nthe Fermi surface. The stationary phase conditions for the\n(standard or improved)\nSPM evaluation\nof the integral taken from $G_1$ over the spatial coordinates ${\\bf r}$\nare the PO equations.\nAs the result, one arrives at the sum over PO sum for this\noscillating level density\n\\cite{gutz,strutmag,bt,sclbook},\n\\begin{eqnarray}\\label{dlevdenscl}\n&&\\delta g_{\\rm scl}(\\varepsilon)\n= {\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)\\quad \\mbox{with}\\qquad\n\\nonumber\\\\\n&&\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)= \\mathcal{B}_{\\rm PO}\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar}\\; S_{\\rm PO}(\\varepsilon) -\ni \\frac{\\pi}{2}\\;\\sigma_{{}_{\\! {\\rm PO}}} -i \\phi_{d}\\right],\\qquad\n\\end{eqnarray}\nwhere $B_{\\rm PO}$ is an amplitude of the oscillating PO terms, see\n\\cite{gutz,strutmag,bt,creglitl,sclbook,migdalrev,spheroidptp}.\nThe above sum runs over the isolated POs and, in the case of\ndegeneracies owing to the symmetries of\n a given potential well, over all families of POs.\n $\\mathcal{B}_{\\rm PO}$ is the oscillation amplitude depending on the\n stability factors, $S_{\\rm PO}(\\varepsilon)$ the action integral along a given\n PO, and $\\sigma_{{}_{\\! {\\rm PO}}}$ is\nthe Maslov phase associated with the turning and\n caustic points along the PO, see\n\\cite{sclbook,maf,migdalrev} for the detailed explanations.\n\nThe semiclassical free-energy shell corrections,\n$\\delta F_{\\rm scl}$ at finite temperature\n($T \\siml \\hbar \\Omega \\ll \\varepsilon_{{}_{\\! {\\rm {\\rm F}}}}$),\ncan be expressed through the PO components of the energy shell corrections\n$\\delta U_{\\rm scl}$ \\cite{strutmag,sclbook,migdalrev}\n(see Appendix E.1),\n\\begin{eqnarray}\\label{descl}\n\\delta U_{\\rm scl} &=& {\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta U_{\\rm PO},\n\\nonumber\\\\\n\\delta U_{\\rm PO}&=&d_s \\; \\frac{\\hbar^2}{t^{2}_{\\rm PO}} \\;\n\\delta g_{{}_{\\! {\\rm PO}}}(\\mu),\n\\end{eqnarray}\nwith the exponentially decreasing temperature-dependent factor\n\\cite{strutmag,kolmagstr,richter,fraukolmagsan,sclbook,mskbPRC2010},\n\\begin{eqnarray}\\label{fpotau}\n\\delta F_{\\rm scl}&=&{\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta F_{\\rm PO} \\nonumber\\\\\n&=&\n{\\mbox {\\rm Re}} \\sum_{\\rm PO}\\frac{\\pi t_{{}_{\\! {\\rm PO}}}T\/\\hbar}{\n\\hbox{sinh}\\left(\\pi t_{{}_{\\! {\\rm PO}}}T\/\\hbar\\right)}\\;\\delta U_{\\rm PO}.\n\\end{eqnarray}\nFinally through (\\ref{descl}), the shell corrections\n$\\delta F_{\\rm scl}$ and $\\delta U_{\\rm scl}$ are determined by\nthe PO level-density shell-correction components\n$\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ of (\\ref{dlevdenscl}) at the chemical potential,\n$\\varepsilon=\\mu \\approx \\varepsilon_{{}_{\\! {\\rm F}}}$.\nIn (\\ref{descl}), one has the additional factor,\n$\\propto 1\/t^2_{\\rm PO}$,\nwhich yields the convergence of the PO sum (without averaging of\n$\\delta g(\\varepsilon)$\nover the s.p.\\ spectrum), $t_{{}_{\\! {\\rm PO}}}$ is the time of motion along\nthe PO (PO period). Another (exponential) convergence of\n$\\delta F_{\\rm scl}$\n(\\ref{fpotau}) with increasing the period $t_{{}_{\\! {\\rm PO}}}$ and temperature\n$T$ is giving by the temperature-dependent factor in front of\n$\\delta U_{\\rm PO}$.\n\n\n\\subsubsection{FROM CRANKING MODEL TO THE RIGID BODY ROTATION}\n\\label{relperprigrot}\n\nSubstituting (\\ref{Ggsplit}) into (\\ref{thetaxdefG}), one has a sum\nof several terms,\n\\bel{thetaxsum}\n\\Theta_{x\\; {\\rm scl}} \\approx \\Theta_x^{00} + \\Theta_x^{01} +\n\\Theta_x^{10} + \\Theta_x^{11},\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{thetaxnnp}\n\\Theta_x^{n n'}&=&\\frac{2 d_s}{\\pi}\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r}_1 \\int \\hbox{d} {\\bf r}_2\\;\n\\ell_x\\left({\\bf r}_1\\right)\\;\\ell_x\\left({\\bf r}_2\\right)\n\\nonumber\\\\\n&\\times&\n{\\mbox {\\rm Re}} \\left[G_{n}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right]\\;\n{\\mbox {\\rm Im}} \\left[G_{n'}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right].\n\\end{eqnarray}\nIndexes $n $ and $n'$ run independently\nthe two integers 0 and 1. As shown in Appendix E.2a,\nthe main smooth part of the semiclassical\nMI $\\Theta_{x\\; {\\rm scl}}$ (\\ref{thetaxsum})\nis associated with the TF (ETF)\nrigid-body component through the first term $\\Theta_x^{00}$\naveraged over the phase-space variables; see section \\ref{etfmi}, also\n\\cite{bartelpl,gzhmagsit,mskbPRC2010}, and previous publications\n\\cite{bloch,amadobruekner,rockmore,jenningsbhbr}.\nThe statistical averaging over phase space coordinates removes\nthe non-local long-length correlations.\nThe $\\hbar$ corrections of the smooth ETF approach to the TF\napproximation were obtained in\n\\cite{jenningsbhbr,bartelnp,bartelpl},\nsee Sect. \\ref{etfmi} for the review of these\nworks.\n\n\n Using the transformation of the coordinates\n${\\bf r}_1$ and ${\\bf r}_2$ to the center-of-mass and relative ones\n${\\bf r}$ and ${\\bf s}_{{}_{\\! 12}}$,\n\\bel{newcoord}\n{\\bf r}=({\\bf r}_1+{\\bf r}_2)\/2 \\qquad {\\rm and}\\qquad {\\bf s}_{{}_{\\! 12}}={\\bf r}_2-{\\bf r}_1,\n\\end{equation}\nin (\\ref{thetaxnnp}), respectively, one simplifies much the calculations of\nthe oscillating terms,\n$\n\\Theta_x^{01} +\n\\Theta_x^{10} + \\Theta_x^{11}~$.\nIn this way, one finds that the shell\ncomponent $\\delta \\Theta_x^{01}$ of $\\Theta_x^{01}$\n[see (\\ref{thetaxnnp}) at $n=0$\nand $n'=1$]\nis dominating in the MI shell correction\n$\\delta \\Theta_{x\\; {\\rm scl}}$ within the NLLLA (\\ref{nllla}),\nsee Appendix E.2b.\nIndeed, in this approximation, substituting the\ncomponents, $G_0$ and $G_1$, of the Green's function (\\ref{Ggsplit})\n[see (\\ref{G0}) for $G_0$] into\n(\\ref{thetaxnnp}) for $\\Theta_x^{01}$,\nand using the averaging over the phase-space\nvariables in the fluctuating (shell) part $\\delta\n\\Theta_{x}$ of $\\Theta_{x}$,\none results in the relationship for the\ncorresponding shell corrections (see Appendix E.2b):\n\\bel{dmilocapproach}\n\\delta \\Theta_{x\\; {\\rm scl}} \\approx\n\\delta \\Theta_{x}^{01} \\approx \\delta \\Theta_{x}^{\\rm (RB)}.\n\\end{equation}\nHere, $\\delta \\Theta_x^{\\rm (RB)}$ is\nthe shell correction to the rigid-body\nMI $\\Theta_x^{\\rm (RB)}$,\n which is related to the semiclassical particle-density\n$\\rho({\\bf r})$ through\n\\bel{rigbodmomgen}\n\\Theta_x^{\\rm (RB)} =\nm \\int \\hbox{d} {\\bf r}\\;r_{\\perp x}^2\\; \\rho\\left({\\bf r}\\right),\n\\end{equation}\nwith\n\\bel{rperpcoord}\nr_{\\perp x}^2=y^2+z^2.\n\\end{equation}\nThe particle density $\\rho({\\bf r})$, and therefore, the MI\n(\\ref{rigbodmomgen}), can be expressed in terms of the Green's\nfunction $G$,\n\\bel{denpartgen}\n\\rho({\\bf r}) = -\\frac{d_s}{\\pi}\\; {\\mbox {\\rm Im}} \\int \\hbox{d} \\varepsilon\\; n\\left(\\varepsilon\\right)\\;\n\\left[G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right]_{{\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}}.\n\\end{equation}\nWith the splitting of the Green's function (\\ref{Ggsplit}),\none obtains the semiclassical sum of the smooth and oscillating (shell)\ncomponents \\cite{strutmagvvizv1986,brackrocciaIJMPE2010}:\n\\bel{denpartscl}\n\\rho({\\bf r})\\approx \\rho_{\\rm scl}({\\bf r})=\\rho_{{}_{\\! {\\rm ETF}}}({\\bf r}) +\n\\delta \\rho_{\\rm scl}({\\bf r}).\n\\end{equation}\nThe integration over $\\varepsilon$ in (\\ref{denpartgen}) is performed\nover the whole s.p.\\ energy spectrum.\nFor the Green's function $G$, we applied the semiclassical expansion\n(\\ref{GRdefsem}) in terms of the sum (\\ref{Ggsplit}) of CTs in the\nlast equation for the semiclassical particle density $\\rho_{\\rm scl}({\\bf r})$.\nThe first term in (\\ref{denpartscl}) is the (extended) Thomas--Fermi component\n(see Appendix E.2a).\nSubstituting the particle density splitting (\\ref{denpartscl}) into\n(\\ref{rigbodmomgen}), one has the corresponding semiclassical\nexpression\nof the rigid-body MI,\n\\bel{rigmomsplit}\n\\Theta_x^{\\rm (RB)} \\approx \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)} =\n\\Theta_{x\\; {\\rm ETF}}^{\\rm (RB)} +\n\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)}.\n\\end{equation}\nWe introduced the shell corrections $\\delta \\rho$\n(see \\cite{brackrocciaIJMPE2010}) to the particle\ndensity $\\rho$ and $\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}$ to the rigid-body MI\n$\\Theta_x^{\\rm (RB)}$, and their semiclassical counterparts,\n\\bel{drigbodmomgen}\n\\delta \\Theta_{x}^{\\rm (RB)} \\approx\n\\delta \\Theta_{x \\; {\\rm scl}}^{\\rm (RB)} =\nm \\int \\hbox{d} {\\bf r}\\;r_{\\perp x}^2\\; \\delta \\rho_{\\rm scl}\\left({\\bf r}\\right),\n\\end{equation}\nwhere\n\\bel{ddenpart}\n\\delta \\rho_{\\rm scl}\\left({\\bf r}\\right)=-\\frac{d_s}{\\pi}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1}\\int \\hbox{d} \\varepsilon\\; n\\left(\\varepsilon\\right)\\;\nG_{{\\rm CCT}_1}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right),\n\\end{equation}\nwhere $G_{{\\rm CCT}_1}$ is given by (\\ref{Gct}) with ${\\rm CT}$ being the\nclosed ${\\rm CT}_1$, i.e.,\n${\\rm CCT}_1$ (${\\bf r}_1\\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}$).\nWith the smooth (extended) TF MI component (\\ref{TFrig}),\nsee also the section \\ref{etfmi}, the equation (\\ref{dmilocapproach})\nyields semiclassically\n\\bel{milocapproach}\n\\Theta_{x\\; {\\rm scl}} \\approx \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)},\n\\end{equation}\nthat is in agreement with the adiabatic picture\nof the statistically equilibrium rotation \\cite{mskbPRC2010}.\nNote that the non-adiabatic MI at arbitrary rotation\nfrequencies for the HO mean field by Zelevinsky \\cite{zelev} was extended\nto the finite\ntemperatures in \\cite{mskbPRC2010}.\n\n\nWe emphasize that due to\nan averaging over the phase space variables, one survives with\nthe NLLLA.\nNote also that the classical angular-momentum projection (\\ref{l2}) in the\nrotating body-fixed coordinate system is caused by the global rotation\nwith a given frequency $\\omega$ rather than by\nthe motion of particles along the trajectories inside the nucleus\nwith respect to this system,\nconsidered usually in the cranking model.\nAccording to the time-reversible symmetry of the Routhian,\nthe particles are, indeed, moving in the non-rotating coordinate system\nalong these trajectories in both opposite directions.\nTheir contributions to the total angular momentum of the nucleus turns out to\nbe zero. Performing then\nthe integration over ${\\bf s}$ in (\\ref{dthetax01new})\n in the spherical coordinate system,\none obtains the rigid-body shell correction\n$\\delta \\Theta_x^{\\rm (RB)}$ in the NLLLA\nas explained in Appendix E.2.\nNote that\nthe cranking model for the nuclear rotation\nimplies that the correlation (non-local) corrections to\n(\\ref{dthetax01new}) and (\\ref{rigbodmomgen}) should be\nsmall enough with respect to the main rigid-body shell component\n$\\delta \\Theta_x^{\\rm (RB)}$ to be neglected within the adiabatic picture\nof separation of the global rotation of the Fermi system\nfrom its vibration and then,\nboth from the internal motion of particles.\nOther contributions,\nexcept for a smooth rigid-body part coming from $\\Theta_x^{00}$, like\n$\\Theta^{10}_{x}$ and $\\Theta^{11}_{x}$, as\nreferred to the fluctuation (non-local) correction\nto the rigid body MI are found semiclassically to be negligibly small\nin the NLLLA\ndue to the averaging\nover phase-space variables, see Appendix E.2b.\nIn particular, for the HO Hamiltonian, it was shown that there is almost\nno contribution of the\n$\\delta \\Theta_x^{11}$ at leading order in $\\hbar$ in \\cite{mskbPRC2010}.\nThus, with the semiclassical precision, from the\nadiabatic cranking model expression (\\ref{thetaxdefG}) we come to the\nMI of the statistically equilibrium rotation (\\ref{milocapproach}),\nwhich must be the rigid-body MI, according to\nthe general theorem of the statistical physics. This is in agreement\nwith the ETF approach of section \\ref{etfmi}.\nOur semiclassical derivations, valid for the rotation frequencies\n$\\hbar \\omega \\ll \\hbar \\Omega$, are beyond the quantum criterion\nof the application of the standard\n2nd order perturbation approach within the cranking model where\n$\\hbar \\omega $ is small as compared to the distance between the\nneighboring levels of quantum spectra. We point out that this weakness\nof the perturbation theory criterion is similar to that with the statistical\naveraging in the heated Fermi systems and with accounting for the pairing\ncorrelations \\cite{belyaevhighspin,belyaevbif}, where the role of the\ndistance between the quantum neighboring energy levels plays the\ntemperature and the pairing gap, as distance between gross shells\n$\\hbar \\Omega$ (\\ref{hom}) in the POT \\cite{strutmag}, respectively.\n\n\n\\medskip\n\n\\subsubsection{\nSHELL CORRECTIONS TO THE RIGID-BODY MI}\n\\label{SCperprigrot}\n\nUsing (\\ref{ddenpart}) for calculations of\nthe MI rigid-body shell correction\n$\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)}$ (\\ref{drigbodmomgen}),\none may exchange the order of integrations over the coordinate ${\\bf r}$\nand energy $\\varepsilon$.\nBy making use also of the semiclassical\ntrajectory expansion (\\ref{GRdefsem}) for the oscillating Green's\nfunction component\n$G_1\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)$ of the sum (\\ref{Ggsplit}), one finds\n\\begin{eqnarray}\\label{dTxrigSCL}\n&&\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)} =\n-\\frac{m d_s}{\\pi}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} \\varepsilon\\;\nn(\\varepsilon) \\qquad \\nonumber\\\\\n&\\times& \\int \\hbox{d} {\\bf r}\\;\\left\\{r_{\\perp x }^2 \\mathcal{A}\\left({\\bf r},{\\bf r};\\varepsilon\\right)\n\\right.\\qquad\\nonumber\\\\\n&\\times& \\left.\n\\hbox{exp}\\left[\\frac{i}{\\hbar}\\;\n S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right) -\n\\frac{i \\pi}{2}\\sigma - i\\phi_{d}\\right]\n\\right\\}_{{\\rm CCT}_1}.\\qquad\n\\end{eqnarray}\nAs usually, with the semiclassical precision, we evaluate\nthe spatial integral by the SPM extended to continuous symmetries\n\\cite{strutmag,sclbook,migdalrev} and the\nbifurcation phenomena (ISPM)\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar}.\nThe SPM (ISPM) condition writes\n\\begin{eqnarray}\\label{spmcond}\n&&\\left[\\frac{\\partial S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)}{\\partial {\\bf r}_1}\n+ \\frac{\\partial S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)}{\\partial {\\bf r}_2}\n \\right]^{\\ast}_{{\\rm CCT}_1}\n\\nonumber\\\\\n &\\equiv&\n\\left(-{\\bf p}_1+{\\bf p}_2\\right)^{\\ast}_{{\\rm CCT}_1},\n\\end{eqnarray}\nwhere the asterisk means the SPM value of the spatial coordinates\nand momenta,\n$~{\\bf r}_j={\\bf r}_j^{\\ast}~$ and $~{\\bf p}_j={\\bf p}_j^{\\ast}~$ ($~j=1,2$)\nat the closed ${\\rm CT}_1$s\nin the phase space,\n${\\bf r}^{\\ast}_1={\\bf r}^{\\ast}_2$ and ${\\bf p}^{\\ast}_1={\\bf p}^{\\ast}_2$. Thus, with the\nstandard relations for the canonical variables by using the action as a\ngenerating function, one arrives\nat the PO condition on right of (\\ref{spmcond}).\nWithin the simplest ISPM\n\\cite{spheroidptp,maf,migdalrev,magvlasar},\nthe other smooth factors $r_{\\perp x}^2$ and\n$\\mathcal{A}_{{\\rm CT}_1}\\left({\\bf r},{\\bf r},\\varepsilon\\right)$\nof the integrand in (\\ref{dTxrigSCL})\n can be taken off the\nintegral over ${\\bf r}$ at these stationary points. Assuming that the\nquantum averages\n$\\langle (y^2+z^2)^2 \\rangle\/\\varepsilon$ are smooth enough functions of $\\varepsilon$\nas compared to other factors, for instance, $\\delta n$,\none may take them approximately also off the integral\nover $\\varepsilon$ at the chemical potential, $\\varepsilon=\\mu$. For example,\nfor the HO potential (see \\cite{mskbPRC2010}),\nthey are simply exact constants.\nTherefore, the main contribution into the integral\nin (\\ref{dTxrigSCL}) is coming from\nthe PO stationary-phase points, determined by (\\ref{spmcond}),\n as for calculations of the\nlevel-density shell corrections $\\delta g_{\\rm scl}$ (\\ref{dlevdenscl})\n\\cite{sclbook,strutmag,migdalrev,mskbPRC2010}.\nThe SPM condition (\\ref{spmcond}) is\n identity for any stationary point of the classically accessible\nspatial region for a particle motion filled by PO families\nin the case of their high\ndegeneracy\n$\\mathcal{K}\\geq 3$.\nFor instance, it is the case for the contribution of\nthe three dimensional (3D) orbits in the axially symmetric HO-potential well\nwith commensurable frequencies, $\\omega _x=\\omega _y=\\omega _\\perp$ and $\\omega _z$\n\\cite{magosc,mskbPRC2010}.\nThe stationary points occupy\nsome spatial subspace for a smaller degeneracy $\\mathcal{K}$.\nIn the latter case of the equatorial orbits (EQs) ($\\mathcal{K}=2$)\nin this HO potential well,\nthe SPM condition is\nidentity in the equatorial plane $z=0$.\nFollowing\nsimilar derivations of the oscillating component $\\delta g_{\\rm scl}$\n(\\ref{dlevdenscl})\nof the level density $g_{\\rm scl}(\\varepsilon)$ (\\ref{totdensscl})\nand free-energy shell correction $\\delta F_{\\rm scl}$ (\\ref{fpotau}),\none expands the\nsmooth amplitudes and action phases of the\nMI shell corrections\n$\\delta \\Theta_{\\kappa {\\rm scl}}^{\\rm rig}$ (\\ref{dTxrigSCL}) up to the\nfirst nonzero terms\n(see Appendix C of \\cite{mskbPRC2010} and Appendix E.2\nhere).\nFinally, from (\\ref{dTxrigSCL}), one obtains \\cite{mskbPRC2010}\n\\bel{dTxrigSCLgen}\n\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}=\\frac{m}{\\mu}\\;{\\mbox {\\rm Re}} \\sum_{\\rm PO}\n\\langle r_{\\perp x}^2 \\rangle_{{}_{\\! {\\rm PO},\\mu}}\\;\\delta F_{\\rm PO},\n\\end{equation}\nwhere $\\langle r_{\\perp x}^2 \\rangle_{{\\rm PO},\\mu}$\nis the average given by\n\\bel{avsqcoor}\n\\langle r_{\\perp x}^2 \\rangle_{{\\rm PO},\\varepsilon} =\n\\frac{\\int \\hbox{d} {\\bf r}\\;\\mathcal{A}_{\\rm PO}\\left({\\bf r},{\\bf r};\\varepsilon\\right)\\;r_{\\perp x}^2\n }{\\int \\hbox{d} {\\bf r}\\;\\mathcal{A}_{\\rm PO}\\left({\\bf r},{\\bf r};\\varepsilon\\right)}\n\\end{equation}\nat $\\varepsilon=\\mu$, $\\mathcal{A}_{\\rm PO}({\\bf r},{\\bf r};\\varepsilon)$ are the\nGreen's function amplitudes\nfor a closed ${\\rm CT}_1$ in the phase space, i.e., PO.\nIntegration over ${\\bf r}$ is performed over the classically\naccessible region of the spatial coordinates.\nSemiclassical expression (\\ref{dTxrigSCLgen})\nis general for any potential well.\nShorter POs are\ndominating in the PO sum (\\ref{dTxrigSCLgen})\n\\cite{sclbook,strutmag,kolmagstr,mskbPRC2010,migdalrev},\nsee\n(\\ref{fpotau}), (\\ref{descl}). Therefore,\naccording to (\\ref{fpotau}) for $\\delta F_{\\rm scl}$,\nwe obtain approximately the relation\n\\bel{dTxrigSCLgen1}\n\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}\\approx\n\\frac{m}{\\mu}\\;\n\\langle r_{\\perp x}^{2}\\rangle_{\\mu}\\;\n\\delta F_{\\rm scl},\n\\end{equation}\nwhere $\\langle r_{\\perp x}^2 \\rangle_{\\mu}$ is an\naverage value of the quantity (\\ref{avsqcoor}),\nindependent of the specific PO, at $\\varepsilon=\\mu$ over short\ndominating POs.\n\n\n\nFor the axially symmetric\n HO potential well with the commensurable frequencies $\\omega _\\perp$\nand $\\omega _z$, as the simplest example,\nthe integration in (\\ref{avsqcoor}) over ${\\bf r}$\nfor the 3D contribution means over the 3D volume occupied by\nthe 3D families of orbits.\nFor the EQ component\nthe integral is taken\nover the 2D spatial region filled by the EQ families\nin the equatorial ($z=0$) plane \\cite{mskbPRC2010}.\nIn the incommensurable-frequency\ncase (irrational $\\omega _\\perp\/\\omega _z$), one has the only EQ-orbit contributions.\nThe average (\\ref{avsqcoor}) can be easily calculated\nby using the Green's function amplitudes $\\mathcal{A}_{\\rm PO}$\nfor 3D and\nfor EQ orbits, which are given in \\cite{magosc,mskbPRC2010}.\nFinally, for the considered HO potential, one may arrive at\n\\bel{dthetaxdehoscl}\n\\delta \\Theta_{x\\; {\\rm scl}} \\approx \\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}=\\frac{1 +\n\\eta_{\\rm HO}^2}{3 \\omega _\\perp^2}\\;\\delta F_{\\rm scl},\n\\end{equation}\nwhere $\\delta F_{\\rm scl}$ is the semiclassical PO sum\n(\\ref{fpotau}), (\\ref{descl})\nfor the semiclassical free-energy shell-corrections,\n$\\eta_{{}_{\\!{\\rm HO}}}=\\omega _\\perp\/\\omega _z$ is the deformation parameter.\nFor the parallel (alignment) rotations around the symmetry axis,\n one finds similar relations of the MI through the rigid-body MI to\nthe free-energy shell corrections.\nMoreover, one has such relations\nfor the smooth TF parts, in particular for the HO case, see Appendices E.2.1\nhere and D1 in\n\\cite{mskbPRC2010}.\nThus, for the total\nmoment $\\Theta_x$ [see (\\ref{thetaxsum})], one may prove semiclassically\nwithin the POT, up to the same $\\hbar$ corrections in a\nsmooth TF part, that\nthe shell MI and free-energy shell corrections are approximately\nproportional, in particular exactly\n for that\nHO Hamiltonian \\cite{mskbPRC2010}:\n\\bel{thetaxhoscl}\n\\Theta_{x\\; {\\rm scl}}=\\frac{1 + \\eta^2}{3 \\omega _\\perp^2}\\;F_{\\rm scl},\\qquad\nF_{\\rm scl}=F_{{\\rm ETF}}+\\delta F_{\\rm scl}.\n\\end{equation}\nWe emphasize that the POT expressions (\\ref{dthetaxdehoscl})\nfor $\\delta \\Theta_{x\\; {\\rm scl}}$ and\n(\\ref{thetaxhoscl})\nof $\\Theta_{x\\; {\\rm scl}}$ were derived\nwithout a direct use of the statistically equilibrium rotation\ncondition \\cite{bohrmot,mskbPRC2010}.\n\n\nSubstituting the semiclassical PO expansion\n(\\ref{descl}) for the free-energy shell correction\n$\\delta F_{{\\rm scl}}$ (\\ref{fpotau}) (after \\cite{spheroidptp})\nfor 3D orbit families and\nfor EQ POs into (\\ref{dthetaxdehoscl}),\none arrives finally at the explicit POT expressions for the MI\nshell corrections $\\delta \\Theta_x$\nin terms of the characteristics of the classical\nPOs. For the mean field with the spheroidal shapes and sharp edges\n(spheroid cavity), these derivations can be performed similarly\nas for the HO Hamiltonian in \\cite{mskbPRC2010}\nbut with accounting\nfor the specific PO degeneracies.\nNote that\nthe parallel, $\\delta \\Theta_z$,\nand perpendicular, $\\delta \\Theta_x$,\nMI shell components are expressed through the 3D and EQ POs\nthrough the free-energy shell correction which contains generally\nspeaking both them for the deformations larger the bifurcation ones.\nThe dominating\n contributions\nof one of these families or coexistence of both together\ndepend on the surface deformation parameter (semi-axis ratio\nof spheroid). For the critical deformations and on right of them, one observes\nthe significant enhancement of the MI shell corrections through the PO\nlevel-density amplitudes $\\mathcal{B}_{\\rm POT}$\n[see (\\ref{dlevdenscl})] of the free-energy shell corrections\n(\\ref{fpotau}), (\\ref{descl}).\n\n\n\\subsubsection{COMPARISON OF SHELL STRUCTURE CORRECTIONS WITH\\\\ QUANTUM RESULTS}\n\\label{qmsclcomparison}\n \n\nFig.\\ \\ref{fig15} shows the\nsemiclassical free-energy shell correction $\\delta F_{\\rm scl}$,\n[(\\ref{fpotau}), (\\ref{descl}), see also \\cite{magosc,mskbPRC2010}]\nvs the particle-number variable, $A^{1\/3}$,\nat a small temperature of $T=0.1\\; \\hbar \\omega_{{}_{\\! 0}}$ for different critical\nsymmetry-breaking and bifurcation deformations\n$\\eta_{{}_{\\! HO}}=1$, $6\/5$, and $2$ of the HO potential\n\\cite{sclbook,mskbPRC2010} with\nthe corresponding\nquantum SCM results for the same\ndeformations.\nThis comparison also shows practically a perfect agreement\nbetween the\nsemiclassical,\n(\\ref{fpotau}) and (\\ref{descl}),\nand quantum results.\nFor the spherical case ($\\eta_{{}_{\\! HO}}=1$),\none has only contributions of the families\nof 3D orbits with the highest degeneracy $\\mathcal{K}=4$.\nAt the bifurcation points $\\eta_{{}_{\\! HO}}=6\/5$ and $2$ the relatively simple\nfamilies of these 3D POs appear\nalong with EQ orbits of smaller degeneracy.\nFor $\\eta_{{}_{\\! HO}}=6\/5$, one mainly has the contributions from EQ POs\nbecause the 3D\norbits are generally too long in this case.\nFor the bifurcation point $\\eta_{{}_{\\! HO}}=2$,\n one finds an interference of the two\ncomparably large contributions of EQ and 3D\norbits\nwith essentially the different time periods $t_{{}_{\\! EQ}}$ and\n$t_{{}_{\\! 3D}}$, respectively.\n\n\nThe quantum (QM) and semiclassical\n(SCL) shell corrections to\nthe MI $\\delta \\Theta^{}_x$ of (\\ref{dthetaxdehoscl})\nare compared in Fig.\\ \\ref{fig16}.\nAn excellent agreement is observed\nbetween the semiclassical and quantum results as for the free-energy shell\ncorrections $\\delta F$.\n It is not really astonishing\nbecause of the proportionality of the $\\delta \\Theta^{}_x$ to\n$\\delta F$ [see (\\ref{dthetaxdehoscl})].\nOne finds in particular the same clear interference of contributions of\n3D and EQ POs in the shell corrections to the MI at $\\eta_{{}_{\\! HO}}=2$.\nThe exponential decrease of shell oscillations with increasing temperature,\ndue to the temperature factor\nin front of the PO energy-shell correction components $\\delta E_{\\rm PO}$\nin (\\ref{fpotau})\nis clearly seen in Fig.\\ \\ref{fig16}. As the MI and free-energy\nshell corrections are basically proportional [see (\\ref{dTxrigSCLgen})]\nfor any mean potential well, we may emphasize\nthe amplitude enhancement of the MI near the bifurcation\ndeformations due to that\nfor the energy-shell corrections found in\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar}.\nThe\ncritical temperature for a disappearance of shell effects in the MI\nis found for prolate deformations ($\\eta>1$)\nand particle numbers\n$A \\sim 100-200$, approximately at $T^{}_{cr}=\n\\hbar \\omega^{}_{EQ}\/\\pi \\sim \\hbar \\omega^{}_{0}\/\\pi \\approx 2-3$ MeV\njust as for $\\delta F$, see \\cite{sclbook,strutmag,mskbPRC2010}. This effect\nis also general for any potentials.\nThe particle-number dependence of the shell corrections\n $\\delta \\Theta^{}_z$ to the total MI $\\Theta^{}_z$\n(alignment)\nis not shown because it is similar to that of $\\delta \\Theta^{}_x$\nthrough their approximate relations,\n$\\delta \\Theta^{}_z \\propto \\delta \\Theta^{}_x \\propto \\delta F$.\n\n\n\n\\section{Conclusions}\n\\label{concl}\n\nWe derived the dynamical equations of motion, such as the conservation\nof the particle number, momentum and energy as well as the\ngeneral transport equation for the entropy for \\textit{low}\nfrequency excitations in nuclear matter within the Landau\nquasiparticle theory of \\textit{heated} Fermi-liquids.\nOur approach is based essentially on the\nLandau--Vlasov equation for the distribution function, and it\nincludes all its moments in phase space, in contrast to several\ntruncated versions of fluid dynamics\nsimilar to the hydrodynamic description\nin terms of a few first moments. From the dynamics of the\nLandau--Vlasov equation for the distribution function, linearized\nnear the \\textit{local} equilibrium, we obtained the momentum flux\ntensor and heat current in terms of the shear modulus, viscosity,\nin-compressibility and thermal conductivity coefficients as for\nvery viscose liquids called sometimes \\textit{amorphous solids}.\nWe obtain the dependence of these coefficients on the\n temperature, the frequency and the Landau interaction parameters.\nWe derived the \\textit{temperature\nexpansions} of the density-density and\ntemperature-density response functions for nuclear matter and got\ntheir \\textit{specific expressions for small temperatures} as\ncompared to the chemical potential. The \\textit{hydrodynamic limit}\nof normal liquids for these response functions \\textit{within the\nperturbation theory} was obtained from the Landau--Vlasov equation for\nboth distribution function and sound\n velocity, as for an eigenvalue problem.\nIn this way we found the Landau--Placzek and first sound peaks in\nthe corresponding strength functions as the hydrodynamic limit of\nthe Fermi-liquid theory for heated Fermi-systems. The former (heat\npole) peak was obtained only because of the use of the local\nequilibrium in the Landau--Vlasov linearized dynamics instead of\nthe global static Fermi-distribution of the giant multipole-resonance\nphysics. This is very important for the dispersion equation and\nits wave velocity solutions.\n\nWe got the \\textit{isolated, isothermal and adiabatic\nsusceptibilities for the Fermi-liquids} and showed that they\nsatisfy the \\textit{ergodicity} condition of equivalence of the\nisolated and adiabatic susceptibilities as well as the general\nKubo inequality relations. We found the \\textit{correlation function}\nusing the fluctuation-dissipation theorem and discussed its\nrelation to the susceptibilities and Landau--Placzek \"heat pole\" in\nthe hydrodynamic limit.\n\nWe applied the theory of heated Fermi-liquids to the Fermi-liquid\ndrop model of finite nuclei within the Landau--Vlasov dynamics in the\nnuclear interior and macroscopical boundary conditions in the\neffective sharp surface approximation. Solutions of this\nproblem in terms of the response functions and transport\ncoefficients were obtained. We considered the hydrodynamic limit\nof these solutions and found the ``heat pole'' correlation function\nfor frequencies smaller than some critical frequency. The latter\nwas realized only because of using the local equilibrium for the\ndistribution function. The isolated, isothermal and adiabatic\nsusceptibilities for finite nuclei within the FLDM in the ESA\nwere derived. We showed that the ergodicity condition is satisfied\nalso for finite Fermi-systems as for infinite nuclear\nmatter in the same ESA.\n\nWe found a three-peak structure of the collective strength\nfunction: the \"heat\", standard hydrodynamic and essentially\nFermi-liquid peaks. The conditions for the existence of such modes\nwere analyzed and the temperature dependence of their transport\ncoefficients such as friction, stiffness and inertia were\n obtained\nin particular, in the hydrodynamic limit.\nWe arrived at the increasing temperature dependence of the\nfriction coefficient for the specific Fermi-liquid mode which\nexist due to the Fermi-surface distortions. At enough large\ntemperatures, we showed a nice agreement with the results for the\nfriction which were obtained earlier within the microscopic\nshell-model approach of \\cite{hofbook}. The correlation functions\nfound in the FLDM and quantum shell models were discussed in\nrelation to the susceptibilities and ergodicity properties of\nfinite nuclei.\n\nThe expression for the surface symmetry-energy constant $k_{{}_{\\! S}}$\nwas derived from simple isovector solutions\nof the particle density and energies in the leading ES\napproximation. We used them for the calculations\nof the energies, sum rules of the IVGDR strength and the transition\ndensities\nwithin the HDM and FLDM \\cite{kolmagsh} for several Skyrme-force parameters.\nThe surface symmetry-energy constant\ndepends much on the fundamental\nwell-known parameters of the Skyrme forces, mainly through the\ncoefficient in the density gradient terms of the isovector part of the\nenergy density.\nThe value of this isovector constant\n is rather sensitive also\non the SO interaction.\nThe IVGDR strength is split into the two main and satellite peaks.\nThe mean energies and EWSRs within both HDM and FLDM are in fairly\ngood agreement with the\nexperimental data.\n\nSemiclassical functional expressions were derived\n in the framework of the Extended Thomas--Fermi approach. We used\n these analytical expressions to obtain a self-consistent description\n of rotating nuclei where the rotation velocity impacts on the structure\n of the nucleus. It has been shown that such a treatment leads, indeed,\n to the Jacobi phase transition to triaxial shapes as already predicted\n in \\cite{CPS74} within the rotating LDM. We emphasize\n that the rigid-body moment of inertia gives a quite accurate\n approximation for the full ETF value. Being aware of the mutual influence\n between rotation and pairing correlations\n \\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev}, it would be\n especially interesting to work on an approach that is able to\ndetermine the nuclear structure depending on its angular velocity,\nas we have done here in the ETF approach, but taking pairing correlations\nand their rotational quenching into account.\n\nWe derived also the shell corrections\nof the MI\nin terms of free-energy shell corrections\nwithin the nonperturbative extended POT through those of the rigid-body MI of\nthe equilibrium rotations,\nwhich is exact for the HO potential.\nFor the HO, we extended\nto the finite temperature case the Zelevinsky\nderivation of the non-adiabatic\nMI at any rotation frequency.\nFor the deformed HO potential, one finds a perfect agreement\nbetween the semiclassical POT and quantum\nresults for the free-energy\nand the MI shell corrections\nat several critical deformations and temperatures.\nFor larger temperatures, we show that the short EQ orbits are\nmostly dominant.\nFor small temperatures, one observes a remarkable interference of the\nshort 3D and EQ orbits in the superdeformed region.\nAn exponential decrease of all shell corrections\nwith increasing temperature is observed, as expected. We point out\nalso the amplitude enhancement of the MI shell corrections due to\nthe bifurcation catastrophe phenomenon.\n\nAs further perspectives, it would be worth to apply our results to\ncalculations of the IVGDR structure within the Fermi-liquid droplet\nmodel to determine the value of the fundamental surface symmetry-energy\nconstant from comparison with experimental data for the pygmy resonance\n\\cite{adrich,wieland} and theoretical calculations\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2,BMR}.\nFor further extensions to\nthe description of the isovector low-lying collective states, one has first to\nuse the POT for including semiclassically the shell effects\n\\cite{strutmag,sclbook,gzhmagfed,blmagyas,BM}.\nIt would be also worth to apply this semiclassical theory\nto the shell corrections of the MI for the spheroid cavity\nand for the inertia parameter of the\nlow-lying collective excitations in nuclear dynamics involving\nmagic nuclei\n\\cite{dfcpuprc2004,magvvhof,yaf,gzhmagfed}. One of the most attractive\nsubject of the semiclassical periodic orbit theory, in line of the main\nworks of S.T. Belyaev\n\\cite{belyaevfirst,belyaevzel,belzel,belsmitolfay87},\nis its extension\nto the {\\it pairing} correlations\n\\cite{brackquenNPA1981,brackrocciaIJMPE2010},\nand their influence on the collective\nvibrational and rotational excitations in heavy deformed neutron-rich\nnuclei\n\\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev} (see also\n\\cite{abrpairing} for the semiclassical phase-space dynamical approach\nto the Hartree--Fock--Bogoliubov theory).\n\n\\bigskip\n\n\\section*{Acknowledgement}\n\n\\label{aknow}\n\n\\medskip\n\n\nAuthors gratefully acknowledge \nH.\\ Hofmann for many suggestions and fruitful discussions, also \n S.\\ Aberg, V.\\ I.\\ Abrosimov, J.\\ Blocki, R.\\ K.\\ Bhaduri, M.\\ Brack, \nV.\\ Yu.\\ Denisov, S.\\ N.\\ Fedotkin,\nH.\\ Heisenberg, F.\\ A.\\ Ivanyuk,\nV.\\ M.\\ Kolomietz, M.\\ Kowal, J.\\ Meyer, V.\\ O.\\ Nesterenko, V.\\ V.\\ Pashkevich,\nM.\\ Pearson, V.\\ A.\\ Plujko, P.\\ Ring, V.\\ G.\\ Zelevinsky, \nA.\\ I.\\ Sanzhur, S.\\ Siem, J.\\ Skalski,\nand X.\\ Vinas for many useful discussions. One of us (A.G.M.) is\nalso very gratitude for a nice hospitality during his working visits of the\nTechnical Munich University in Garching and the University \nof Regensburg in Germany, the Interdisciplinary \nHubert Curien Institute of the Louis Pasteur University \nin Strassburg of France, and \nNational Centre for Nuclear Research in Otwock-Swierk of Poland.\n\n\n\\vspace{1cm}\n\n\\noindent\n\\begin{appendix}\n\\setcounter{equation}{0}\n\\renewcommand{\\thesubsection}{\\Alph{section}.\\arabic{subsection}}\n\\renewcommand{\\theequation}{\\mbox{\\Alph{section}.\\arabic{equation}}}\n\\noindent\n\\section{Elements of Landau theory for equilibrized systems}\n\\label{app1}\n\\subsection{Thermodynamic relations}\n\\label{app1thermrel} Let us begin recalling the fundamental\nequations $TdS = dE-\\mu dN +{\\cal P} \\hbox{d} \\mathcal{V}$ and $-S\\hbox{d} T = \\hbox{d} F-\\mu\n\\hbox{d} N +{\\cal P} \\hbox{d} \\mathcal{V}$, which are related to each other by the\nLegendre transformation $F=E-TS$. They imply the following\nrelations for the chemical potential $\\mu$ and pressure\n${\\cal P}$: \n\\bel{chempot} \n\\mu =-T\\left(\\frac{\\partial S }{\\partial\nN}\\right)_{E,{\\mathcal{V}}}= \\left(\\frac{\\partial E }{\\partial\nN}\\right)_{S,{\\mathcal{V}}}= \\left(\\frac{\\partial F }{\\partial\nN}\\right)_{T,{\\mathcal{V}}}, \n\\end{equation} \n\\bel{pressure} {\\cal P} =\n-\\left(\\frac{\\partial E }{\\partial \\mathcal{V}}\\right)_{S,N}=\n-\\left(\\frac{\\partial F }{\\partial \\mathcal{V}}\\right)_{T,N}. \n\\end{equation} \nFor\nhomogeneous systems the {\\it intensive} quantities depend only on\ntwo independent variables. For instance, the entropy per particle\n$S\/N=\\varsigma(E\/N,\\mathcal{V}\/N)$ only depends on the energy and\nvolume per particle, $E\/N $ and $\\mathcal{V}\/N$ respectively. For\nsuch systems, the adiabadicity condition may simply be expressed as\n$\\varsigma = {\\rm const}$. \nCommonly in\nnuclear physics, one uses the particle density $\\rho=N\/\\mathcal{V}$,\nin which case the chemical potential can be expressed as\n\\bel{chempothom} \n\\mu = \\left({\\partial \\phi \\over\n\\partial \\rho}\\right)_T \n\\end{equation}\n with $\\phi = F\/\\mathcal{V}$ being the\nfree internal energy per unit volume.\n\nFor differential quantities there exist various variants of the \nGibbs-Duheim relation \n\\bel{giduvar} \n\\hbox{d} \\phi = -\\varsigma \\rho \\hbox{d} T + \\mu \\hbox{d} \\rho \\qquad \\mbox{or} \n\\end{equation}\n\\begin{eqnarray}\\label{gibbsduh} \n&&\\hbox{d}{\\cal P} = \\varsigma \\rho \\hbox{d} T + \\rho \\hbox{d} \\mu~\n\\quad{\\rm implying} \\quad \\left({\\partial \\varsigma \\over\n\\partial \\mu}\\right)_T \\nonumber\\\\\n&=& \\frac{1}{ \\rho} \\left[\\left(\\frac{\\partial \\rho\n}{\\partial T}\\right)_\\mu- \\varsigma \\left(\\frac{\\partial \\rho\n}{\\partial \\mu}\\right)_T \\right], \n\\end{eqnarray}\n as follows from\nLegendre transformations. Thus for the derivatives of the pressure\n$\\mathcal{P}$, considered as functions of $T$ and $\\rho$, one gets from\n(\\ref{gibbsduh}):\n\\begin{eqnarray}\\label{dpdmt}\n &&\\left(\\frac{\\partial {\\cal P} }{\\partial \\rho} \\right)_T =\n \\rho \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T,\n \\qquad\n \\left(\\frac{\\partial {\\cal P} }{\\partial T} \\right)_\\rho =\\nonumber\\\\\n &=&\\rho\\left(\\varsigma-\n \\left(\\frac{\\partial \\rho } {\\partial T}\\right)_{\\mu}\n \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T\\right).\n\\end{eqnarray}\n In deriving such relations, it is useful to employ special\nproperties of the Jacobian, which allows\none to perform transformations between different variables (see\ne.g.,\n\\cite{brenig}).\nThese relations will be used below to get the specific heats as\nwell as the isothermal and adiabatic compressibilities, together\nwith the corresponding susceptibilities. At first, we shall look at\n{\\it in}-compressibilities defined by the\nderivative of the pressure over the particle density\n (multiplied by a factor of 9). At {\\it\nconstant entropy per particle} $\\varsigma$, the {\\it adiabatic}\nin-compressibility $K^{\\varsigma}$ writes \n\\bel{incompraddef}\nK^{\\varsigma} \\equiv 9 \\left(\\frac{\\partial {\\cal P} }{\\partial\n\\rho}\\right)_\\varsigma = 9 \\left(\\frac{\\partial {\\cal P} }{\\partial\n\\mu}\\right)_\\varsigma \\left(\\frac{\\partial \\mu }{\\partial\n\\rho}\\right)_\\varsigma. \n\\end{equation} \nTo get the corresponding quantity\nat constant temperature $K^{T}$, one only needs to replace\n$\\varsigma$ by $T$. According to (\\ref{dpdmt}) and\n(\\ref{chempothom}), one obtains \n\\bel{incomprTdef} \nK^{T} \\equiv 9\n\\left(\\frac{\\partial {\\cal P} }{\\partial \\rho}\\right)_T = 9 \\rho\n\\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T = 9 \\rho\n \\left(\\frac{\\partial^2 \\phi }{\\partial \\rho^2}\\right)_T.\n\\end{equation}\n\nNext we turn to the specific heats at constant volume and constant\npressure. If measured per particle, they can be defined in terms\nof the entropy per particle $\\varsigma$ as \n\\bel{specifheatpdef}\n{\\tt C}_{\\mathcal{V}} =\n T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_{\\mathcal{V}}=\n T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_\\rho,\n\\quad {\\tt C}_{\\cal P} =\n T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_{\\cal P}.\n\\end{equation} \nThey obey the following, well known relation to the\nin-compressibilities \\cite{forster,brenig} \n\\bel{cvcpkSkT} \n\\left(\\frac{{\\tt C}_{\\cal P} \n}{{\\tt C}_{\\mathcal{V}}}\\right)= \n\\frac{\\left(\\partial {\\cal P} \/\\partial\n\\rho\\right)_\\varsigma }{\\left(\\partial {\\cal P} \/\\partial \\rho \\right)_T} = \n\\frac{K^{\\varsigma} }{K^{T}}. \n\\end{equation}\n\nFor the variation of the entropy $\\varsigma$ per particle, one\nfinds %\n\\bel{entropypart}\n \\hbox{d} \\varsigma = - \\frac{1}{\\rho}\n\\left[\\varsigma + \\left(\\frac{\\partial \\mu }{\\partial\nT}\\right)_\\rho\\right] \\hbox{d} \\rho + \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\hbox{d} T,\n\\end{equation} \nafter using (\\ref{giduvar}) and the specific heat ${\\tt\nC}_{\\mathcal{V}}$ of (\\ref{specifheatpdef}). To get the first term we\napplied \n\\bel{derphi} -\\left(\\frac{\\partial (\\varsigma\n\\rho) }{\\partial \\rho}\\right)_T= \\left(\\frac{\\partial \\mu \n}{\\partial T}\\right)_\\rho \\equiv \\frac{\\partial^2 \\phi }{\\partial\n\\rho \\partial T}, \n\\end{equation} \nwhich is a consequence of (\\ref{giduvar}).\n\n\n\\subsection{Landau theory proper}\n\\label{landtheorprop}\n\nIn the following, we will repeat some important relations discussed\nin \\cite{heipethrev} without arguing much about their proofs.\nThese relations will be needed to derive some specific\nthermodynamic properties for quantities, as the entropy or the\nspecific heats.\nA basic element in Landau theory is the microscopic\nexpression for the entropy per particle, \n\\begin{eqnarray}\\label{entropydef} \n\\varsigma &=&\n - \\frac{1}{\\rho} \\int \n\\frac{2 d {\\bf p} }{\\left(2 \\pi \\hbar \\right)^3}\n \\left[f_{\\bf p} \\ln f_{\\bf p} + \\left(1-f_{\\bf p}\\right)\n \\ln \\left(1-f_{\\bf p}\\right) \\right]\\nonumber\\\\\n &=& \\left\\langle \\frac{p^2 }{3 m^*}\n \\left(\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\\right) \\right\\rangle \/\n\\left\\langle \\frac{p^2 }{3 m^*} \\right\\rangle. \n\\end{eqnarray}\n in terms of\nthe Fermi distribution $f_{{\\bf p}}$ [c.f. (\\ref{fgeq})]. The (static)\nquasiparticle density $\\rho$ in (\\ref{entropydef}) may be\nexpressed as \n\\bel{densstat} \n\\rho =\\frac{N}{\\mathcal{V}} = \\frac{p_{\\rm F}^3\n}{3 \\pi^2 \\hbar^3}=\n \\int \\frac{2 \\hbox{d} {\\bf p} }{\\left(2 \\pi \\hbar \\right)^3} f_{\\bf p}\n = \\mathcal{N}(T) \\left\\langle \\frac{p^2 }{3 m^*} \\right\\rangle,\n\\end{equation} \nwith the density of states $\\mathcal{N}(T)$ \n(\\ref{enerdensnt}). The additional factor 2 in the integration\nmeasure accounts for the spin degeneracy. The expressions on the\nright in both (\\ref{entropydef}) and (\\ref{densstat}) are obtained\nafter integrating by parts. The brackets $<\\cdots>$ denote some\nkind of average, which if written for any quantity $A({\\bf r},{\\bf p},t)$ is\ndefined as \n\\bel{averag} \n\\langle A({\\bf r},{\\bf p},t) \\rangle\n =\\frac{1}{\\mathcal{N}(T)} \\int \\frac{2 \\hbox{d} {\\bf p}^\\prime\n }{(2 \\pi \\hbar)^3}\n \\left(-\\frac{\\partial f_{{\\bf p}^\\prime}\n }{\\partial \\varepsilon_{{\\bf p}^\\prime}}\\right)\n A({\\bf r},{\\bf p}^\\prime,t).\n\\end{equation}\n In addition to the $\\mathcal{N}(T)$, one needs \n\\bel{mcapt} \n\\mathcal{M}(T) = \\mathcal{N}(T) \\left\\langle\n\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n \\right\\rangle.\n\\end{equation}\n From (\\ref{densstat}) one may derive (see (2.9) and\n(2.11) of \\cite{heipethrev}) \n\\bel{drhomt} \n\\hbox{d} \\rho =\n \\frac{\\mathcal{N}(T) }{\\mathcal{G}_0}~ \\hbox{d} \\mu +\n\\frac{\\mathcal{M}(T)}{\\mathcal{G}_0}~\\hbox{d} T,\n\\end{equation}\n[see also (\\ref{effmass}) for $\\mathcal{G}_0$] which allows\none to express the isothermal in-compressibility $K^{T}$ \n(\\ref{incomprTdef}) by (\\ref{isotherk}).\nFor the variation of the pressure with temperature, one gets\nfrom (\\ref{dpdmt}) and (\\ref{drhomt}) \n\\bel{dpdtr}\n \\left(\\frac{\\partial {\\cal P} }{\\partial T} \\right)_{\\rho} =\n \\left(\\varsigma- \\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right) \\rho =\n \\frac{2}{3} \\rho {\\tt C}_{\\mathcal{V}}.\n\\end{equation} \nFor the proof of the second equation, we refer to (3.35) of\n\\cite{heipethrev} (mind however a difference in the notations for\nthe specific heat: Our $\\rho {\\tt C}_{\\mathcal{V}}$ is identical to the\n$ {\\tt C}_{\\mathcal{V}}$ of \\cite{heipethrev}). For our \n$ {\\tt C}_{\\mathcal{V}}$, one may derive the formula (see (3.34) of\n\\cite{heipethrev}) \n\\bel{specheatveq} \n{\\tt C}_{\\mathcal{V}} = \n\\frac{T \\mathcal{N}(T)}{\\rho} \\left\\langle \n\\left[\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n -\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)} \\right]^2 \\right\\rangle.\n\\end{equation} \nCollecting (\\ref{incomprTdef}), (\\ref{isotherk}) and\n(\\ref{dpdtr}), one can write the variation of the pressure \nas\n\\bel{dpress} \n{\\rm d}{\\cal P} = \\rho \n\\left(\\frac{\\mathcal{G}_0}{\\mathcal{N}(T)} \n{\\rm d} \\rho + \\frac{2}{3} {\\tt C}_{\\mathcal{V}} {\\rm d} T\\right). \n\\end{equation}\n\n \nThermodynamic quantities such as\nin-compressibilities and susceptibilities are calculated under\ndifferent conditions as fixed temperature or entropy. As one\nknows (see, e.g., \n\\cite{forster}), these\n(in)-compressibilities may be associated to different sound\nvelocities. To make use of the adiabaticity condition mentioned\nearlier, we need the derivatives of the entropy per particle\n$\\varsigma(\\rho,T)$. The ones arising in (\\ref{entropypart}) can\nbe simplified by exploiting the specific Fermi-liquid expressions\ngiven in (\\ref{drhomt}) and the second relation of (\\ref{dpdtr})\nbetween the entropy per particle $\\varsigma$ and the specific heat\n${\\tt C}_{\\mathcal{V}}$,\n\\bel{dspdmt}\n \\hbox{d} \\varsigma =\n - \\frac{2}{3 \\rho} {\\tt C}_{\\mathcal{V}} \\hbox{d} \\rho +\n \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\hbox{d} T.\n\\end{equation} \nNext we turn to the adiabatic in-compressibility\n$K^{\\varsigma}$ (\\ref{incompraddef}). It may be expressed by\nthe isothermal one $K^{T}$ given in (\\ref{isotherk}), \nsee (\\ref{Kadiabat}).\nTo derive this relation, \nthe Jacobian transformation from $(\\rho,\\varsigma)$ to $(\\rho,T)$ for\nthe derivatives of the pressure in (\\ref{incompraddef}) has been applied\n[mind also\n(\\ref{dpdtr}), (\\ref{incomprTdef}) and (\\ref{dspdmt})]. Finally,\nfor the ratio of the specific heats, we find from\n(\\ref{cvcpkSkT}), (\\ref{isotherk}) and (\\ref{Kadiabat})\n\\bel{cpcvkakt} \n\\left(\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}\\right)= \n1 + \\frac{4 T {\\tt C}_{\\mathcal{V}} \\mathcal{N}(T)}{9 \\rho\n\\mathcal{G}_0}. \n\\end{equation}\n\n\\subsection{Low temperature expansion}\n\\label{lowtempexp}\n\nIn this subsection, we address the temperature dependence of the\nquantities introduced above. It may be derived as discussed in\n\\cite{heipethrev} and conveniently\nexpressed by expansions in terms of ${\\bar T} = T\/\\varepsilon_{{}_{\\! {\\rm F}}}$; with\n$\\varepsilon_{{}_{\\! {\\rm F}}}$ being the Fermi energy at zero temperature, \n$\\varepsilon_{{}_{\\! {\\rm F}}}=p_{\\rm F}^2\n\/ (2m^*)=(3\\pi^2\\hbar^3 \\rho)^{2\/3}\/(2m^*)$. For some of the\nquantities discussed below we shall include terms of third order\nin ${\\bar T} = T\/\\varepsilon_{{}_{\\! {\\rm F}}}$, which are not considered in \\cite{heipethrev}.\n\nFrom (\\ref{densstat}) one gets for the particle density\n$\\rho(\\mu,T)$ \n\\bel{densexp} \n\\rho =\n \\frac{\\left(2 m^* \\mu\\right)^{3\/2} }{3 \\pi^2 \\hbar^3}\n \\left(1 + \\frac{\\pi^2 \\tbar^2 }{8} \\right)\n\\end{equation} \nas function of the chemical potential $\\mu$ and the\ntemperature $T$. For the chemical potential $\\mu$, one obtains\n\\bel{chemicpot} \\mu = \\varepsilon_{{}_{\\! {\\rm F}}} \\left(1-\\frac{\\pi^2 \\tbar^2 }{12}\n\\right),\n\\end{equation} \nwhich is typical for a system of independent fermions.\n At this stage it may be worth while to mention\nthat the formulas presented here remain largely unchanged in case\nof the presence of a density dependent potential $V(\\rho)$. As\nlong as such a potential does not depend on the momentum, we may\njust change our s.p.\\ energy $\\varepsilon_{\\bf p}^{\\rm g.e.}$ to $\np^2\/(2 m^*) + V(\\rho)$, and the chemical potential $\\mu$ to the\n$\\mu'=\\mu-V(\\rho)$ of \\cite{heipethrev}.\n\n\nFor the density of states $\\mathcal{N}(T)$ of the quasiparticles, one\nfinds from (\\ref{enerdensnt}) \n\\bel{capnexp} \n\\mathcal{N}(T) = \n\\mathcal{N}(0)\\left(1-\\frac{\\pi^2 \\tbar^2 }{12} \\right), \n\\end{equation} \nwhere $\\mathcal{N}(0)$ is given by (\\ref{nzero}).\nSimilarly, for \n$\\mathcal{M}(T)$ defined in (\\ref{mcapt}), one gets \n\\bel{mcapexp} \n\\mathcal{M}(T) = \n\\frac{\\pi^2}{6}\\;\\mathcal{N}(0)\\; {\\bar T}\n \\left(1 + \\frac{13\\pi^2 \\tbar^2 }{60}\\right).\n\\end{equation} \nAs different to \\cite{heipethrev}, we include a temperature\ncorrection here, which is of interest for some of the quantities\ndescribed in the text. The specific heat \n${\\tt C}_{\\mathcal{V}}$ (\\ref{specheatveq}) per particle for the constant volume\nbecomes \n\\bel{specifheatv} \n{\\tt C}_{\\mathcal{V}}=\n \\frac{\\pi^2{\\bar T}}{2}\n \\left(1-\\frac{3 \\pi^2 \\tbar^2 }{10} \\right).\n\\end{equation} \nFor the isothermal in-compressibility $K^{T}$, one\ngets from (\\ref{isotherk}) and (\\ref{capnexp}) \n\\bel{isotherkexp}\nK^{T} = 6 \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_0\n \\left(1 + \\frac{\\pi^2 \\tbar^2 }{12} \\right).\n\\end{equation} \nLikewise, for the in-compressibility modulus $K^{\\varsigma}$\n(\\ref{Kadiabat}) at constant entropy $\\varsigma$ per particle, one\nobtains \n\\bel{incompradexp} \nK^{\\varsigma} =\n 6 \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_0 \\left[1 + \\frac{\\pi^2 \\tbar^2 }{12}\n \\left(1 + \\frac{4 }{\\mathcal{G}_0}\\right) \\right]\\;.\n\\end{equation} \nUsing (\\ref{incompradexp}), the adiabatic sound velocity\n$v^{(\\varsigma)}$ (cf. \n\\cite{forster,brenig}) \ncan be expressed as\n\\bel{speedad} \nv^{(\\varsigma)} =\n \\sqrt{\\frac{K^{\\varsigma} }{9 m}} = v_{{}_{\\! {\\rm F}}} s^{\\varsigma},\n\\end{equation} \nwhere \n\\bel{velocad} \ns^{\\varsigma} = \\sqrt{{\\mathcal{G}_0 \n\\mathcal{G}_1 \\over 3}\n \\left[1 + \\frac{\\pi^2 \\tbar^2 }{12} \\left(1 + \n\\frac{4 }{\\mathcal{G}_0}\\right)\\right]}.\n\\end{equation} \nThe ratio of the specific heats (\\ref{cvcpkSkT}) may be\ncalculated using either the expansions of the in-compressibilities\n(\\ref{incompradexp}) and (\\ref{isotherkexp}) or\n(\\ref{cpcvkakt}) together with (\\ref{capnexp}) and\n(\\ref{specifheatv}). Finally, one gets \n\\bel{cpcvexp} \n\\left(\\frac{{\\tt C}_{\\cal P}\n}{{\\tt C}_{\\mathcal{V}}}\\right) =\n 1 + \\frac{\\pi^2 \\tbar^2 }{3 \\mathcal{G}_0}.\n\\end{equation} \nThus, from (\\ref{specifheatv}) and (\\ref{cpcvexp}),\n\\bel{specifheatp} C_{\\cal P} =\n \\frac{\\pi^2 {\\bar T}}{2}\n \\left[1-\\frac{3 \\pi^2 \\tbar^2 }{10}\n \\left(1-\\frac{10 }{9 \\mathcal{G}_0} \\right)\\right]\n\\end{equation} \nis the specific heat at the fixed pressure.\n\n\n\\subsection{Thermodynamic relations for a finite Fermi-liquid drop}\n\\label{thermgibbs}\n\nIn this subsection, we apply the formulas derived above to extend\nthe derivations of the boundary conditions in\n\\cite{strutmagbr,magstrut,magboundcond} to the case of equilibrium\nat a finite $T$. Like in these papers, the finite Fermi-liquid drop\nis treated in the effective sharp surface approximation, see\nsubsection \\ref{boundary} and Appendix D.\nApplying to the standard thermodynamic relations $~{\\rm d}E=T{\\rm d}S\n-{\\cal P} {\\rm d}\\mathcal{V}-{\\cal P}_Q{\\rm d}Q~$ and $~{\\rm d}G=-S{\\rm\nd}T + \\mathcal{V} {\\rm d}{\\cal P}-{\\cal P}_Q{\\rm d}Q~$, we include the\nchange of the collective variable $Q$\n(see, e.g., \\cite{hofmann,hofbook}). $G$ is the Gibbs free\nenergy $G=F+{\\cal P} \\mathcal{V}=E+TS+{\\cal P} \\mathcal{V}$, defined\nsimilarly to the free energy $F$ with only the volume \n$\\mathcal{V}$\nreplaced by the pressure ${\\cal P}$.\nFor the FLDM \nit is more convenient to use $G$\nrather than $F$, simply because in general volume may not be\nconserved but the pressure has to be fixed by the boundary\ncondition (\\ref{bound2}). The Gibbs free energy is used for\nderiving these boundary conditions as well as for the\ncalculations of the coupling constants and susceptibilities\nassociated to the operator ${\\hat F}({\\bf r})$ (\\ref{foperl}).\n\nFor the following derivations, we need the relations for the\nthermodynamical potentials per particle. The Gibbs free energy per\nparticle $G\/N$ which is identical to the chemical potential\n$\\mu$ \nis related to the corresponding\nfree energy $F\/N$ by the relation $G\/N \\equiv\n\\mu=F\/N+{\\cal P}\/\\rho$~. For a finite Fermi-liquid drop where the\nparticle density $\\rho$ is function of the coordinates (smooth\ninside and sharp decreasing in the surface region) they are\nwritten as in \\cite{strutmagbr,magstrut,magboundcond} through the\nvariational derivatives $\\delta g\/\\delta \\rho$ and $\\delta\n\\phi\/\\delta \\rho$ with the thermodynamical potential densities\n$g$ and $\\phi$ per unit of volume, respectively, and this\nrelation reads now \n\\bel{gibbspart} \n\\frac{\\delta g }{\\delta \\rho}\n\\equiv \\mu= \\frac{\\delta \\phi }{\\delta \\rho} +\n\\frac{{\\cal P} }{\\rho}.\n\\end{equation}\n These densities depend on the coordinates through \n$\\rho$ and its gradients. Their calculation is carried out\nfrom the variations of the corresponding total integral\nquantities $G$ and $F$ with the following integration by parts,\nsee \\cite{strutmagbr,magstrut,magboundcond} for details. Taking\ninto account also that the particles in the Fermi-liquid drop\nmove in a mean field $V$ with the coordinate dependence similar to\nthe density $\\rho$, one gets from (\\ref{gibbspart})\n\\begin{eqnarray}\\label{thermrelmu} \n{\\rm d} \\frac{\\delta g }{\\delta \\rho} &\\equiv&\n{\\rm d} \\mu= -\\varsigma {\\rm d} T + \\frac{1}{\\rho} {\\rm d}{\\cal P}\n+{\\rm d} V\\quad{\\rm with}\\nonumber\\\\ {\\rm d}\nV&=&-\\left({\\cal P}_Q\/N\\right) {\\rm d} Q. \n\\end{eqnarray}\nFrom (\\ref{thermrelmu}) one has \n\\bel{gradrelmu} \n{\\bf \\nabla} \\mu\n=-\\varsigma {\\bf \\nabla} T + \\frac{1}{\\rho} {\\bf \\nabla}{\\cal P}\n+{\\bf \\nabla} V. \n\\end{equation}\n\nFor the derivation of the boundary condition (\\ref{bound2}), we\nused (\\ref{gradrelmu}) for the transformations of\n(\\ref{momenteq}) instead of (17) of \\cite{magstrut}. The\none-to-one correspondence of this derivation with that explained in\n\\cite{strutmagbr,magstrut,magboundcond} becomes obvious if we note\nthat equation (17) \nwas found from \n\\bel{gradrelener} \n{\\bf \\nabla}\n\\varepsilon =T{\\bf \\nabla} \\varsigma + \\frac{1}{\\rho} {\\bf\n\\nabla}{\\cal P} +{\\bf \\nabla} V, \n\\end{equation}\n for the adiabatic condition of\na constant entropy per particle ($\\varepsilon$ here is the same as\n$\\delta \\varepsilon \/ {\\delta \\rho}$ in the notation of\n\\cite{magstrut}). The variational derivative $\\delta g \/ {\\delta\n\\rho}$ (\\ref{gibbspart}) (or the chemical potential $\\mu$)\nappears now in the following key equation for the derivation of\nthe surface condition (\\ref{bound2}): \n\\bel{keybound}\n\\rho_\\infty\\left(\\frac{\\delta g }{\\delta\n\\rho}\\right)_{S}^{\\rm vol}= -b_{\\mathcal{V}}\\rho_\\infty +2 \\alpha \\mathcal{H},\n\\end{equation} \nwhere $b_{\\mathcal{V}}$ is the nucleon binding energy in the infinite\nnuclear matter, $\\mathcal{H}$ the mean curvature of the nuclear\nsurface, $\\mathcal{H}=1\/R_0$ for the spherical shape at equilibrium.\nIndex \"vol\" means that the Gibbs free energy per particle is\nconsidered as that found in the nuclear \n interior. Hence, is a\nsmooth quantity taken at the nuclear surface as the quantities\nin the l.h.s. of the boundary conditions (\\ref{bound1}) and\n(\\ref{bound2}) within the precision of the ESA. \n\n\nThe temperature $T$ and chemical potential $\\mu$ in\n(\\ref{thermrelmu}) and (\\ref{keybound}) are constants as\nfunction of the coordinates ${\\bf r}$ within our Fermi-liquid-drop\ninterior at equilibrium. \nWith these properties, one gets \n\\bel{derVderP}\n{\\bf \\nabla} V =-\\frac{1 }{\\rho} {\\bf \\nabla}{\\cal P} =\n\\frac{K }{9\n\\rho_{{}_{\\! 0}}} {\\bf \\nabla} \\rho. \n\\end{equation} \nIn the second equation, we applied\n(\\ref{dpress}) which shows that the expression in the middle\nof (\\ref{derVderP}) is proportional to the gradient of the\nparticle density with some smooth coefficient related to the\nin-compressibility $K$. The relation (\\ref{derVderP}) will be\nused in the Appendix C \nfor the calculation of several coupling\nconstants and susceptibilities for the constant temperature and\nentropy, as well as for the static limit $\\omega \\to 0$, with the\ncorresponding in-compressibility modulus and particle density in\nthe last equation (\\ref{derVderP}).\n\nFor the derivations of the susceptibilities in Appendix C \nand ratio of the surface energy constants (\\ref{bstbs}), we need \nhere also the following thermodynamic relation:\n\\begin{eqnarray}\\label{ctcaddif}\n\\left(\\frac{\\partial^2 G}{\\partial Q^2}\\right)_T &-& \n\\left(\\frac{\\partial^2 E}{\\partial Q^2}\\right)_S=\n\\left[\\left(\\left(\\frac{\\partial^2 G}{\\partial T^2}\\right)_Q\\right)^{-1}\n\\!\\!\\left(\\frac{\\partial^2 G}{\\partial T\\partial Q}\\right)^2\\right]_{Q=0}\n\\nonumber\\\\ \n&=&-\\left[\\left(\\left(\\frac{\\partial S}{\\partial\nT}\\right)_Q\\right)^{-1} \\left(\\frac{\\partial S}{\\partial\nQ}\\right)^2\\right]_{Q=0}. \n\\end{eqnarray}\n\nWe obtained this relation as explained in Appendix A1 in\n\\cite{hofmann} with the only one change of the free energy $F$ to\nthe Gibbs free energy $G$. \nThe derivatives in\nthese equations should be considered for the constant pressure\ninstead of the volume of the Fermi-liquid drop.\n\n\n\n\\setcounter{equation}{0}\n\\section{Stress tensor and heat current}\n\\label{app2}\nWe shall derive\nthe specific expressions for the shear modulus and viscosity\nin the stress tensor\n$\\sigma_{\\alpha\\beta}$ (\\ref{presstens}) representing it in the form\n(\\ref{prestensone}) with (\\ref{presslamb}) and (\\ref{pressnu})\nin subsection B.2. \nWe are going also to obtain the expression for\nthe thermal conductivity in the heat current in subsection B.1. \nThe next subsection B.3 \nis devoted to the long wave approximation\nfor the above mentioned coefficients. \nIn the latter subsection,\nwe derive some basic formulas for this approximation which are\nused for the response function in whole section \\ref{longwavlim}, \nbeside\nthe above mentioned coefficients, in particular equations\nfor poles of the response function.\n\n\\subsection{Stress tensor, shear modulus and viscosity}\n\\label{app2stress}\n\nFor the calculation of the stress tensor $\\sigma_{\\alpha\\beta}$\n(\\ref{presstens}), we shall show first that it really has\nthe form given in (\\ref{prestensone}), (\\ref{presslamb}), (\\ref{pressnu})\nwith some coefficients $\\lambda$ and $\\nu$ \nin Appendix B.1a, \nand then, find their specific expressions in B.1b. \n\n\\subsubsection{STRESS TENSOR FOR FERMI LIQUIDS}\n\\label{stresscontfliq}\n\nFirst, after a short calculation of the r.h.s. of (\\ref{presslamb})\nand (\\ref{pressnu}), one simply gets \n\\bel{prestensone1}\n{\\tilde \\sigma}_{\\alpha\\beta}= -\\left(\\frac{\\lambda \n}{\\omega } -i \\nu\\right)~ \\left(q_\\beta {\\tilde u}_\\alpha + q_\\alpha\n{\\tilde u}_\\beta -\\frac{2}{3}{\\bf q}{\\tilde {\\bf u}}\n\\delta_{\\alpha\\beta}\\right) .\n\\end{equation}\nTo simplify more these expressions we\nnote now, that the Fourier components\n${\\tilde \\sigma}_{\\alpha\\beta}$ (\\ref{prestensone1}) of the stress tensor\n$\\sigma_{\\alpha\\beta}$ (\\ref{presstens}) is a symmetric\ntensor with the two independent components ${\\tilde\n\\sigma}_{zz}$ and ${\\tilde \\sigma}_{xz}$ in the Cartesian\ncoordinate system $(x,y,z)$ with the axis $z$ directed to the wave\nvector ${\\bf q}$ because of axial symmetry. The tensor (\\ref{prestensone1})\nhas also zero trace. Hence, from the set of\nequations (\\ref{prestensone1}) only two independent ones \nsurvive, namely, \n\\bel{prestensone2} \n{\\tilde \\sigma}_{zz}=\n-\\frac{4}{3} \\left(\\frac{\\lambda }{\\omega } - i \\nu\\right)~ q\n{\\tilde u}_z,\\quad {\\tilde \\sigma}_{xz}= \n-\\left(\\frac{\\lambda }{\\omega } - i \\nu\\right)~q{\\tilde u}_x, \n\\end{equation}\n with\n\\bel{prestensone2b} \n{\\tilde \\sigma}_{xx}= {\\tilde\n\\sigma}_{yy}= -\\frac{1}{2} {\\tilde \\sigma}_{zz}, \\quad\n {\\tilde \\sigma}_{yz}= {\\tilde \\sigma}_{xz}\n \\quad {\\rm and} \\quad {\\tilde \\sigma}_{xy}=0.\n\\end{equation}\n\n\nOn the other hand, the stress tensor $\\sigma_{\\alpha \\beta}$ (\\ref{presstens})\nin the l.h.s. of (\\ref{prestensone2})\nis determined by the distribution function \n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ in the\nplane-wave representation, see\n(3.10) from \\cite{heipethrev},\n\\begin{eqnarray}\\label{basiceq}\n&&\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )= \n\\left(\\frac{\\partial f_{{\\bf p}} \n}{\\partial \\varepsilon_{{\\bf p}}}\\right)_{\\rm g.e.} \n\\left\\{\\frac{\\omega }{\\mathcal{D}_{{\\bf p}}}\n\\left[\\delta {\\tilde \\mu} +\\frac{m }{m^*} {\\bf p}{\\tilde\n{\\bf u}}\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left. \\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu }{T}\\right)_{\\rm g.e.} \n\\delta {\\tilde T}\n- \\frac{\\mathcal{F}_0 }{\\mathcal{N}(T)} \\delta {\\tilde\n\\rho}\\right]\\right.\\nonumber\\\\ \n&-&\\left.\n\\frac{{\\bf q} {\\bf v}_{{\\bf p}} }{\\mathcal{D}_{{\\bf p}}} \n\\left[\\delta {\\tilde \\mu}+{\\bf p}{\\tilde {\\bf u}}+\n\\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu }{T}\\right)_{\\rm g.e.} \\delta {\\tilde\nT}\\right]\\right\\}.\n\\end{eqnarray}\nThis expression can be easy derived from (\\ref{landvlas})\nafter not too lengthy and simple transformations \n\\cite{heipethrev}, besides of the adaptation to our notations.\nWe substitute then the distribution function\n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ given by (\\ref{basiceq})\nto the l.h.s. of (\\ref{prestensone2}) through\n(\\ref{presstens}) in the considered representation.\nIn this way, we easy realize that the stress tensor (\\ref{presstens})\nhas the above mentioned symmetry properties, and\nits components ${\\tilde \\sigma}_{zz}$, and $\n{\\tilde \\sigma}_{xz}$ of l.h.s. of (\\ref{prestensone2})\nwith some shear modulus $\\lambda$ and viscosity $\\nu$\nare indeed proportional to $q {\\tilde\nu}_z$ and $q {\\tilde u}_x$, respectively.\nAs result, these stress tensor components can be represented for convenience\nin terms of the\ntwo dimensionless quantity $\\chi _{zz}$ and $\\chi _{xz}$ independent of\nthe mean velocity ${\\bf u}$, \n\\bel{dpresszz}\n {\\tilde \\sigma}_{zz}=\n -\\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}} \\chi _{zz} {\\tilde u}_z , \\qquad\n {\\tilde \\sigma}_{xz}=\n -\\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}} \\chi _{xz} {\\tilde u}_x,\n\\end{equation} \nwhere \n\\bel{chizzfun} \n\\chi _{zz} = \\mathcal{J}_1 \\frac{\\rho_{{}_{\\! 0}} }{\\mu}\n \\frac{\\delta {\\tilde T} }{\\delta {\\tilde \\rho}} + \\mathcal{J}_2,\n\\end{equation}\n \\bel{J1def} \n\\mathcal{J}_1=\\frac{2i \\mathcal{N}(T) }{s \\tau \\varepsilon_{{}_{\\! {\\rm F}}}\n\\mathcal{N}(0)} \\left\\langle P_2\\left({\\hat p}_z\\right)~\n\\frac{\\varepsilon_{\\bf p} \n}{\\mathcal{D}_{\\bf p}} \\left(\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n-\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right)\\right\\rangle_{\\rm g.e.}, \n\\end{equation}\n\\begin{eqnarray}\\label{J2def}\n\\mathcal{J}_2&=&-\\frac{4 \\mathcal{N}^2(T) }{3 s \n\\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{N}^2(0)} \n\\left\\langle \\varepsilon_{\\bf p} P_2\\left({\\hat p}_z\\right) \n\\left[\\frac{\\omega }{\\mathcal{D}_{\\bf p}}~\\left(\n\\frac{\\mathcal{N}(0) }{\\mathcal{N}(T)} + \n\\frac{3 p \\omega }{2 \\mathcal{G}_1 q \\varepsilon_{{}_{\\! {\\rm F}}}}~ {\\hat p}_z \\right) \n\\right.\\right. \\nonumber\\\\\n&-&\\left.\\left. \\frac{{\\bf q} {\\bf v}_{\\bf p} }{\\mathcal{D}_{\\bf p}}~ \n\\left(\\frac{\\mathcal{G}_0 \n\\mathcal{N}(0) }{\\mathcal{N}(T)} + \\frac{3 p\\omega \n}{2q\\varepsilon_{{}_{\\! {\\rm F}}}}~{\\hat p}_z\\right)\\right] \\right\\rangle_{\\rm g.e.},\n\\end{eqnarray}\n$P_2(x)$ is the Legendre polynomial, ${\\hat p}_\\alpha$ is the\n$\\alpha$ component of the unit vector ${\\hat p}$ defined in\n(\\ref{intampfpp}). Other quantities were defined in Secs.\\\n\\ref{eqmotion}, \\ref{respfunsec} and Appendix D, \nsee also (\\ref{dfgeqrpt}),\n(\\ref{enerdensnt}), (\\ref{mcapt}), (\\ref{nzero}),\n(\\ref{domindp}) and (\\ref{som}). $\\chi _{xz}$ in\n(\\ref{dpresszz}) is given by \n\\begin{eqnarray}\\label{chixzdef}\n&&\\chi _{xz}=-\\frac{3 \\omega \\mathcal{N}(T) }{s p_{\\rm F}^3 \\mathcal{N}(0)} \n\\left\\langle p^3\n\\frac{{\\hat p}_x^2{\\hat p}_z }{\\mathcal{D}_{\\bf p}} \n\\left(\\frac{s }{\\mathcal{G}_1 } -\n\\frac{p }{ p_{{}_{\\! {\\rm F}}}} {\\hat p}_z\\right)\\right\\rangle\\quad\n\\nonumber\\\\\n&=&-\\frac{3 \\omega \\mathcal{N}(T) }{2 s p_{\\rm F}^3 \\mathcal{N}(0)} \n\\left\\langle p^3\n\\frac{(1-{\\hat p}_z^2){\\hat p}_z }{\\mathcal{D}_{\\bf p}} \\left(\n\\frac{s }{\\mathcal{G}_1 }-\\frac{p}{p_{{}_{\\! {\\rm F}}}} {\\hat p}_z\\right)\n\\right\\rangle. \\quad\n\\end{eqnarray} \nIn the second equation, we used the invariance of the average\nin the first equation with respect to the replace ${\\hat p}_x\n\\rightarrow {\\hat p}_y$, due to the axial symmetry and the equation\n$\\sum_\\alpha {\\hat p}_\\alpha^2=1$ for the unit vector ${\\hat p}$. We\napplied also the thermodynamic relation (\\ref{dmu}) for \n$\\delta {\\tilde \\mu}$ in the distribution function (\\ref{basiceq})\nin these derivations.\n\n\\subsubsection{THE SHEAR MODULUS AND VISCOSITY}\n\\label{calclambnu}\n\nThe shear modulus $\\lambda$ and viscosity $\\nu$ can be now found\nfrom the comparison of (\\ref{prestensone2}) for continuous matter\nand explicit expressions\n(\\ref{dpresszz}) obtained above from the Fermi-liquid distribution function\n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ (\\ref{basiceq}) for the same\nstress tensor components ${\\tilde \\sigma}_{zz}$ and ${\\tilde \\sigma}_{xz}$.\nIndeed, substituting (\\ref{dpresszz}) to the l.h.s. of\n(\\ref{prestensone2}) and canceling the velocity field\ncomponents from their both sides, one finds \n\\bel{lamviseq} \n\\mathcal{J}_1~\\frac{\\rho_{{}_{\\! 0}}}{\\mu_{\\rm g.e.}}~ \n\\frac{\\delta {\\tilde T} }{\\delta\n{\\tilde \\rho}} +\\mathcal{J}_2\n =\\frac{4}{3} \\chi _{xz},\\quad\n\\lambda-iv_{{}_{\\! {\\rm F}}} qs\\nu=\\rho_{{}_{\\! 0}}\\varepsilon_{{}_{\\! {\\rm F}}} s~\\chi _{xz}\\;. \n\\end{equation}\nFrom the first equation one has the ratio \n\\bel{dtdrho}\n\\frac{\\delta {\\tilde T} }{\\delta {\\tilde \\rho}} = \\frac{\\mu_{\\rm g.e.}\n}{\\rho_{{}_{\\! 0}} \\mathcal{J}_1}~ \\left(\\frac{4}{3} \\chi _{xz}-\n\\mathcal{J}_2\\right). \n\\end{equation}\nSeparating real and imaginary parts in the second\nequation, one obtains the shear modulus $\\lambda$ and viscosity\n$\\nu$: \n\\bel{shearmod} \n\\lambda=\n \\frac{\\mid s \\mid ^2 \\chi _{xz} ^\\prime }{s ^\\prime}\\rho_{{}_{\\! 0}}\n\\varepsilon_{{}_{\\! {\\rm F}}}q\n\\end{equation}\n and \n\\bel{viscos} \\nu =\n -\\frac{s ^{\\prime\\prime} \\chi _{xz} ^\\prime +\n s ^\\prime \\chi _{xz} ^{\\prime\\prime} }{s^\\prime }~\n \\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}}.\n\\end{equation} \nWith these constants $\\lambda$ and $\\nu$, the equations\n(\\ref{prestensone}), (\\ref{presslamb}), (\\ref{pressnu}) and \n(\\ref{presstens}) are identities.\n\nThe aim of the following derivations of the shear modulus and the viscosity\nis to simplify $\\mathcal{J}_1$\n(\\ref{J1def}), $\\mathcal{J}_2$ (\\ref{J2def}) and $\\chi _{xz}$\n(\\ref{chixzdef}). For this aim, we make use of\ntransformations of the averages like $\\langle p^k{\\hat p}_z^l\n\\varepsilon_{\\bf p}^m ({\\bf q}{\\bf v}_{\\bf p})^n\/\\mathcal{D}_{\\bf p} \\rangle_{\\rm g.e.}$ with\nsome integer numbers $0 \\leq k \\leq 4$, $0 \\leq l \\leq 4$, $m=0,1$\nand $n=0,1$ in terms of more simpler functions $\\chi _n$ \n($n=0,1,2$) introduced\nin \\cite{heipethrev} for the response functions, see\n(\\ref{chinfun}). For these functions, one has simple temperature and\nhydrodynamic expansions presented below at the end subsection\nof this Appendix B. Using\nsuch enough lengthy and simple algebraic derivations, one\nfinally gets\n\\begin{eqnarray}\\label{J1}\n\\mathcal{J}_1&=&\n \\frac{1}{s (1-i s \\tau_q)}\n \\left[\\left(1+ {\\bar T} \\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)}\n +\\frac{3}{\\tau_q ^2}\n (1-i s \\tau_q)^2\\right)\\frac{\n\\chi _1 }{\\mathcal{N}(0)}\\right.\\nonumber\\\\\n &-&\\left.{\\bar T}\\left(\\frac{\\pi^2 }{3} \n{\\bar C}_V-\\frac{\\chi _2 }{\\mathcal{N}(0)}\n \\right)\\right],\n\\end{eqnarray}\n\\begin{eqnarray}\\label{J2}\n && \\mathcal{J}_2=\n \\frac{2i \\mathcal{N}^2(T) }{3s \\tau_q (1-is \\tau_q) \\mathcal{N}^2(0)}\n \\left[3s\\left(1-i s \\tau_q\\right)^2\\right.\\nonumber\\\\\n&+& \\left.{3i \\tau_q \\over {\\mathcal{G}_1}}\n \\left(1-i s \\tau_q\\right)\n\\left(s^2-\\frac{\\mathcal{G}_0\n \\mathcal{G}_1 \\mathcal{N}(0) }{3 \\mathcal{N}(T) }\\right)\n +\\frac{s \\mathcal{N}(0)\\tau_q ^2 }{\\mathcal{N}(T)}\\right]\\nonumber\\\\\n &\\times& \\left[ \\left(\\frac{3 \\mathcal{N}(0)\n }{ \\mathcal{N}(T)\\tau_q ^2}(1-i s \\tau_q)^2 \n + 1 + {\\bar T}\\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)} \\right)\n \\frac{\\chi _0 }{\\mathcal{N}(T)} \\right.\\nonumber\\\\\n &-&\\left. 1\n -{\\bar T}\\frac{\\mathcal{M}(T) }{ \\mathcal{N}(T)}\n +{\\bar T}\\frac{\\chi _1 }{\\mathcal{N}(T)} \\right],\n\\end{eqnarray}\n\\begin{eqnarray}\\label{chixz}\n&&\\chi _{xz} = -\\frac{3i}{\\tau_q}\n \\left(1-\\frac{i \\mathcal{F}_1 s \\tau_q }{3 \\mathcal{G}_1}\\right)\n \\left[\\left(\\frac{(1-i s \\tau_q)^2\\mathcal{N}(0)\n }{ \\tau_q ^2 \\mathcal{N}(T) } +1\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.{\\bar T}\n \\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)} \\right)\n \\frac{\\chi _0 }{\\mathcal{N}(0)} \n + {\\bar T} \\frac{\\chi _1 }{\\mathcal{N}(0)}\n -\\frac{1}{3}\\left(1+{\\bar T}\n\\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)}\\right)\n \\frac{\\mathcal{N}(T) }{\\mathcal{N}(0)} \\right]\\nonumber\\\\\n &=& \\chi _{xz} ^\\prime + i\\chi _{xz} ^{\\prime\\prime}.\n\\end{eqnarray} \nNote that the shear\nmodulus $\\lambda$ (\\ref{shearmod}) and viscosity\n$\\nu$ (\\ref{viscos})\ndepend on the sound velocity $s$, and hence,\non the solution of the Landau-Vlasov\nequation (\\ref{landvlas}) for $s$ [(\\ref{J1}),(\\ref{J2})\nand (\\ref{chixz})].\n\n\\subsection{Heat current}\n\\label{app2heat}\n\nFor the following derivations of\nthe thermal conductivity $\\kappa$ in Fermi liquids, we need to\nderive the equation for the temperature $T$ from the general\ntransport equation (\\ref{entropyeq}). The latter equation\n(\\ref{entropyeq}) in the linear approximation with respect to the\ndynamical variations $\\delta f$ in terms of the moments, such as the\nvelocity field ${\\bf u}$ (\\ref{veloc}), particle density $\\delta \\rho$,\nentropy density per particle $\\delta \\varsigma$ and so on, writes\n\\bel{entropeqlin} \n\\rho T {\\partial \\varsigma \\over\n{\\partial t}}\n +{\\bf \\nabla} \\cdot {\\bf j}_T = 0,\n\\end{equation} \nwhere ${\\bf j}_T$ is the heat current given in terms of the\nthermal conductivity $\\kappa$ and temperature gradient by\n(\\ref{currheat}). By making use of the thermodynamic relation\nfor the entropy $\\varsigma$ per particle, \n\\bel{thermeqspt} \n\\hbox{d} \\varsigma=\n \\left(\\frac{\\partial \\varsigma }{\\partial {\\cal P}}\\right)_T\n \\hbox{d} {\\cal P}\n +\\left(\\frac{\\partial \\varsigma }{\\partial T}\\right)_{\\cal P}\n \\hbox{d} T,\n\\end{equation} \nand the well known arguments\nto get the\nthermal conductivity equation, we consider the process with the\n{\\it constant pressure} rather than the constant of particle density.\n(We again omitted the symbol variation $\\delta$ as in Sec.\\ \\ref{conserveqs}).\nWith the help of (\\ref{thermeqspt}), one then results in the\nFourier thermal conductivity equation \n\\bel{fouriereq}\n \\rho {\\tt C}_{\\cal P} \\frac{\\partial T }{\\partial t}-\n \\kappa \\triangle T = 0,\n\\end{equation} \nwhere ${\\tt C}_{\\cal P}$ is the specific heat for the\nconstant pressure per particle, see (\\ref{specifheatp}).\n(Equation (\\ref{currheat}) was also used in (\\ref{fouriereq}) for the\nheat current ${\\bf j}_T$). Solving equation (\\ref{fouriereq}) for the\ntemperature $T({\\bf r},t)=T_{\\rm g.e.}+\\delta T$ in terms of the plane\nwaves for the dynamical part of the temperature $\\delta T({\\bf r},t)$\nas in (\\ref{planewave}) and using the relations\n(\\ref{som}), one gets \n\\bel{kappadef} \n\\kappa=i \\rho {\\tt C}_{\\mathcal{P}} v_{{}_{\\! {\\rm F}}}s\/q. \n\\end{equation} \nNotice, the thermal conductivity $\\kappa$\n(\\ref{kappadef}) depends on the sound velocity $s$ as the shear\nmodulus $\\lambda$ (\\ref{shearmod}) and viscosity\n$\\nu$ (\\ref{viscos}), and therefore, on the solution of the Landau-Vlasov\nequation (\\ref{landvlas}) for $s$.\n\n\n\\subsection{Long wave-length limit}\n\\label{app2exp}\n\nAs shown in section \\ref{respfunsec} and subsections \nB.1a and B.1b, \nmany physical quantities, such as the response\nfunctions, see (\\ref{ddresp}), the shear modulus (\\ref{shearmod})\nand viscosity (\\ref{viscos}) can be expressed in terms of the\nsame helpful functions $\\chi _n$ (\\ref{chinfun}). By this reason,\nit is easy to get their \nLWL limit \nby expanding the only $\\chi _n$ in small parameter $\\tau_q$.\n\nFor small $\\tau_q$, one can use asymptotic expansions \nfor the Legendre function of second kind\n$Q_1(\\zeta)$ \nand its derivatives\nwhich\nenter $\\chi _n$ with its derivatives, according to (\\ref{chitemzero}),\n(\\ref{chitemone}) and (\\ref{chitemtwo}). This approximation is\nvalid for large arguments $\\zeta$. Substituting these expansions\ninto the functions $\\chi _n$ (\\ref{chinfun}) and $\\wp$\n(\\ref{alphas}), one gets to fourth order in $\\tau_q$:\n\\begin{eqnarray}\\label{chiexpzero} \n\\chi _0 &=&\n \\frac{\\tau_q^2 }{3} \\left[1 + 2 i s_0 \\tau_q\n -\\left(2 s_1 + 3 s_0^2 +\\frac{3}{5}\\right) \\tau_q^2\\right.\n\\nonumber\\\\\n &-& \\left. \\frac{\\pi^2 \\tbar^2 }{4} \\tau_q^2 \\right] \\mathcal{N}(0),\n\\end{eqnarray} \n\\bel{chiexpone} \n\\chi _1 =\n \\frac{\\pi^2 {\\bar T} \\tau_q^2 }{9}\n \\left[1+2 i s_0 \\tau_q\n -\\left(2 s_1 + 3 s_0^2 +\\frac{6}{5}\\right) \\tau_q^2\\right] \n\\mathcal{N}(0),\n\\end{equation} \n\\begin{eqnarray}\\label{chiexptwo}\n&&\\chi _2 =\n \\frac{\\pi^2 \\tau_q^2 }{9}\n \\left[1+2 i s_0 \\tau_q\n -\\left(2 s_1 + 3 s_0^2 +\\frac{3}{5}\\right) \\tau_q^2 \n -\\frac{\\pi^2 \\tbar^2 }{60}\\right.\\nonumber\\\\\n &\\times&\\left.\\left(1+2 i s_0 \\tau_q\n -\\left(2 s_1 + 3 s_0^2 - 60 \\right) \\tau_q^2 \\right) \\right] \n\\mathcal{N}(0).\n\\end{eqnarray}\nWith these expressions, \none obtains the collective response\nfunction $\\chi _{DD}^{\\rm coll}$ of (\\ref{ddresp}), (\\ref{despfunc}) \nthrough\n\\begin{eqnarray}\\label{ampsexp}\n&&\\aleph(s) \\equiv \\aleph(\\tau_q,s_0,s_1) = \\frac{\\pi^2 \\taubar^3 }{27}\n \\left\\{-3 i s_0 + \\left(1 + 6 s_0^2 +3 s_1\\right) \\tau_q\n\\right.\\nonumber\\\\\n &+& \\left.\\frac{\\pi^2 \\tbar^2 }{120}\n \\left[93 i -\\left(2+186 s_0^2 + 93 s_1\\right)\n \\tau_q \\right]\\right\\} \\mathcal{N}^2(0),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{dszexpzero}\n&&D_0(s) \\equiv D_0(\\tau_q,s_0,s_1) =\n \\frac{\\pi^2 \\taubar^3 }{9}\n \\left\\{-i s_0 \\left(1 - 3 s_0^2\\right)\\right.\\nonumber\\\\\n &+& \\left. \\frac{1}{15}\n \\left[5 + 3 s_0^2 \\left(1-30 s_0^2\\right)+\n 15 s_1\\left(1 -9 s_0^2 \\right)\\right] \\tau_q ~~~\\right.\\nonumber\\\\\n &+& \\left. \\frac{\\pi^2 \\tbar^2 }{120}\n \\left[-i s_0 \\left(19+93 s_0^2\\right)\n+\\frac{1}{15}\n \\left(-160 +18 s_0^2\\right.\\right.\\right.\\nonumber\\\\\n&\\times&\\left.\\left.\\left.\\left(54 +155 s_0^2\\right) \n+ \n 5 s_1 \\left(57 +837 s_0^2 \\right)\\right)\\tau_q\n \\right]\\right\\} \\mathcal{N}(0)~~~\n\\end{eqnarray}\n[also for the temperature-density response function (\\ref{dtresp})].\nThese two quantities determine the expansion of the function $D(s)\n\\equiv D(\\tau_q,s_0,s_1)$ (\\ref{despfunc}) in powers of $\\tau_q$,\nand then approximately, the excitation modes given by the\ndispersion relation (\\ref{despeq}). Indeed, equaling zero the\ncoefficients which appear in front of the each power of $\\tau_q $\nin this expansion of $D(\\tau_q,s_0,s_1)$, we get \nequations for the unknown quantities $s_0$ and $s_1$ of\n(\\ref{somexp}), \n\\begin{eqnarray}\\label{eqzero} \n&&\\frac{i s_0 }{\\mathcal{G}_1}\n\\left[s_0^2 -\\frac{\\mathcal{G}_0 \\mathcal{G}_1 }{3}\n + \\frac{\\pi^2 \\tbar^2 }{120}\n \\left(-40 \\mathcal{G}_1 + 21 \\mathcal{G}_0 \\mathcal{G}_1\n\\right.\\right.\\nonumber\\\\ \n&-&\\left.\\left. 63 s_0^2\n - 30 \\mathcal{G}_1 s_0^2\\right)\\right] = 0,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eqone}\n&&\\frac{1}{45 \\mathcal{G}_1}\\left\\{5 \\mathcal{G}_0 \\mathcal{G}_1 - 3 s_0^2\n\\left[5 +\n 2 \\mathcal{G}_1 \\left(2 - 5 \\mathcal{G}_0 \\right) + 30 s_0^2 \\right] +\n 15 s_1 \\right.\\nonumber\\\\\n&\\times&\\left.\n\\left(\\mathcal{G}_0 \\mathcal{G}_1 -\n 9 s_0^2\\right) \n +\\frac{\\pi^2 \\tbar^2 }{120}\n \\left[40 \\mathcal{G}_1\\left( \\mathcal{G}_0 -5\\right)\n - 6s_0^2 \\left(60 - 287 \\mathcal{G}_1 \\right.\\right.\\right.\\nonumber\\\\\n&+& \\left.\\left.\\left.\n 105 \\mathcal{G}_0 \\mathcal{G}_1 +\n 15 s_0^2 \\left(21 + 10 \\mathcal{G}_1 \\right)\\right)\n + 15 s_1 \\right.\\right.\\nonumber\\\\\n&\\times&\\left.\\left.\n\\left(40 \\mathcal{G}_1 - \n21 \\mathcal{G}_0 \\mathcal{G}_1 +\n 9 s_0^2\\left(21 + 10 \\mathcal{G}_1 \\right) \\right)\\right]\\right\\} \n = 0.\n\\end{eqnarray}\nSolving these equations, one obtains the position of the poles as\ngiven in (\\ref{shp}) and (\\ref{sfirst}).\n\nThe shear modulus ($\\lambda$) and viscosity ($\\nu$) coefficients\nenter the response function $\\chi_{FF}^{\\rm coll}$ (\\ref{chicollfldm})\nand (\\ref{glx}) in terms of the sum $(\\lambda - i \\nu \\omega )\/\\rho_{{}_{\\! 0}}\n\\varepsilon_{{}_{\\! {\\rm F}}}$. The \nLWL expansion of this sum can be\nobtained with help of (\\ref{chiexpzero}), (\\ref{chiexpone})\nand (\\ref{somexp}) and expansions of all static quantities in\n${\\bar T}$ there, see Appendix A, but with taking into account fourth\norder terms,\n\\begin{eqnarray}\\label{chixzexp}\n&&(\\lambda - i \\nu \\omega )\/\\rho_{{}_{\\! 0}} \\varepsilon_{\\rm F}= s \\chi_{xz}({\\bar T},\\omega \\tau)=\n-i \\frac{2}{5}\\left(1 + \\frac{5 \\pi^2 \\tbar^2 }{12} \\right) \\omega \\tau +\n\\nonumber\\\\\n&& \\frac{\\pi^4 {\\bar T}^4 }{2160}\n \\left[13 \\left(\\mathcal{G}_0 - \\frac{4}{5} \\mathcal{G}_1\n - \\mathcal{G}_0 \\mathcal{G}_1\\right) -\n \\frac{13 i \\mathcal{G}_0 \\mathcal{G}_1 }{\\omega \\tau} + \n\\right.\\nonumber\\\\\n && \\left. \\frac{i \\omega \\tau }{25 \\mathcal{G}_0}\n \\left(780 + 2815 \\mathcal{G}_0 + 52 \\mathcal{G}_1\n + 260 \\mathcal{G}_0 \\mathcal{G}_1\\right)\\right].\n\\end{eqnarray}\nSeparating the real and imaginary parts in these equations, one\ngets the LWL approximation of the both {\\it real}\ncoefficients $\\lambda$ and $\\nu$. \nThe terms linear in\n$\\omega \\tau$ determine the hydrodynamic viscosity $\\nu^{(1)}$\n(\\ref{shearvisone}), and the terms proportional to $1\/\\omega \\tau$ are\nrelated to $\\nu^{(2)}$ (\\ref{shearvishp}), see \ndiscussions of the \"heat pole\" for the FLDM transport coefficients \nin Sec.\\ \\ref{transprop}.\n\nThe LWL approximation for\nthe thermal conductivity $\\kappa$ is determined by \nequation (\\ref{kappadef})\nand solutions (\\ref{shp}) for the heat pole \nand (\\ref{sfirst0}) for the sound velocity\n$s$ of the\ndispersion equations (\\ref{eqzero}) and\n(\\ref{eqone}).\n\n\nThe explicit final expressions for the viscosity $\\nu$ and the\nthermal conductivity $\\kappa$ are presented and discussed in\nsubsection \nII D in the LWL limit in\nconnection with the first sound and overdamped (heat pole) modes,\nsee (\\ref{shearvisonehp}) and (\\ref{kappaexp}). As seen\nimmediately from (\\ref{chixzexp}), the linear terms in\n$\\tau_q$ for the shear modulus $\\lambda$ appear as high\ntemperature corrections proportional to ${\\bar T}^4$. They are\nregular in $\\omega \\tau$, and therefore, are totally immaterial, see\nmore discussions in the subsection mentioned above.\nIn the\nlinear approximation in $\\tau_q$ of the \nLWL limit,\nit is easy to check that the\nderivative $\\delta {\\tilde T} \/ \\delta {\\tilde \\rho}$ \n(\\ref{dtdrho}) is the same as obtained in terms of the response\nfunctions in (\\ref{dtdrhoexp}), and therefore, the in-compressibility\n$K_{\\rm tot}$ (\\ref{incomprtot}) turns into the adiabatic one.\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Coupling constants and\nsusceptibilities} \\label{app4}\n\nLet us consider the change of average $\\langle {\\hat F}({\\bf r})\n\\rangle$ of the operator ${\\hat F}({\\bf r})$ (\\ref{foper}) due to a\nquasistatic variation of the particle density $\\rho_{\\rm qs}({\\bf r},Q,T)$,\n\\bel{quasavf} \n\\delta \\langle {\\hat F}({\\bf r}) \\rangle_{\\rm qs}^{\\tt X} =\n\\int~{\\rm d} {\\bf r} ~{\\hat F}^{\\tt X}({\\bf r})~ \\delta \\rho_{\\rm qs}^{\\tt X}\n=-\\chi_{FF}^{\\tt X}\\; \\delta Q\\;, \n\\end{equation}\nwhere \n\\bel{quasdens} \n\\delta \\rho_{\\rm qs}^{\\tt X}\n=\\left[\\left(\\frac{\\partial \\rho_{\\rm qs} }{\\partial Q}\\right)_T\n+\\left(\\frac{\\partial \\rho_{\\rm qs} }{\\partial T}\\right)_Q \n\\frac{\\delta T}{\\delta Q}\\right]^{\\tt X} \\delta Q \\;.\n\\end{equation} \nThe index \"${\\tt X}$\" shows one of the conditions of the constant temperature \n(${\\tt X}=T$), entropy \n(${\\tt X}$ is \"ad\") or static limit $\\omega \\to 0$ (${\\tt X}$ is \"$\\omega =0$\").\nWe shall follow the notations of \\cite{hofmann,hofbook} omitting the\nindex $\\omega =0$ for the coupling constant ($k_{\\omega =0} \\equiv k$),\nsurface energy constant ($b_{{}_{\\! S}}^{\\omega =0} \\equiv b_{{}_{\\! S}}$), and\nin-compressibility [$K^{\\omega =0} \\equiv K=K_{\\rm tot}$ for $\\omega =0$, see\n(\\ref{incomprtot})]. We write it as the zero argument for the\nisolated susceptibility, $\\chi^{\\omega =0} \\equiv \\chi(0)$, and\nstiffness coefficient, $C^{\\omega =0}=C(0)$. $f^{\\tt X}$ and\n$\\delta f^{\\tt X}$ denote the quantity $f$ and its variation\nprovided that the \n${\\tt X}$ condition is\ncarried out. The index \"qs\" stands for the quasistatic quantities\nas in \\cite{hofmann} and will be omitted within this Appendix. \nNote that the\noperator ${\\hat F}({\\bf r})$ (\\ref{foperl}) depends in the FLDM on \n${\\tt X}$ through the derivatives of the particle density, and by\nthis reason, the upper index ${\\tt X}$ appears in \n${\\hat F}^{\\tt X}({\\bf r})$ of (\\ref{quasavf}).\nFrom the first of (\\ref{quasavf}) with (\\ref{foperl}) and\n(\\ref{quasdens}), one gets the self-consistency condition\n(\\ref{selfconsist}),\n\\bel{Favfld} \n\\delta \\left\\langle {\\hat F}({\\bf r})\n\\right\\rangle^{\\tt X} =k_{\\tt X}^{-1} \\delta Q,\n\\end{equation}\n with the following expression for the\ncoupling constant, \n\\begin{eqnarray}\\label{couplfld} \nk_{\\tt X}^{-1}\n&=& - R_0 \\int~{\\rm d} {\\bf r} \\left\\{\\frac{\\partial V }{\\partial r}\nY_{L0}({\\hat r}) \\left[\\left(\\frac{\\partial \\rho }{\\partial\nQ}\\right)_T \\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.\\left(\\frac{\\partial \\rho }{\\partial T}\\right)_Q\n\\frac{\\delta T }{\\delta Q}\\right]\\right\\}_{Q=0}^{\\tt X}\\mathcal{O}_1, \n\\end{eqnarray} \nwithin the ESA parameter of smalleness\n$a\/R \\sim A^{-1\/3}\\ll 1$,\n$\\mathcal{O}_1=1+\\mathcal{O}\\left(a\/R\\right)=\n1+\\mathcal{O}\\left(A^{-1\/3}\\right)$.\nFor the susceptibilities $\\chi_{FF}^{\\tt X}$ defined by (\\ref{quasavf}) \nup to small corrections of the\norder of $A^{-1\/3}$ in the same approximation, from\n(\\ref{couplfld}) for $k_{\\tt X}^{-1}$ one gets \n\\bel{chix} \n\\chi^{\\tt X}=-k_{\\tt X}^{-1}\\mathcal{O}_1.\n\\end{equation}\nWe omit also the low indexes \"FF\"\nfor the susceptibilities. Note that we have not identities of\n$-k_{\\tt X}^{-1}$ to $\\chi^{\\tt X}$ because we neglected earlier\nhigh order $A^{-1\/3}$ corrections in the derivation of the\noperator ${\\hat F}$ (\\ref{foperl}), in particular, in the FLDM\napproximation (\\ref{denseq}) for the quasistatic particle density\n$\\rho_{\\rm qs}$. \nThe equation (\\ref{chix}) is in\nagreement with (\\ref{kstiffC0chi0}) (identical to equation (3.1.26)\nof \\cite{hofmann}), see also\n(\\ref{smallpar}), (\\ref{isorespk}), for the specific relation\nbetween the coupling constant $k^{-1}$ and isolated susceptibility\n$\\chi(0)$ {\\it with presence of the stiffness term} $C(0)$ in \"the\nzero frequency limit\" within the FLDM. As shown in \nSec.\\ \\ref{zerofreqlim} through (\\ref{chicollfldm}) by using the\nexpansion in small parameter $kC$ (\\ref{smallpar}) up to the\nsecond order terms in $kC$, the isolated susceptibility\n$\\chi(0)$, see (\\ref{respintr}) at $\\omega =0$, is related to the\ncoupling constant $k^{-1}$ by (\\ref{kstiffC0chi0}) with the\nstiffness term $C(0)$. The correction related to the stiffness\n$C(0)$ appears in (\\ref{chix}) in a \nhigher order than\n$A^{-1\/3}$ because it is of the order of the small parameter $kC\n\\sim A^{-2\/3}$, see (\\ref{smallpar}) and discussion near this\nequation. The zero frequency stiffness $C(0)$ is equal\napproximately to the liquid drop one $C$ (\\ref{stiffness0}) in the\nFLDM for the considered enough large temperatures for which the\nquantum shell effects can be neglected.\n\n\nThe derivatives of the quasistatic particle density in\n(\\ref{quasdens}), (\\ref{Favfld}) and (\\ref{couplfld}) can be\nfound from (\\ref{denseq}), \n\\begin{eqnarray}\\label{drdqdt} \n\\left(\\frac{\\partial \\rho }{\\partial Q}\\right)_T^{\\tt X}\n&=& -\\left(\\frac{\\partial \\rho \n}{\\partial r}\\right)^{\\tt X} R_0 Y_{L0}({\\hat r}),\n\\nonumber\\\\\n\\left(\\frac{\\partial \\rho }{\\partial T}\\right)_Q^{\\tt X} &=& \n\\frac{1}{\\rho_0^{\\tt X}}\\frac{\\partial \\rho_0^{\\tt X} }{\\partial T}\n\\rho^{\\tt X}+\\frac{R_0 }{3\\rho_\\infty} \\frac{\\partial \\rho_\\infty\n}{\\partial T} \\left(\\frac{\\partial \\rho }{\\partial\nr}\\right)^{\\tt X}, \n\\end{eqnarray}\n for $Q=0$ with \n\\bel{rho0x} \n\\rho_0^{\\tt X}=\n\\rho_\\infty \\left(1+\\frac{6 b_{{}_{\\! S}}^{\\tt X} r_{{}_{\\! 0}} }{K^{\\tt X} R_0\n}\\right), \n\\end{equation}\n as in (\\ref{rho0}). We emphasize\nthat the surface energy constant $b_{{}_{\\! S}}^{\\tt X}$ (or the\nsurface tension coefficient $\\alpha^{\\tt X}$) depends also on the\ntype of the process specified by index ${\\tt X}$ as the\nin-compressibility $K^{\\tt X}$ because of the ${\\tt X}$-dependence\nof the particle density derivative in the integrand of\n(\\ref{tensionconst}) for the tension coefficient. The total\nquasistatic energy is the sum of the volume and surface parts\ndetermined by the in-compressibility $K^{\\tt X}$ and surface\n$b_{{}_{\\! S}}^{\\tt X}$ constants, respectively. The in-compressibility\nmodulus $K^{\\tt X}$ (responsible for the change of the volume\nenergy) is given by (\\ref{incomprTdef}) for ${\\tt X}=T$, and\n(\\ref{incompraddef}) for ${\\tt X}$ is \"ad\", see also\n(\\ref{isotherk}), (\\ref{Kadiabat}) or \n(\\ref{isotherkexp}), (\\ref{incompradexp}) of their more \nspecific expressions for\nnuclear matter.\nThe in-compressibility $K$ equals the adiabatic one\n$K^{\\varsigma}$ as shown through (\\ref{dtdrhoexp}) and\n(\\ref{incompradexp}), $K=K_{\\rm tot}(\\omega =0)=K^{\\varsigma}$. In the\nderivations of (\\ref{drdqdt}), we took into account that \n$\\rho_0^{\\tt X}$ (\\ref{rho0x}) does not depend on $Q$, and the\ndensity $\\rho_\\infty$ (or $r_0$) is assumed to be approximately\nindependent of index \"${\\tt X}$\" in (\\ref{drdqdt}).\n\n\nSubstituting (\\ref{drdqdt}) into (\\ref{couplfld}) for the\ncoupling constant $k_{\\tt X}^{-1}$, one writes \n\\begin{eqnarray}\\label{couplfld1}\n&&k_{\\tt X}^{-1} = R_0^2 \\int~{\\rm d} {\\bf r} \\left\\{{\\partial V \\over\n{\\partial r}} \\frac{\\partial \\rho }{\\partial r} Y_{L0}({\\hat r})~\\qquad\n\\right.\\nonumber\\\\\n&\\times & \\left.\\left[Y_{L0}({\\hat r}) -\\frac{1}{3 \\rho_\\infty} \n\\frac{\\partial\n\\rho_\\infty }{\\partial T} \\frac{\\delta T }{\\delta\nQ}\\right]\\right\\}_{Q=0}^{\\tt X} \\mathcal{O}_1.\n\\end{eqnarray}\nThe\nfirst term proportional to the density in $\\partial \\rho\/\n\\partial T$ of (\\ref{drdqdt}) leads to small $A^{-1\/3}$\ncorrections to the coupling constant $k_{\\tt X}^{-1}$\n(\\ref{couplfld1}) with respect to the second component depending\non the coordinate derivative $\\partial \\rho\/\\partial r$. However,\nall terms including these corrections related to the variation of\nthe temperature $\\delta T$ in (\\ref{couplfld}) [or\n(\\ref{couplfld1})] can be neglected as compared to the first\nterm in the square brackets. (It comes from the variation of the\ncollective variable $\\delta Q$ up to the same relatively small\ncorrections of the order of $A^{-1\/3}$.) Indeed, for the\nisothermal case \"${\\tt X}=T$\" one has it exactly by its\ndefinition. For other \"${\\tt X}$\" the quantity $\\delta T\/\\delta\nQ$ in (\\ref{Favfld}) and (\\ref{couplfld1}) with the density\n(\\ref{denseq}) can be transformed within the ES precision\n\\bel{dtdqx} \n\\left(\\frac{\\delta\nT }{\\delta Q}\\right)^{\\tt X} =\\left(\\frac{\\delta T }{\\delta\n\\rho} \\frac{\\partial \\rho }{\\partial Q}\\right)^{\\tt X}\n=\\left(\\frac{\\partial T }{\\partial r}\\right)^{\\tt\nX}R_0Y_{L0}({\\hat r}).\n\\end{equation}\n For\ninstance, for the constant entropy (adiabatic) condition $S=\\int\n{\\rm d}r \\rho \\varsigma=S(\\rho,T)={\\rm const.}$, see\n(\\ref{entropydef}) with the quasistatic particle density\n$\\rho$ (\\ref{denseq}), the derivative $\\delta T\/\\delta Q$ can be\ncalculated through a variation of this density $\\rho$ as shown in\nthe middle of (\\ref{dtdqx}). In the quasistatic limit $\\omega \n\\to 0$ all quantities of the equilibrium state can be considered\nalso as a functional of the only density $\\rho$ (\\ref{denseq})\nin the ESA \nand one has again\n(\\ref{dtdqx}). We have used already this property in the\nderivation of the operator ${\\hat F}({\\bf r})$ for transformations of\nthe derivatives of a mean field $V$ in (\\ref{foperl}). As noted\nand used for the derivations in Appendix A.1, \nthe\ntemperature $T({\\bf r})$ is approximately independent of the spatial\ncoordinates ${\\bf r}$ at equilibrium. \nTherefore, according to (\\ref{dtdqx}),\nthe second terms in (\\ref{Favfld}) and (\\ref{couplfld1}),\nwhich appear due to the temperature variation $\\delta T$, turn\ninto zero with the FLDM precision. \n\nAfter the simple integration over the angles ${\\hat r}$ in\n(\\ref{couplfld1}) for the coupling constant $k_{\\tt X}^{-1}$,\none then arrives at\n\\bel{couplfld2}\nk_{\\tt X}^{-1}\n= R_0^4 \\int_0^\\infty~{\\rm d} r \\left[ \\frac{\\delta V }{\\delta\n\\rho} \\left(\\frac{\\partial \\rho }{\\partial r}\\right)^2\n\\right]_{Q=0}^{\\tt X} \\mathcal{O}_1.\n\\end{equation}\nAccording to (\\ref{denseq}),\nthe integrands in (\\ref{couplfld2}) contains the sharp bell\nfunction ${\\partial \\rho\/\\partial r}$ of $r$. Therefore, the\nintegrals converges there to a small spatial region near the\neffective nuclear surface defined as the positions of maxima of\nthis derivative at $r=R_0$ (Appendix D). We use these properties of the\nintegrand in the derivation of (\\ref{couplfld2}) taking\nsmooth quantities as $r^2$ at the nuclear surface point $r=R_0$\noff the integrals up to small corrections of the order of\n$A^{-1\/3}$ within the same ESA. [This is like for the\nderivations of the boundary conditions\n(\\ref{bound1}), (\\ref{bound2}), see\n\\cite{strutmagbr,magstrut,strutmagden}, and of (\\ref{foperl})\nfor the operator ${\\hat F}({\\bf r})$.] In this way, we get the expansion\nof the coupling constant $k_{\\tt X}^{-1}$ (\\ref{couplfld}), in\npowers of the $A^{-1\/3}$ with the leading term shown in the second\nequation there, see (\\ref{couplfld2}).\n\nFor the following derivations, we specify now the quasistatic\nderivative $\\left(\\delta V \/ \\delta \\rho\\right)^{\\tt X}$ at $Q=0$\ntaken it from (\\ref{gradrelmu}), \n\\bel{derVderP1} \n{\\bf \\nabla}\nV^{\\tt X}~ =~ \\frac{K^{\\tt X} }{9 \\rho_0^{\\tt X}}~ {\\bf \\nabla}\n\\rho^{\\tt X}, \n\\end{equation}\n where index ${\\tt X}$ in ${\\bf \\nabla} f^{\\tt\nX}$ means the gradient of the quantity $f$ taken for the condition\nmarked by ${\\tt X}$ as in the variation $\\delta f^{\\tt X}$. The\nproportionality of the gradients in (\\ref{derVderP1}) shows the\nself-consistency within the ESA precision, see\n\\cite{magstrut} for more general relations of the self-consistency\nin the FLDM.\n\nUsing (\\ref{derVderP}) and (\\ref{tensionconst}) in \n(\\ref{couplfld2}) for the coupling\nconstants and (\\ref{chix}) for the susceptibilities, one obtains\nthe identical results for these quantities with \nsmall corrections\nof the order of $A^{-1\/3}$, \n\\bel{couplxchix} \n\\chi^{\\tt X}= -k_{\\tt X}^{-1}\\mathcal{O}_1=\n\\frac{K^{\\tt X} b_{{}_{\\! S}}^{\\tt X} R_0^4 }{72 \\pi\nr_0^2 \\rho_\\infty^{\\tt X} \\mathcal{C}} \\mathcal{O}_1.\n\\end{equation}\n\nWe shall show now from (\\ref{couplxchix}) that the adiabatic\nsusceptibility $\\chi ^{\\rm ad}$ and coupling constant $k_{\\rm ad}^{-1}$\nare equal to the isolated ($\\chi(0)$) and quasistatic ($k^{-1}$)\nones, respectively, up to small $A^{-1\/3}$ corrections within the\nESA.\n\nAs noted above, for the adiabatic ($K^{\\varsigma}$) and\nquasistatic ($K$) in-compressibility modula, we got\n$K^{\\varsigma}=K$, see after (\\ref{dtdrhoexp}). \nThe surface energy constant $b_{{}_{\\! S}}$ equals\nthe adiabatic one $b_{S}^{\\rm ad}$, according to\n(\\ref{keybound}). Indeed, the volume energy per particle is\nalso approximately the same for these cases, \n$b_{\\mathcal{V}}^{\\rm ad} = b_{{}_{\\! \\mathcal{V}}}$,\nbecause of its relation $b_{{}_{\\! \\mathcal{V}}} \\approx K\/18$ to the\nin-compressibility modulus ($b_{\\mathcal{V}}^{\\rm ad}=K^{\\varsigma}\/18$)\nwithin the ESA \\cite{strutmagden}\nand equivalence of the corresponding in-compressibility modula. The\nfunctional derivative in (\\ref{keybound}) is the quasistatic\nchemical potential $\\mu$ which does not depend obviously on the\ntype of the process ${\\it X}$.\nFrom (\\ref{keybound}), one gets now $\\alpha^{\\rm ad}=\\alpha$ for\nthe surface tension coefficient or\n$b_{{}_{\\! S}}=b_{S}^{\\rm ad}$ for the surface energy constant. Namely,\nthis quantity should be identified with the experimental value\n$b_{{}_{\\! S}}=17-19~\\mbox{MeV}$ in the FLDM computations. \nThus, from (\\ref{couplxchix}) one obtains the ergodicity condition\n(\\ref{ergodicity1}) for the susceptibilities within the ESA precision,\n\\bel{chi0adfld} \n\\chi(0)=\\chi^{\\rm ad}=\\frac{K b_{{}_{\\! S}} r_0^5\n}{54 \\mathcal{C}} A^{4\/3}, \n\\end{equation}\n up to small $A^{-1\/3}$ corrections.\nAs seen from (\\ref{couplxchix}), one gets also\n$k^{-1}=k_{\\rm ad}^{-1}$ for the coupling constants within the same\napproximation , see (\\ref{kfld}) for $k^{-1}$. The index \"ad\"\nfor the coupling constant will be omitted below in line of\n\\cite{hofmann}.\n\nWe are interested also in the discussion of the difference between\nthe susceptibilities $\\chi^T$ and $\\chi^{\\rm ad}$. From\n(\\ref{chix}), one has \n\\bel{chiTchiad}\n\\chi^T-\\chi^{\\rm ad}=\\left(k_T^{-1}-k^{-1}\\right) \\mathcal{O}_1. \n \\end{equation} \nIt is useful to re-derive this\nrelation by applying Appendix A1 of \\cite{hofmann} for the\nspecific Fermi-liquid drop thermodynamics, see\n(\\ref{ctcaddif}). As noted in Appendix A.4, \nit is\nmore convenient to use the Gibbs free energy $G$ instead of the\nfree energy $F$. As the derivations of the $\\chi^T-\\chi^{\\rm ad}$ in\nAppendix A1 of \\cite{hofmann} do not contain any change in the\nvolume and pressure variables, we can use all formulas in A1 of\n\\cite{hofmann} here with the replace of the free energy $F$ by\nthe Gibbs free energy $G$, in particular (\\ref{ctcaddif}).\nThere is a specific property of the FLDM with respect to the\nmicroscopic shell model with the residue interaction of\n\\cite{hofmann}. The second derivatives of the Hamiltonian\n$\\langle\\left(\\partial ^2 {\\hat H} \/ {\\partial\nQ^2}\\right)_{Q=0}\\rangle_0$ in equations (A.1.6) and (A.1.7) of\n\\cite{hofmann} \ndepend in the FLDM on the type of the\nprocess ${\\tt X}$, isothermal and adiabatic one, relatively. The\nfirst derivative of the Hamiltonian ${\\partial H\/\\partial Q}$ in\nequations (A.1.2) and (A.1.3), see \\cite{hofmann}, is proportional to\nthe derivatives of the density ${\\partial \\rho\/\\partial Q}$ like\nin the susceptibilities (A.1.8), \n\\bel{dhdq} \n\\left(\\frac{\\partial H\n}{\\partial Q}\\right)^{\\tt X} \n=\\left(\\frac{\\delta V }{\\delta\n\\rho} \\frac{\\partial \\rho }{\\partial Q}\\right)^{\\tt X}\n=-\\left(\\frac{\\partial V }{\\partial r}\\right)^{\\tt\nX}R_0Y_{L0}({\\hat r}). \n\\end{equation}\nWe used also this self-consistent\ndependence of mean potential $V$ through the particle density\n$\\rho$ in the derivations of the operator ${\\hat F}({\\bf r})$\n(\\ref{foperl}) in the FLDM: The derivatives of $V$ are\nproportional to the ones of the density $\\rho$ [see (\\ref{derVderP1})], \nwhich both depend\non ${\\tt X}$, i.e., whether we consider the latter for the fixed temperature\nor entropy. Therefore, (A.1.6) and (A.1.7) with the definitions\n(A.1.8) of \\cite{hofmann} as applied to the FLDM, see\n(\\ref{ctcaddif}), should be a little modified to \n\\begin{eqnarray}\\label{d2GEdq2}\n\\left(\\frac{\\partial^2 G }{\\partial Q^2}\\right)_T&=&\n\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^T -\\chi^T ,\\nonumber\\\\\n\\left(\\frac{\\partial^2 E }{\\partial Q^2}\\right)_S&=&\n\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^{\\rm ad} -\\chi^{\\rm ad} .\n\\end{eqnarray} \nThe derivatives of\nthe thermodynamic potential G are considered for the constant\npressure instead of the constant volume as used for the free\nenergy case. Similar calculations of the average value of the\nsecond derivative of the Hamiltonian $\\langle\\partial ^2 {\\hat H}\/\n\\partial Q^2\\rangle^{\\tt X}$ as for the coupling constants lead\nto\n\\begin{eqnarray}\\label{secderiv}\n&&\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^{\\tt X}= \\int {\\rm d} {\\bf r} \n\\left(\\frac{\\partial\n^2 V }{\\partial Q^2}\\right)_{Q=0}^{\\tt X} \\rho \\qquad\\nonumber\\\\\n&=& -R_0^4\\int_0^{\\infty}\n{\\rm d} r \\left(\\frac{\\partial V }{\\partial r}~ \\frac{\\partial \\rho\n}{\\partial r}\\right)_{Q=0}^{\\tt X} \\mathcal{O}_1. \n\\end{eqnarray}\nWe integrated in the second equation of (\\ref{secderiv}) by\nparts. Taking then (\\ref{denseq}) for the quasistatic density\n$\\rho_{\\rm qs}$, one gets \n\\bel{d2hdq2} \n\\left\\langle\\left(\\frac{\\partial ^2\n{\\hat H} }{\\partial Q^2}\\right)_{Q=0}\\right\\rangle_0^{\\rm qs}=\n-k^{-1} \\mathcal{O}_1\n\\end{equation} \nwith the\ncoupling constant $k$ (\\ref{kfld}). Using the same transformations\nof the thermodynamic derivatives as in Appendix A1 of\n\\cite{hofmann}, see (\\ref{ctcaddif}),\nfrom (\\ref{d2GEdq2}) and (\\ref{d2hdq2}) up\nto relatively small $A^{-1\/3}$ corrections of the ESA, one gets\n\\begin{eqnarray}\\label{chiTchiad1}\n&&\\chi^T-\\chi^{\\rm ad}= \n-\\left[\\left(\\left(\\frac{\\partial^2 G}{\\partial\nT^2}\\right)_Q\\right)^{-1} \\left(\\frac{\\partial^2 G}{\\partial T\\partial\nQ}\\right)^2\\right]_{Q=0}\\nonumber\\\\\n&+& k_T^{-1}-k^{-1} \\;.\n\\end{eqnarray}\nApplying then the relation of the Gibbs free energy $G=A \\mu$ to\nthe chemical potential $\\mu$, we note that there is the factor $A^{-1}$ which\nsuppress much the contribution of the first term compared to\nsecond one, $k_T^{-1}-k^{-1}$, see (\\ref{couplxchix}),\n\\bel{ktk} \nk_T^{-1}-k^{-1}= \\frac{r_0^5 A^{4\/3} K b_{{}_{\\! S}} }{56\n\\mathcal{C}} \\left(1-\\frac{K^{T} }{K} \\frac{b_{{}_{\\! S}}^T \n}{b_{{}_{\\! S}}}\\right) \\mathcal{O}_1.\n\\qquad\\qquad \n\\end{equation} \nMoreover, the terms in\nthe square brackets of\n(\\ref{chiTchiad1}) are zero because the second derivative\n$~\\left(\\partial \\mu\/\\partial Q\\right)_T~$ is zero within the\nprecision of the FLDM. To show this, let us take the equation as\nfor the temperature (\\ref{dtdqx}) with the only replace of the\ntemperature $T$ by the chemical potential $\\mu$. \nThe above mentioned statement becomes\nnow obvious because the chemical\npotential $\\mu$ is a constant as function of the spatial\ncoordinates at the equilibrium as the temperature $T$\nindependently on the type of the process ${\\tt X}$\nwithin the FLDM.\nAs the result, we obtain the same\nrelation (\\ref{chiTchiad}) with the difference of the coupling\nconstants shown in (\\ref{ktk}).\n\nWe can evaluate the ratio of the surface energy coefficients\n$b_{S}^T\/b_{{}_{\\! S}}$ of (\\ref{couplxchix}) using in\n(\\ref{chiTchiad}) the fundamental relation\n(\\ref{chitchiskSkT}) for the ratio of the susceptibilities\n$\\chi^T\/\\chi^{\\rm ad}$ in terms of the in-compressibility modula\n$K\/K^T$ ($K=K^{\\varsigma}=K^{\\rm ad}$), \n\\bel{bstbs} \n\\frac{b_{S}^T }{b_{{}_{\\! S}}}= \n\\frac{K }{K^T}~\\left(2-\\frac{K }{K^T}\\right)\n\\approx 1+\\frac{2 \\pi^2 {\\bar T}^2 }{3 \\mathcal{G}_0}. \n\\end{equation} \nIn the last\nequation, we used the temperature expansions for the\nin-compressibilities $K$ (\\ref{incompradexp}) and $K^T$\n(\\ref{isotherkexp}). Thus, the surface energy constant $b_{S}^T$\nfor the constant temperature is larger than adiabatic (or\nquasistatic) $b_{{}_{\\! S}}$ and their difference is small as\n${\\bar T}^2$.\n\n\n\\setcounter{equation}{0}\n\n\\section{Symmetry-energy density functional and boundary conditions}\n\\label{endenfun}\n\nThe nuclear energy, \n$E=\\int \\hbox{d} {\\bf r}\\; \\rho_{+}\\;{\\cal E}\\left(\\rho_{+},\\rho_{-}\\right)\\;$, \nin the local density approach\n\\cite{brguehak,chaban,reinhard,bender,revstonerein,ehnazarrrein,pastore}\ncan be calculated through the energy density \n${\\cal E}\\left(\\rho_{+},\\rho_{-}\\right)$ per particle, \n\\begin{eqnarray}\\label{enerden}\n&&{\\cal E}\\left(\\rho_{+},\\rho_{-}\\right) =\n- b_{{}_{\\! \\mathcal{V}}} \n+ J \\mathcal{I}^2 \n+ \n\\frac{K}{18}\\epsilon_{+}(\\rho_{+}) -\nJ \\mathcal{I}^2\\epsilon_{-}(\\rho_{+},\\rho_{-}) +\n\\quad\\nonumber\\\\\n&& \n\\left(\\frac{\\mathcal{C}_{+}}{\\rho_{+}} +\\mathcal{D}_{+} \n\\right) \n\\left(\\nabla \\rho_{+}\\right)^2 \n+ \\left(\\frac{\\mathcal{C}_{-}}{\\rho_{+}} + \n\\mathcal{D}_{-} \n\\right) \n\\left(\\nabla \\rho_{-}\\right)^2.\\quad\n\\end{eqnarray}\nHere, $\\rho_{\\pm}=\\rho_n \\pm \\rho_p$ are the isoscalar, $\\rho_{+}$, and \nisovector, $\\rho_{-}$, particle densities, $\\mathcal{I}=(N-Z)\/A$ \nis the asymmetry parameter, \n$N$ and $Z$ are the neutron and proton numbers in the nucleus, $A=N+Z$. \nThe particle separation energy \n$b_{{}_{\\! \\mathcal{V}}} \\approx$ 16 MeV \nand the symmetry energy constant of nuclear matter $~J \\approx$ \n30 MeV are introduced also in (\\ref{enerden}). \nThe in-compressibility modulus of \nthe symmetric nuclear matter \n $K \\approx 220-260$ MeV is shown in Table I of \n\\cite{chaban,reinhard,BMRV}). \nEquation (\\ref{enerden})\ncan be applied approximately to the\nrealistic Skyrme forces \\cite{chaban,reinhard}, in particular\nby neglecting small semiclassical $\\hbar$ corrections \nand Coulomb terms \\cite{brguehak,strtyap,strutmagbr,strutmagden,magsangzh}. \n$\\mathcal{C}_{\\pm}$ and $\\mathcal{D}_{\\pm}$ are constants defined by the \nbasic Skyrme force parameters. The isoscalar \nsurface energy-density part, independent explicitly \nof the density gradient terms, is determined by the dimensionless function \n$\\epsilon_{+}(\\rho_{+})$ satisfying\nthe standard saturation conditions \\cite{strutmagden,magsangzh,BMRV}.\nFor the derivation of the explicitly analytical results, we \nuse the quadratic approximation $\\epsilon_{+}(\\rho_{+}) = \n(1-\\rho_{+}\/\\rho_\\infty)^2=(1-w_{+})^2$, where $\\rho_\\infty\\approx$ 0.16 fm$^{-3}$ \nis the density of infinite nuclear matter [see around \n(\\ref{denseq})].\nThe isovector component can be simply evaluated as \n$\\epsilon_{-}=1 - \\rho_{-}^2\/(\\mathcal{I}\\rho_{+})^2=1-w_{-}^2\/w_{+}^2$.\nIn both these energies $\\epsilon_{\\pm}$, \n$w_{\\pm}=\\rho_{\\pm}\/(\\mathcal{I}_{\\pm} \\rho_\\infty)$ are the dimensionless\nparticle densities, $\\mathcal{I}_{+}=1$ and \n$\\mathcal{I}_{-}= \\mathcal{I}$.\nThe isoscalar SO gradient terms in (\\ref{enerden}) are defined with a\nconstant: \n$\\mathcal{D}_{+} = -9m W_0^2\/16 \\hbar^2$, where\n$W_0 \\approx$100 - 130~ MeV$\\cdot$fm$^{5}$ \n and $\\mathcal{D}_{-}$ is relatively small \n\\cite{brguehak,chaban,reinhard}.\n\n\nFrom the condition of the minimum energy $E$ \nunder the constraints\nof the fixed particle number $A=\\int \\hbox{d} {\\bf r}\\; \\rho_{+}({\\bf r})$ and\n neutron excess $N-Z= \\int \\hbox{d} {\\bf r}\\; \\rho_{-}({\\bf r})$,\none arrives at the Lagrange equations for $\\rho_{\\pm}$ with the corresponding \nmultipliers being the \nisoscalar and isovector chemical potentials\nwith the surface corrections at the first order, \n$\\Lambda_\\pm \\propto \\mathcal{I}_\\pm a\/R \\sim A^{-1\/3}$ \n\\cite{strutmagbr,strutmagden,magsangzh,BMRV}. \n\n\nThe isoscalar and isovector \nparticle densities $w_\\pm$ \ncan be derived from (\\ref{enerden})\nfirst at the leading approximation in a small parameter \n$a\/R$. For the isoscalar particle density $w_{+}=w_{+}(\\xi)$ \n[$\\xi$ is the distance\nof the given point ${\\bf r}$ from the ES in units of the diffuseness\nparameter $a$ in the local ES coordinates, see (\\ref{denseq}), \n$\\xi = (r-R)\/a$ for the spherical nuclei], \none finds (Appendix B of \\cite{BMRV} and\n\\cite{strutmagden,magsangzh}),\n\\bel{ysolplus}\n\\xi=-\\int_{w_r}^{w}\\hbox{d} y\\; \\sqrt{\\frac{1 +\\beta y}{y\\epsilon(y)}}\\;,\\qquad\n\\end{equation}\nbelow the turning point $~\\xi(w=0)~$ and $~w=0~$ for $~\\xi \\geq \\xi(w=0)~$,\n$\\beta=\\mathcal{D}_{+}\\rho_\\infty\/\\mathcal{C}_{+}$ is the dimensionless SO\nparameter (for simplicity of the notations, we omit the low index\n``+'' in $w_{+}$). \nFor $w_r=w(\\xi=0)$,\none has the boundary condition,\n$\\hbox{d}^2 w(\\xi)\/\\hbox{d} \\xi^2=0$\nat the ES ($\\xi=0$):\n\\bel{boundeq}\n\\epsilon(w_r)+w_r(1 +\\beta w_r) \\left[\\hbox{d} \\epsilon(w_r)\/\\hbox{d} w\\right]=0.\n\\end{equation}\n(see Appendix B of \\cite{BMRV} where the specific solutions for $\\xi(w)$\nin the quadratic approximation for $\\epsilon_{+}(w)$ in terms of \nelementary functions were derived).\nFor $\\beta=0$ (i.e. without SO terms), it simplifies to\nthe solution $w(\\xi)=\\tanh^2\\left[(\\xi-\\xi_0)\/2\\right]$ for\n$\\xi\\leq \\xi_0=2{\\rm arctanh}(1\/\\sqrt{3})$\nand zero for $\\xi$ outside the nucleus ($\\xi>\\xi_0$).\nFor the same leading term of the isovector density, $w_{-}(w)$,\none approximately (for large enough constants $c_{sym}$ \nof all desired Skyrme forces \n\\cite{chaban,reinhard}) \nfinds (Appendix A of \\cite{BMRV}) \n\\bel{ysolminus}\nw_{-} = w\\;\\left(1-\\frac{\\widetilde{w}^2(w) \\left[1\n+ \\tilde{c}\\widetilde{w}(w)\\right]^2}{2\\left(1+\\beta\\right)}\n\\right).\n\\end{equation}\nwhere\n\\bel{defpar}\n\\widetilde{w}=\\frac{1-w}{c_{sym}}, \\quad \nc_{sym}=a \\sqrt{\\frac{J}{\\rho_\\infty\\vert\\mathcal{C}_{-}\\vert}},\\quad\n\\widetilde{c}=\\frac{\\beta c_{sym}\/2-1}{1+\\beta},\n\\end{equation}\nand $a \\approx 0.5 - 0.6$ fm is the \ndiffuseness parameter [see (\\ref{denseq})]. \n\n\nSimple expressions for the constants \n$b_S^{(\\pm)}$ \n(\\ref{bsplusminus}) can be easily\nderived in terms of\nthe elementary functions \nin the quadratic approximation to $\\epsilon_{+}(w)$,\ngiven explicitly in Appendix A \\cite{BMRV}.\nNote that in these derivations we neglected curvature terms \nand being of the same order shell corrections. \nThe isovector energy terms were obtained within the ES \napproximation with high accuracy up to the product of two\nsmall quantities, $\\mathcal{I}^2$ and $(a\/R)^2$.\n\n\n\nWithin the improved ES approximation accounting also \nfor next order corrections in a small parameter \n$a\/R$, we derived \nthe macroscopic boundary conditions \n(Appendix B of \\cite{BMRV})\n\\begin{eqnarray}\\label{macboundcond}\n&&\\delta \\mathcal{P}_{\\pm}\\Big|_{ES} \n\\equiv \\left(\\rho_\\infty\\;\\mathcal{I}_{\\pm}\\;\n\\Lambda_{\\pm}\\right)_{ES} = \\delta P_{S}^{\\pm},\\nonumber\\\\ \n&&\\mbox{where}\\quad \\delta P_{S}^{\\pm} \\equiv 2 \\alpha_{\\pm} \n\\delta \\mathcal{H} \n\\end{eqnarray}\n are the\nisovector and isoscalar surface-tension (capillary) pressures, \n$\\delta \\mathcal{H} \\approx -\\delta R_\\pm\/R_\\pm^2$ are small \nvariations of \nmean ES curvatures $\\mathcal{H}$ (\\ref{keybound}), \n$\\delta R_\\pm $\nare radius variations (\\ref{surface}),\nand $\\alpha_{\\pm}$ are the tension coefficients, respectively,\n\\begin{eqnarray}\\label{sigma}\n\\alpha_{\\pm}&=&b_S^{(\\pm)}\/4 \\pi r_0^2,\\quad b_{S}^{(\\pm)} \\approx\n\\frac{8 \\pi}{a}\\left(\\rho_\\infty \\mathcal{I}_\\pm\\right)^2 \n\\mathcal{C}_{\\pm}\\nonumber\\\\\n&\\times&\\int_{-\\infty}^{\\infty} \\hbox{d} \\xi\\;\n\\left(1 + \\frac{\\mathcal{D}_{\\pm}\\rho_\\infty}{\\mathcal{C}_{\\pm}} w_{+}\\right)\n\\left(\\frac{\\partial w_{\\pm}}{\\partial \\xi}\\right)^2.\\quad\n\\end{eqnarray}\nThe conditions (\\ref{macboundcond}) ensure \nthe equilibrium through the equivalence\nof the volume and surface pressure \n(isoscalar or isovector) variations, see detailed derivations \nin Appendix B of \\cite{BMRV}.\nAs shown in Sec.\\ III \\cite{strutmagbr,strutmagden,kolmagsh}, \nthe pressures \n$\\delta \\mathcal{P}_{\\pm}$\ncan be obtained through moments of dynamical variations of the\ncorresponding phase-space distribution functions \n$\\delta f_\\pm({\\bf r},{\\bf p},t)$ (\\ref{planewave}) in the nuclear volume. \n\n\nFor the nuclear energy $E$ \nin this improved ESA (Appendix C of \\cite{BMRV}), \none obtains\n\\bel{EvEs}\nE \\approx -b^{}_V\\; A + J (N-Z)^2\/A + E_S^{(+)} + E_S^{(-)},\n\\end{equation}\nwith the following \nisoscalar (+) and isovector (-) surface energy components,\n\\bel{Espm}\n E_S^{(\\pm)}= \\alpha_{\\pm}\\mathcal{S}=b_{S}^{(\\pm)} \\mathcal{S}\/(4\\pi r_0^2),\n\\end{equation}\nand the ES area $\\mathcal{S}$. \nThese energies \nare determined by the\nisoscalar and isovector \nsurface energy constants $b_{S}^{(\\pm)} \\propto \\alpha_{\\pm}$ (\\ref{sigma}) \nthrough the solutions \nfor $w_{\\pm}(\\xi)$ taken \n at the leading order in $a\/R$. \n\n\nFor the energy surface coefficients\n $b_{S}^{(\\pm)}$ (\\ref{sigma}) with $\\mathcal{D}_{-} \\approx 0$] \nin the quadratic approximation $\\epsilon_{+}(w)=(1-w)^2$,\nwe finally arrived at the following explicit analytical expressions\nin terms of\nthe Skyrme force parameters \n(Appendix C of \\cite{BMRV})\n\\bel{bsplusminus} \nb_{S}^{(+)}=\\frac{6 \\mathcal{C}_{+}\n\\rho_\\infty \\mathcal{J}_{+}}{ r_0 a},\\quad\nb_{S}^{(-)}=k^{}_S \\mathcal{I}^2,\\quad\nk_{{}_{\\! S}}= 6 \\rho_\\infty \\mathcal{C}_{-}\\mathcal{J}_{-}\/(r_0 a), \n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{Jp}\n&&\\mathcal{J}_{+}=\\int_0^1 \\hbox{d} w\\; \\sqrt{w(1+\\beta w)}\\;(1-w)\\nonumber\\\\\n&=&\\frac{1}{24}\\;(-\\beta)^{-5\/2}\\;\n\\left[\\mathcal{J}_{+}^{(1)}\\; \\sqrt{-\\beta(1+\\beta)}\n+\\mathcal{J}_{+}^{(2)}\\; \\arcsin\\sqrt{-\\beta}\\right],\\nonumber\\\\\n&&\\mathcal{J}_{+}^{(1)}=3 + 4 \\beta(1+\\beta),\\quad\n\\mathcal{J}_{+}^{(2)}=-3-6\\beta,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{Jm}\n\\mathcal{J}_{-}&=&-\\frac{1}{1+\\beta}\\;\n\\int_0^1 \\hbox{d} w\\; \\sqrt{\nw(1+\\beta w)}\\;(1-w)(1+\\widetilde{c} \\widetilde{w})^2\n\\nonumber\\\\\n&=&\\frac{\\widetilde{c}^2}{1920 (1+\\beta) (-\\beta)^{9\/2}}\\;\n\\left[\\mathcal{J}_{-}^{(1)}\\left(c_{sym}\/\\widetilde{c}\\right)\\;\n\\sqrt{-\\beta(1+\\beta)}\\right.\\nonumber\\\\ \n&+& \\left.\n\\mathcal{J}_{-}^{(2)}\\left(c_{sym}\/\\widetilde{c}\\right)\\;\n\\arcsin\\sqrt{-\\beta}\\right],\n\\nonumber\\\\\n\\mathcal{J}_{-}^{(1)}(z)&=& 105- 4 \\beta \\left\\{95 +75 z +\n\\beta \\left[119+10z (19+6z) \\right.\\right.\\nonumber\\\\\n&+& \\left.\\left. 8 \\beta^2\n\\left(1+ 10z(1+z)\\right)\n+ 8 z \\left(5 z (3 +2 z) -6\\right)\\right]\\right\\},\n\\nonumber\\\\\n\\mathcal{J}_{-}^{(2)}(z)&=&15 \\left\\{7+2\\beta \\left[\n5 (3 + 2 z)\\right.\\right.\\nonumber\\\\ \n&+& \\left.\\left. 8 \\beta (1+z)\n\\left(3 +z +2 \\beta (1+z)\\right)\\right]\\right\\},\n\\end{eqnarray}\nsee also (\\ref{defpar})\nfor $\\widetilde{c}$, $c_{sym}$ and $\\widetilde{w}$.\nFor the limit $\\beta \\rightarrow 0$ from (\\ref{Jp}) and (\\ref{Jm}),\none has\n$\\mathcal{J}_{\\pm} \\rightarrow 4\/15$. In the limit\n$\\mathcal{C}_{-} \\rightarrow 0$, one obtains $k^{}_S \\rightarrow 0$. \n\n\n\n\\setcounter{equation}{0}\n\\section{POT calculations of the MI}\n\\label{semcalmi}\n\n\\subsection{Energy shell corrections}\n\\label{enshcor}\n\nThe energy shell corrections $\\delta E$ can be expressed approximately \nthrough the oscillating level density component \n$\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)$, \naveraged locally by using the convolution (folding) integral \nwith a small averaging parameter $\\Gamma$ of the \nGaussian weight function\n \\cite{strut,fuhi}.\nAs shown in \\cite{fuhi}, neglecting small corrections of the order of \nthe squares of the Fermi energy shell fluctuations \n$(\\delta \\varepsilon_{{}_{\\! {\\rm F}}})^2$ at \n$\\Gamma \\ll \\hbar \\Omega \\sim \\varepsilon_{{}_{\\! {\\rm F}}}\/A^{1\/3}$ (see (\\ref{hom}), \nalso \\cite{strutmag}), one has\n\\begin{eqnarray}\\label{dedge}\n&&\\delta E = \\int \\hbox{d} \\varepsilon\\; n(\\varepsilon) \n\\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right)\\; \\delta g_{{}_{\\! \\Gamma}}(\\varepsilon),\\nonumber\\\\\n&& {\\rm with} \\quad N=\\int \\hbox{d} \\varepsilon\\; n(\\varepsilon),\\quad \nn(\\varepsilon)=\\theta(\\varepsilon_{{}_{\\! {\\rm F}}}-\\varepsilon). \n\\end{eqnarray}\nSubstituting \n\\bel{avdden}\n\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)=\n{\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)\\;\n\\hbox{exp}\\left[-\\left(\\frac{t_{{}_{\\! {\\rm PO}}} \\Gamma}{\\hbar}\\right)^2\\right]\n\\end{equation}\nwith (\\ref{dlevdenscl}) for $\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ into (\\ref{dedge}),\none can expand a smooth action in exponent at the linear order,\n\\bel{actexpef}\nS_{\\rm PO}(\\varepsilon) \\approx S_{\\rm PO}(\\varepsilon_{{}_{\\! {\\rm F}}}) + t_{{}_{\\! {\\rm PO}}} \n\\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right),\n\\end{equation} \nand pre-exponent \namplitude at zero order over $\\varepsilon$ near the Fermi energy\n $\\varepsilon_{{}_{\\! {\\rm F}}}$\n[$t_{{}_{\\! {\\rm PO}}}= \\partial S_{\\rm PO}(\\varepsilon_{{}_{\\! {\\rm F}}})\/\\varepsilon$]\n(see a similar derivation of the averaged density $\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)$\nin \\cite{strutmag,sclbook,migdalrev}). \nThese expansions are valid for a small enough\nwidth $\\Gamma$ mentioned above to get a sharped bell-shaped Gaussian \naveraging function near $\\varepsilon_{{}_{\\! {\\rm F}}}$. \nCalculating then\nsimple Gaussian integrals over the energy $\\varepsilon$ by integration\nby parts, one arrives at \n(\\ref{descl}). In these derivations at the leading order in expansion in\n$(\\delta \\varepsilon_{{}_{\\! {\\rm F}}})^2$, we accounted for the zero value\noriginated by the lower limit $\\varepsilon=0$ in (\\ref{dedge}) \nby using that\n$t_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ is relatively large at small but finite $\\Gamma$.\nThus, one stays with the only contribution (independent of $\\Gamma$)\n at the upper limit\n$\\varepsilon=\\varepsilon_{{}_{\\! {\\rm F}}}$, in line of the basic concepts that\nthe energy shell correction is determined by the quantum s.p.\\ states\nnear the Fermi surface \\cite{fuhi,strut}. \nSimilarly, the same result can be obtained\nby using the Lorentzian weight function [the summand in \n(\\ref{avdden}) is proportional to \n$\\hbox{exp}\\left(-t_{{}_{\\! {\\rm PO}}} \\Gamma\/\\hbar\\right)$ \nin the Lorenzian\ncase, instead of the Gaussian exponent]. In this case, the \nlocal convolution averaging\nof the oscillation level density component with the Lorentzian width\nparameter $\\Gamma$ \nis resulted in a formal shift of the energy \n$\\varepsilon \\to \\varepsilon +i\\Gamma$ ($\\Gamma \\ll \\hbar \\Omega$). Thus, the \nstrightforward\ncalculations by the residue method also gives (\\ref{descl}). \n\n\n\\subsection{Derivation of the rigid-body MI }\n\\label{derrigmi}\n\n\n\\subsubsection{TF COMPONENT }\n\\label{tfmi}\n\nWe substitute approximately the Green's function\n$\\langle G_{0}\\rangle_{\\Gamma_p}$, locally averaged over the momentum $p$\nby using the Gaussian weight function with a finite small width \n$\\Gamma_p$, into $\\Theta^{00}$ [see (\\ref{thetaxnnp}) at $\\nu=\\nu'=0$] \ninstead of $G_0$,\n\\begin{eqnarray}\\label{G0av}\n\\left\\langle G_{0}\\right\\rangle_{\\Gamma_p} &=& \n\\frac{1}{\\Gamma_p\\sqrt{\\pi}}\\int_{-\\infty}^\\infty \\hbox{d} p'\\; G_0(s,p')\\;\n\\hbox{exp}\\left[-\\left(\\frac{p'-p}{\\Gamma_p}\\right)^2\\right]\n\\nonumber\\\\\n&\\approx&\nG_0(s,p)\\; \\hbox{exp}\\left[\n-\\frac{s^2 \\Gamma_p^2}{4\\hbar^2}\\right],\n\\end{eqnarray}\nas in \\cite{strutmag} for the level density.\nTransforming then\nthe integration variables ${\\bf r}_1$ and ${\\bf r}_2$ to the canonical \naverage ${\\bf r}$ and\ndifference ${\\bf s}={\\bf s}_{{}_{\\! 12}}$ ones (\\ref{newcoord}), \nfor the corresponding locally averaged MI component\n$\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}$, one approximately gets \n\\begin{eqnarray}\\label{thetaxnnp00}\n&&\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}\n\\approx\\frac{d_sm^2}{4\\pi^3}\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r} \\int \\frac{\\hbox{d} {\\bf s}}{s^2}\\;\n\\ell_x\\left({\\bf r}+\\frac{{\\bf s}}{2}\\right)\\quad\n\\nonumber\\\\\n&\\times&\n\\ell_x\\left({\\bf r}-\\frac{{\\bf s}}{2}\\right)\\;\\hbox{sin}\\left(\\frac{2 s p}{\\hbar}\\right)\\;\n\\hbox{exp}\\left[- \\frac{s^2 \\Gamma_p^2}{2\\hbar^2}\\right].\\quad\n\\end{eqnarray}\nFor simplicity, we omit here and below the \nindex in $s_{{}_{\\! 12}}$ within this \nAppendix E.2 because it will not interfer with\ndifferent notations. As shown below (at the end of this Appendix E.2a), \nthe final result for $\\langle \\Theta_x^{00} \\rangle_{\\Gamma_p}$\n(\\ref{thetaxnnp00}) does not depend approximately on $\\Gamma_p $,\nthat looks as a plateau of the SCM (without correction polynomials).\nWithin the NLLLA (\\ref{nllla}), \nused already in (\\ref{G0})\n\\cite{gzhmagsit,gzhmagsit,mskbPRC2010} after the averaging over \nthe phase-space variables, the main contribution is given\nby small distance $s_{{}_{\\! 12}}$ with respect to\nthe wave length $\\hbar\/p_{{}_{\\! {\\rm F}}}$ of the particle near the Fermi surface.\nIn this approximation at the leading zero order, due to the \nexponential cut-off factor decreasing with $s$ and $\\Gamma_p$,\none may expand smooth classical quantities in $sp\/\\hbar$ in the argument\nof exponent and pre-exponent amplitude factors in (\\ref{thetaxnnp00})\n at the leading \norder, in particular, applying \n\\bel{l2}\n\\ell_x\\left({\\bf r}+s\/2\\right)\\;\\ell_x\\left({\\bf r}-{\\bf s}\/2\\right) \\approx\n\\ell_x^2({\\bf r}) \\approx p^2 r_{\\perp x}^2.\n\\end{equation}\nIn (\\ref{thetaxnnp00}), we integrate over ${\\bf s}$ in the spherical \ncoordinates, $\\hbox{d} {\\bf s}=s^2\\hbox{d} s\\; \\hbox{sin}\\theta_s\\hbox{d} \\theta_s\\;\\hbox{d} \\varphi_s$,\nwith the polar axis $z_s$\ndirected along ${\\bf p}({\\bf r})$ (see Fig.\\ \\ref{fig14}). Then, \nfor the ${\\rm CT}_0$ momentum ${\\bf p}({\\bf r})$, i.e., \nalong the ${\\bf s}_{{}_{\\! 12}}$, \none takes into account that the integrand and \nlimits of the integration \nover angles $\\theta_s$,\nand $\\varphi_s$ are constants independent of \nother variables. \nTherefore, this integration over all angles\ngives simply $4 \\pi$, and we arrive at \n\\begin{eqnarray}\\label{thetaxnnpint00}\n\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}\n&\\approx&\\frac{d_sm^2}{\\pi^2}\\int \\hbox{d} {\\bf r} \\;r_\\perp^2\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \n\\hbox{d} \\varepsilon \\; \\left[{\\tt I}_{00}(s_{\\rm max},\\varepsilon,\\Gamma_p)\\right.\n\\nonumber\\\\\n&-& \\left.{\\tt I}_{00}(0,\\varepsilon,\\Gamma_p)\\right].\n\\end{eqnarray}\nHere, we exchanged the order of integrations over $\\varepsilon$\nand ${\\bf r}$. The remaining indefinite integral ${\\tt I}_{00}(s,\\varepsilon,\\Gamma_p)$ \nover $s$ as function of $s$, $\\varepsilon$ and $\\Gamma_p$\ncan be approximately (within the NLLLA) taken\nanalytically, \n\\begin{eqnarray}\\label{sinttf}\n&&{\\tt I}_{00}(s,\\varepsilon,\\Gamma)=\\int \\hbox{d} s\\; \\hbox{sin}\\left(\\frac{2 s p}{\\hbar}\\right)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma^2}{2\\hbar^2}\\right] \n\\nonumber\\\\\n&=& \n\\sqrt{\\frac{\\pi}{2}}\\;\\frac{i \\hbar}{2 \\Gamma}\\;\n\\hbox{exp}\\left[-2 \\left(\\frac{p}{\\Gamma}\\right)^2\\right]\\;\n\\left[{\\rm erf}\\left(\\frac{ip \\sqrt{2}}{\\Gamma}+\n\\frac{\\Gamma s}{\\hbar\\sqrt{2}}\\right)\\right.\n\\nonumber\\\\ \n&+&\\left.\n{\\rm erf}\\left(\\frac{ip \\sqrt{2}}{ \\Gamma}-\n\\frac{\\Gamma s}{\\hbar \\sqrt{2}}\\right)\\right],\n\\end{eqnarray}\nwhere \n${\\rm erf}(z)$\nis the standard error function, ${\\rm erf}(z)=\n(2\/\\sqrt{\\pi})\\int_0^z \\hbox{d} t\\; \\hbox{exp}(-t^2)$.\nThis integral, taken at the upper limit\n$s=s_{max}$, is rather a complicated function\nof ${\\bf r}$, especially near the ES of the potential well. \nHowever, the Gaussian factor in the integrand \nwith any small but \na finite Gaussian parameter $\\Gamma_p$,\n\\bel{avcondgp}\n\\hbar\/R \\ll \\Gamma_p \\ll p_{{}_{\\! {\\rm F}}},\n\\end{equation}\nremoves the oscillating contribution\narising from the upper limit $s_{\\rm max}$ ($R$ is the mean \nnuclear radius). The reason is due to the exponential \nasymptotics at a large argument\n$s$, such as\n\\begin{eqnarray}\\label{int00as}\n&&\\hbox{exp}\\left[-\\frac{\\Gamma_p^2 s_{\\rm max}^2}{2 \\hbar^2}\\right], \\qquad\n{\\rm or} \n\\nonumber\\\\\n&&\\hbox{exp}\\left[-\\frac{2 p^2}{\\Gamma_p^2}\\right],\\quad \n{\\rm at}\\quad p\\sim p_{{}_{\\! {\\rm F}}}, \n\\end{eqnarray}\nor even strongly as the product of these exponents.\nThen, according to another asymptotics for small $s \\rightarrow 0$,\n\\begin{eqnarray}\\label{int00as0}\n{\\tt I}_{00}(s,\\varepsilon,\\Gamma) &=& -\\frac{\\hbar}{2p} + \n\\frac{i\\; \\sqrt{2 \\pi}\\;\\hbar}{\\Gamma}\\;\n\\hbox{exp}\\left(-\\frac{p^2}{2\\Gamma^2}\\right)\\nonumber\\\\\n&+&\\frac{p s^2}{\\hbar}\\left\\{1 + \n\\mathcal{O}\\left[\\left(\\frac{ps}{\\hbar}\\right)^2\\right]\\right\\},\n\\end{eqnarray}\nwe are left with the only constant contribution from the lower\nlimit $s=0$, independent of $s$ and of\nthe Gaussian averaging parameter $\\Gamma_p$ satisfying the conditions \n(\\ref{avcondgp}),\n \\bel{sinttffin}\n{\\tt I}_{00}(s,\\varepsilon,\\Gamma) \\approx -\\hbar\/(2 p).\n\\end{equation}\nFinally, from (\\ref{thetaxnnpint00}) and (\\ref{sinttffin})\none obtains \n\\begin{eqnarray}\\label{TFrig}\n\\left\\langle\\Theta_x^{00}\\right\\rangle&=&\n\\frac{d_sm^2}{2\\pi^2 \\hbar^3}\\left\\langle\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r} \\; r_{\\perp x}^2 \\; p({\\bf r})\\right\\rangle\\nonumber\\\\\n&=& d_s m \\int \\hbox{d} {\\bf r}\\; r_{\\perp x}^2\\;\\rho_{{}_{\\! {\\rm TF}}}({\\bf r})=\n\\Theta_{x,{\\rm TF}}^{\\rm (RB)}.\n\\end{eqnarray}\nWe used also the expression for the TF particle density through\n$G_0$ (\\ref{G0}),\n\\bel{tfpartden}\n\\rho_{{}_{\\! {\\rm TF}}}({\\bf r}) = \n-\\frac{1}{\\pi} {\\mbox {\\rm Im}} \\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\; G_0\\Big|_{s\\rightarrow 0}=\n\\frac{m}{2 \\pi^2\\hbar^3}\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\; p({\\bf r}).\n\\end{equation}\nSimilarly, using the Lorentzian weight function for the averaging in \n(\\ref{G0av}) instead of the Gaussian one,\n\\bel{G0avlor}\n\\left\\langle G_{0}\\right\\rangle_{\\Gamma} =\n\\frac{1}{\\pi}\\int_{-\\infty}^\\infty \\hbox{d} p'\\; \n\\frac{\\Gamma G_0(s,p')}{\\left(p'-p\\right)^2 + \\Gamma^2}\n=G_0(s,p+i\\Gamma),\n\\end{equation}\none obtains the same result (\\ref{TFrig}) independently of the choice of\nthe averaging function ($\\Gamma=\\Gamma_p$ in this Appendix E.2a). \nIn these derivations, we used\nthe residue technics for the analytical evaluations of the integrals,\nthat means formally the replace of such a local averaging by the shift of the\nmomentum, $p \\rightarrow p +i \\Gamma_p$ [see (\\ref{thetaxnnp}) \nat $\\nu=\\nu'=0$ and (\\ref{l2})],\n\\begin{eqnarray}\\label{theta00avlor}\n&&\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma}\\approx\n\\frac{d_sm^2}{\\pi^2} {\\mbox {\\rm Im}} \\int \\hbox{d} {\\bf r}\\; \\left(y^2+z^2\\right)\n\\nonumber\\\\ \n&\\times&\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \n\\hbox{d} \\varepsilon \\int_{0}^{s_{\\rm max}} \\hbox{d} s\\;\n\\hbox{exp}\\left[ \\frac{2 i s \\left(p+i \\Gamma\\right)}{\\hbar}\\right]\n\\approx \\frac{d_s m \\hbar}{2 \\pi^2}\n\\nonumber\\\\\n&\\times& \\int \\hbox{d} {\\bf r} \\left(y^2+z^2\\right)\\;\\int_0^{p_{{}_{\\! {\\rm F}}}}\n\\hbox{d} p\\; p^2 \\left\\{1 - \\hbox{exp}\\left[-\\frac{2 \\Gamma s_{\\rm max}}{\\hbar}\\right]\\right.\n\\nonumber\\\\\n&\\times& \\left. \n\\left[\\frac{\\Gamma}{p}\\; \n\\hbox{sin}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right) +\n\\hbox{cos}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\\right]\\right\\}. \n\\end{eqnarray}\nAgain, according to (\\ref{avcondgp}), the second strongly \noscillating term of the integrand coming\nfrom the upper limit $s=s_{\\rm max}$ in the last\nline can be neglected as exponentially small, instead of \nthe Gaussian behavior above. Then, \nwe are left with the main first TF term [see (\\ref{TFrig})] \nindependent of $\\Gamma_p$, as in the case of the Gaussian averaging. \n\n\\bigskip\n\\subsubsection{MI SHELL CORRECTIONS}\n\\label{mishcor}\n\nTo average the oscillating component $\\delta \\Theta_x^{01}$ of the sum \n(\\ref{thetaxsum}) (see \n(\\ref{thetaxnnp}) at $n=0$ and $n'=1$) over the phase space variables,\none may use the Green's function \n$\\left\\langle G_0\\right\\rangle_\\Gamma$ (\\ref{G0av}), locally averaged\nwith a Gaussian weight \ninstead of $G_0$, and similarly, instead of $G_1$ \\cite{strutmag},\n\\begin{eqnarray}\\label{G1av}\n&&\\left\\langle G_1\\right\\rangle_\\Gamma=\n\\frac{1}{\\Gamma \\sqrt{\\pi}} \\int \\hbox{d} \\varepsilon'\\; G_1\\left({\\bf r}_1,{\\bf r}_2,\\varepsilon'\\right)\\;\n\\hbox{exp}\\left[-\\frac{(\\varepsilon'-\\varepsilon)^2}{\\Gamma^2}\\right]=\\qquad\n\\nonumber\\\\\n&=&\n\\sum_{CT_1}\\mathcal{A}_{CT_1}\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{CT_1}\\! -\\! \\frac{i\\pi}{2}\\sigma_{{}_{\\! CT_1}}\n\\!- \\! i \\phi_d \n\\! -\\! \\frac{t_{{}_{\\! CT_1}}^2 \\Gamma^2}{4\\hbar^2}\\right].\\,\\,\\,\n\\end{eqnarray}\nTransforming also the integration variables ${\\bf r}_1$ and ${\\bf r}_2$\nto the canonical ones (\\ref{newcoord}) at zero temperature, one finds\n\\begin{eqnarray}\\label{dthetax01new}\n&&\\left\\langle\\delta \\Theta_x^{01}\\right\\rangle=-\\frac{d_s m}{\\pi^2 \\hbar^2}\n\\sum_{{\\rm CT}_1} {\\mbox {\\rm Im}}\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\int \\hbox{d} {\\bf r} \\int \\frac{\\hbox{d} {\\bf s}}{s} \\;\n\\ell_x\\left({\\bf r}-\\frac{{\\bf s}}{2}\\right)\n\\nonumber\\\\\n&\\times& \\ell_x\\left({\\bf r}+\\frac{{\\bf s}}{2}\\right)\n\\;\\hbox{cos}\\left[\\frac{s}{\\hbar} p({\\bf r})\\right]\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\times& \\mathcal{A}_{{\\rm CT}_1}\\left(\n{\\bf r}-\\frac{{\\bf s}}{2},{\\bf r}+\\frac{{\\bf s}}{2};\\varepsilon\\right)\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CT}_1}\\left(\n{\\bf r}-\\frac{{\\bf s}}{2},{\\bf r}+\\frac{{\\bf s}}{2};\\varepsilon\\right)\\right. \n\\nonumber\\\\\n&-&\\left.\ni \\frac{\\pi}{2} \\sigma_{{}_{\\! CT_1}}- i \\phi_{\\rm d}\n- \\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right].\n\\end{eqnarray}\nWe shall put $\\Gamma$ and $\\Gamma_p$ to be zero in the final \nexpressions\nfor this average $\\left\\langle\\delta \\Theta_x^{01}\\right\\rangle$, as far as\n$\\Gamma$ is much smaller than the distance between gross shells \n$\\hbar \\Omega$ (\\ref{hom}) and $\\Gamma_p$ satisfies inequalities\n (\\ref{nllla}).\nExpanding then the action phase of the second exponent and its \npre-exponent factors \nin small ${\\bf s} p\/\\hbar$ up to first nonzero terms \n(i.e., \nup to the first and zeroth order ones, respectively), due to \nthe first sharp-peaked exponential Gaussian factor in the second line of \n(\\ref{dthetax01new}), \none applies (\\ref{l2}) and \n\\begin{eqnarray}\\label{ampactoc}\n\\mathcal{A}_{{\\rm CT}_1}\\left({\\bf r} - \\frac{{\\bf s}}{2},{\\bf r} + \\frac{{\\bf s}}{2}; \\varepsilon\\right)\n&\\approx& \\mathcal{A}_{{\\rm CCT}_1}\\left({\\bf r},{\\bf r}; \\varepsilon\\right),\\qquad\n\\nonumber\\\\\nS_{{\\rm CT}_1}\\left({\\bf r} - \\frac{{\\bf s}}{2},{\\bf r} + \\frac{{\\bf s}}{2}; \\varepsilon\\right)\n&\\approx& S_{{\\rm CCT}_1}\\left({\\bf r},{\\bf r}; \\varepsilon\\right) +{\\bf p} {\\bf s}.\\qquad\n\\end{eqnarray}\nWith these expansions in (\\ref{dthetax01new}), \nfor the integration over $d{\\bf s}=s^2 \\hbox{d} s\\; \\hbox{d} x_s\\;\\hbox{d} \\varphi_s$ \nin (\\ref{dthetax01new}), we use the \nsame spherical coordinate system ($s, \\theta_s, \\varphi_s$)\nwith the polar axis $z_s$ directed again along the momentum vector \n${\\bf p}({\\bf r})=({\\bf p}_1 +{\\bf p}_2)\/2$, \n$x_s=\\hbox{cos} \\theta_s$ (Fig.\\ \\ref{fig14}). The integral over \nthe azimuthal angle $\\varphi_s$ gives simply $2 \\pi$ due to the azimuthal\nsymmetry. The integration limits over $x_s$ \ncan be considered as from -1 to 1 within the NLLLA (\\ref{nllla})\n(neglecting thus the dependence of limits \nfor the integration over angles $\\theta_s$ on $s_{\\rm max}$ \nand ${\\bf r}$),\none approximately finds from (\\ref{dthetax01new}) \n\\begin{eqnarray}\\label{dthetax01new1}\n\\left\\langle \\delta \\Theta_{x}^{01} \\right\\rangle\n&\\approx& \\frac{2d_s m}{\\pi \\hbar}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} {\\bf r} \\;r_{\\perp x}^2\n\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\mathcal{A}_{{\\rm CCT}_1}\n\\quad\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CCT}_1} - \\frac{i \\pi}{2}\\sigma_{{}_{\\! {{\\rm CCT}_1}}} - i \\phi_{d}\n\\right]\\;I_{01}, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\\label{int01}\nI_{01}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\; s\\; \\int_{-1}^{1} \\hbox{d} x_s\\; \n\\hbox{cos}\\left[s p({\\bf r})\/\\hbar\\right]\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right]\\nonumber\\\\\n&\\times&\\hbox{exp}\\left[\\frac{i}{\\hbar} s p({\\bf r})\\; x_s\\right]\\;\n\\hbox{exp}\\left[-\\frac{t_{{}_{\\! CT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n& \\approx& \\left[{\\tt I}_{00}\\left(s_{\\rm max},\\varepsilon,\n\\frac{\\Gamma_p}{\\sqrt{2}}\\right)\n-{\\tt I}_{00}\\left(0,\n\\varepsilon,\\frac{\\Gamma_p}{\\sqrt{2}}\\right)\\right]\n\\nonumber\\\\\n&\\times& \\hbox{exp}\\left[-\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\approx \\frac{\\hbar}{2p}\\;\n\\hbox{exp}\\left[-\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right].\n\\end{eqnarray}\nThe sum runs all of ${\\rm CCT}_1$s (closed ${\\rm CT}_1$s).\nTaking then the integral over the angle variable $x_s$ \nin the NLLLA (\\ref{nllla}), one then integrate\nover the\nmodulus $s$ within integration limits\nfrom $0$ to $s_{\\rm max}$. Note that with the approximation \n$t_{{}_{\\! CT_1}} \\approx t_{{}_{\\! CCT_1}}$, due to $\\Gamma \\ll \\hbar \\Omega$\n(but significantly larger\nthan a distance between neighboring energy levels), this integral is \nreduced to $s=0$ and $s_{\\rm max}$ boundaries of \n${\\tt I}_{00}(s,\\varepsilon,\\Gamma_p\/\\sqrt{2})$ (\\ref{sinttf}), \nsee the third line in (\\ref{int01}).\nCalculating approximately the integral over $s$ as \nin the subsection E.2a of this Appendix, and using the \nsame asymptotics (\\ref{int00as})\nat large upper integration limit $s=s_{\\rm max}$ \nand\n(\\ref{int00as0}) at small lower one $s=0$, one obtains\nthe nonzero contribution only from the lower integration limit $s=0$\nas in the previous subsection E.2a.\nOther contributions of the upper limit $s_{\\rm max}$ can be neglected \nbecause the integrand over $s$ contains rapidly\noscillating functions, and after a local averaging in \nthe phase space variables (even with a small but finite Gaussian\naveraging parameter), they exponentially disappear under the \ncondition (\\ref{avcondgp})\nfor $\\Gamma_p$ as in the calculations of \nthe Thomas-Fermi MI component\n(Appendix E.2a). Finally, by making use of (\\ref{int01}) \nin (\\ref{dthetax01new1}), \none obtains\n\\begin{eqnarray}\\label{dthetax01new2}\n&&\\left\\langle \\delta \\Theta_{x}^{01}\\right\\rangle \\approx \n-\\frac{d_s m}{\\pi}\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} {\\bf r}\\; r_{\\perp x}^2 \\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\mathcal{A}_{{\\rm CCT}_1}\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CCT}_1} - \n\\frac{i \\pi}{2}\\sigma_{{}_{\\! {{\\rm CCT}_1}}} - i \\phi_{d}\n-\\frac{t_{{\\rm CCT}_1}^2\\Gamma^2}{4\\hbar^2}\\right]\n\\nonumber\\\\\n&=&d_s m\\int \\hbox{d} {\\bf r} \\;\\left(z^2+y^2\\right)\\; \\delta \\rho_{\\rm scl}({\\bf r})=\n\\delta \\Theta_{x,{\\rm scl}}^{\\rm (RB)}.\n\\end{eqnarray}\nIn these derivations, we used\n(\\ref{rperpcoord}) for \nthe perpendicular coordinate $r_{\\perp x}$,\nand (\\ref{dTxrigSCL}) for the oscillating shell component \n$\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm RB}$ of the semiclassical MI \n(\\ref{rigmomsplit}). This component is \nrelated to the oscillating shell part $\\delta \\rho_{\\rm scl}({\\bf r})$\n[see (\\ref{ddenpart}) and (\\ref{Gct}) with a closed ${\\rm CT}_1$] \nin the semiclassical particle\ndensity (\\ref{denpartscl}). \nLike in the previous subsection of Appendix E.2, we obtain the \nsame result (\\ref{dthetax01new2}) \nby using the Lorentzian weight function for the local average\n($\\varepsilon \\rightarrow \\varepsilon+i \\Gamma$). Indeed, using its definition\n(\\ref{G0avlor}) for both Green function components $G_0$ and $G_1$,\nand performing the same integrations in the NLLLA (\\ref{nllla}),\none gets\n\\begin{eqnarray}\\label{dthetax01new2lor}\n&&\\langle \\delta \\Theta_x^{01}\\rangle \n=-\\frac{m d_s}{\\pi}\\int \\hbox{d} {\\bf r} \\;\\left(y^2 +z^2\\right){\\mbox {\\rm Im}}\n\\sum_{{\\rm CCT}_1}\\int \\hbox{d} \\varepsilon \n\\mathcal{A}_{{\\rm CCT}_1} \n\\nonumber\\\\\n&\\times&\\hbox{exp}\\left[\\frac{i}{\\hbar}\\; S_{{\\rm CCT}_1}\n-\\frac{i \\pi}{2}\\sigma_{{}_{\\! {\\rm CCT}_1}} -i \\phi_{\\rm d} -\n\\frac{\\Gamma t_{{}_{\\! {\\rm CCT}_1}}}{\\hbar}\\right]\n\\nonumber\\\\\n&\\times& \\left\\{1 + \\hbox{exp}\\left(-\\frac{\\Gamma_ps_{\\rm max}}{\\hbar}\\right)\n\\left[\\frac{\\Gamma_p}{2p}\\; \\hbox{sin}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\n\\right.\\right.\n\\nonumber\\\\\n&-&\\left.\\left.\\hbox{cos}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\\right]\\right\\}.\n\\end{eqnarray}\nAs transparently seen from this explicit expression, \none has exponential disappearance of the oscillating contributions\non the upper integration limit $s_{\\rm max}$ \nunder the conditions (\\ref{avcondgp})\nfor $\\Gamma_p$, see the second term in figure brackets of the last two lines\nof (\\ref{dthetax01new2lor}). Therefore, \nthe first constant term \nin these brackets (coming from the lower integration limit $s=0$) \nyields immediately the finite $\\Gamma \\rightarrow 0$ rigid-body limit \n(\\ref{dthetax01new2}) for $\\langle \\delta \\Theta_x^{01}\\rangle$. \n \n\nUsing analogous analytical calculations of the other terms \n$\\langle\\delta \\Theta_{x}^{10}\\rangle$ and \n$\\langle\\delta \\Theta_{x}^{11}\\rangle$ [see (\\ref{thetaxnnp})], \none finds the essentially different integrals over $s$, such as\n\\begin{eqnarray}\\label{int10} \nI_{10}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\n\\hbox{sin}^2 (ps\/\\hbar)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\approx& \\hbox{exp}\\left[-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\\;\n\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\n\\left[1-\\hbox{cos}\\left(\\frac{2ps}{\\hbar}\\right)\\right]\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right],\n\\end{eqnarray}\n and \n\\begin{eqnarray}\\label{int11} \nI_{11}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\; s\\; \\hbox{sin}(2 p s\/\\hbar)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\approx& \\hbox{exp}\\left[-\n\\frac{t_{{\\rm CCT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\\;\n\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\ns\\;\\hbox{sin}\\left(\\frac{2ps}{\\hbar}\\right)\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right],\n\\end{eqnarray}\nrespectively. \nIntegrating analytically in (\\ref{int10}) and (\\ref{int11}),\none can see that any contributions coming from the upper limit \n$s_{\\rm max}$ exponentially disappear as shown in (\\ref{int00as}) \n(with the formal replace $\\Gamma_p$ by $\\Gamma_p\/\\sqrt{2}$) as above.\nHowever, in contrast to the calculations \nof $\\left\\langle \\Theta_x^{00}\\right\\rangle$ \nand $\\left\\langle \\Theta_x^{01}\\right\\rangle$, the\n contributions from the lower \nintegration limit\n$s=0$ turn into zero too, according to the asymptotics at the 4th order in \ndistance $s$ in units of the wave-length $\\hbar\/p$:\n\\begin{eqnarray}\\label{intas1011}\nI_{10} &=& \\frac{2 p^2 s^3}{3 \\hbar^2} \\hbox{exp}\\left[-\n\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\\left[1 +\n\\mathcal{O}\\left(\\frac{ps}{\\hbar}\\right)\\right],\\qquad\n\\nonumber\\\\\nI_{11} &=& \\frac{2 p s^3}{3 \\hbar} \\hbox{exp}\\left[-\n\\frac{t_{{}_{\\! CCT_1}}^2 \\Gamma^2}{4 \\hbar^2}\\right]\\left[1 +\n\\mathcal{O}\\left(\\frac{ps}{\\hbar}\\right)\\right].\\qquad \n\\end{eqnarray}\nTherefore, \nin addition to (\\ref{dthetax01new2}), independently of the\nweight function for averaging, the two components associated with \nintegrals (\\ref{int10}) and (\\ref{int11})\ndo not contribute at both integration limits within \nthe NLLLA, as explained above. Thus, for \nall nonzero terms of the oscillating part of the MI,\n$\\left\\langle \\delta \\Theta_x\\right\\rangle$,\none finally approximately arrives\nat the same rigid-body MI shell component \nin the NLLLA (\\ref{nllla}). \nThis result does not depend on the choice of the weight (Gaussian\nand Lorentzian) functions for the local averaging over the phase space.\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe spectroscopic observations of the \\textit{Infrared Space Observatory}\n(ISO) (Kessler et al. \\cite{kes96}) opened a new window for the study of the physical and\nchemical properties of IR-bright, ultraluminous infrared galaxies\n(ULIRG) and active galactic nuclei (AGN). Most of the pioneering work on the mid-to-far infrared spectra of active and starburst galaxies is derived from ISO spectroscopy (Sturm {\\em et al.\\\/} \\cite{stu02}; Spinoglio {\\em et al.\\\/} \\cite{spi05}; Verma {\\em et al.\\\/} \\cite{ver03}; Verma {\\em et al.\\\/} \\cite{ver05}).\n\n\\begin{figure}\n\\includegraphics[width=11cm]{fig1.eps}\n\\caption{Far infrared spectra observed with the LWS onboard of ISO of a sample of ultraluminous infrared galaxies (Fischer {\\em et al.\\\/} \\cite{Fis99})}\n\\end{figure}\n\nThe mid-IR spectral range includes most of the fine-structure lines excited by the hard radiation produced\nby black hole accretion as well as those mainly excited by stellar\nionization (Spinoglio \\& Malkan \\cite{sm92}) and thus represents an essential tool\nto distinguish between the two processes, especially in obscured\nnuclei suffering severe dust extinction. With the advent of the \\textit{Spitzer Space Telescope}, \nwith its powerful mid infrared instrument, the IRS (Houck {\\em et al.\\\/} \\cite{hou04}), systematic spectroscopic \nstudies of samples of galaxies have stated to appear (Dale {\\em et al.\\\/} \\cite{dal06}; Higdon {\\em et al.\\\/} \\cite{hig06}; Brandl {\\em et al.\\\/} \\cite{bra06}; Armus {\\em et al.\\\/} \\cite{arm07}; Farrah {\\em et al.\\\/} \\cite{far07}; Buchanan {\\em et al.\\\/} \\cite{buc06}; Tommasin {\\em et al.\\\/} \\cite{tom07}).\n\nIn a complementary way, the far-IR range contains a large variety of molecular (OH,\nH$_{2}$O, high-J CO) and low excitation ionic\/atomic transitions,\nin emission or in absorption, that can reveal the geometry and\nmorphology of the circumnuclear and nuclear regions in galaxies.\nIn particular far-infrared molecular lines could trace the expected conditions of\nX-UV illuminated dusty tori predicted from the unified models (Antonucci \\cite{ant93} and\nwhose presence in type 2 active galaxies is\nforeseen to reconcile the type1\/type2 dichotomy.\n \n \\begin{figure}\n\\includegraphics[width=12cm]{fig2.eps}\n\\caption{Fine-structure lines in the 3-200 $\\mu$m range. \nThe lines are plotted as a function of their\nionization potential and critical density. Different symbols are\nused for lines from photodissociation regions (squares),\nstellar\/HII region lines (triangles), AGN lines (circles) and\ncoronal lines (stars). The two [OI] lines have been plotted - for\ngraphical reasons - at a ionization potential higher than their\neffective value.}\\end{figure}\n\nThe far-IR spectra of local IR-bright and ULIRG galaxies, as measured by\nISO-LWS (Fischer {\\em et al.\\\/} \\cite{Fis99}), showed an unexpected sequence of features,\nas can be seen in Fig. 1, from strong [OIII]52, 88 $\\mu$m and\n[NIII]57 $\\mu$m line emission to detection of only faint\n[CII]157$\\mu$m line emission and [OI]63 $\\mu$m in absorption. The\n[CII]157 $\\mu$m line in 15 ULIRGs (L$_{IR}\\geq10^{12}L_{\\odot}$)\nrevealed an order of magnitude deficit compared to normal and\nstarburst galaxies relative to the FIR continuum. Non-PDR\ncomponents, such as dust-bounded photoionization regions,\ngenerating much of the FIR continuum but not contributing\nsignificant [CII] emission, can explain the [CII] deficiency. Such\nenvironments may also explain the suppression of FIR\nfine-structure emission from ionized gas and PAHs, and the warmer\nFIR colors of ULIRGs. (Luhman {\\em et al.\\\/} \\cite{Lum03}).\n\n\nLWS observations of Arp 220 show absorption in molecular lines of OH, H$_2$O, CH,\nNH, and NH$_3$, as well as in the [OI]63$\\mu$m line and faint\nemission in the [CII]158$\\mu$m line. The molecular absorption in\nthe nuclear region is characterized by high\nexcitation due to high infrared radiation density (Gonz\\'alez-Alfonso {\\em et al.\\\/} \\cite{go04}).\nNotably, the LWS spectrum of the prototype Seyfert 2 galaxy NGC\n1068, beside the expected ionic fine structure emission lines,\nshows the 79, 119 and 163$\\mu$m OH rotational lines in emission,\nnot in absorption as in every other galaxy yet observed.\nModeling the three FIR lines of OH suggests the gas lies in small\n(0.1pc) and dense clouds ($\\sim 10^4 {\\rm cm^{-3}}$) in\nthe nuclear region (potentially a signature of the torus) with a minor contribution from\nthe circumnuclear starburst ring at 3kpc (Spinoglio {\\em et al.\\\/} \\cite{spi05}).\n\n\\section{Fine-structure emission lines}\n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig3.eps}\n\\caption{[NeVI]7.6$\\mu$m\/[OIV]26$\\mu$m ratio as a function of the \n[NeV]14.3$\\mu$m\/24.3$\\mu$m ratio. The line rations of Seyfert 1's (NGC1365, NGC4151, Tol0109), Seyfert 2's (MKN3, CenA, Circinus, NGC1068, PKS2048) and NLXR galaxies (NGC5503, NGC7582) are presented, from ISO SWS observations (Spinoglio {\\em et al.\\\/} \\cite{spi00}; Sturm {\\em et al.\\\/} \\cite{stu02}). }\n\n\\end{figure}\n\nMid-IR and far-IR spectroscopy of fine-structure emission lines\nare powerful tools to understand the physical conditions in\ngalaxies from the local universe to distant cosmological objects.\nFig. 2 shows the critical density (i.e. the density for which the\nrates of collisional and radiative de-excitation are equal) of\neach line as a function of the ionization potential of its ionic\nspecies. This diagram shows how these lines can measure\ntwo fundamental physical quantities (density and ionization) of\nthe gas. Lines from different astrophysical emission regions in\ngalaxies are shown in the figure with different symbols. The ratio\nof two lines with similar critical density, but different\nionization potential, gives a good estimate of the ionization,\nwhile the ratio of lines with similar ionization potential, but\nwith diffrerent critical density, can measure the density of the\ngas in the region (see, e.g., Spinoglio \\& Malkan \\cite{sm92}).\nFrom Fig. 2 it is clear that infrared spectroscopy has a thorough\ndiagnostic power for gas with densities from about 10$^2$ cm$^{-3}$ \nto as high as 10$^8$ cm$^{-3}$ and ionization potentials up to 350 eV, \nusing, to trace these extreme conditions, the so called coronal lines. \nMoreover, increasing the wavelength \nof the transition used, the spectroscopic diagnostics become more and \nmore insensitive to dust extinction, and can therefore probe regions \nhighly obscured at optical or even near-to-mid infrared wavelengths.\n\n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig4.eps}\n\\caption{[CII]157$\\mu$m\/[OI]63$\\mu$m ratio as a function of the \n[OIII]88$\\mu$m\/[OI]63$\\mu$m ratio. The grid represents starburst photoionization models computed using the CLOUDY code. At the right bottom are shown the gas density values as derived from AGN photoionization models with log U = -2.5, while at the left are given the densities derived from photodissociation region models (Spinoglio {\\em et al.\\\/} \\cite{spi03}).}\n\n\\end{figure}\n\nTo give an example of the diagnostic power of infrared spectroscopy, we show in Fig. 3 \na diagram showing lines excited only in the highly energetic environments of AGN and not from stellar ionization. In this diagram the [NeVI]7.6$\\mu$m\/[OIV]26$\\mu$m ratio is shown as a function of the \n[NeV]14.3$\\mu$m\/24.3$\\mu$m ratio (Spinoglio {\\em et al.\\\/} \\cite{spi00}). The photoionization models using the CLOUDY code (Ferland \\cite{fer00}) have been computed and shown as a grid in the diagram. As expected, the former ratio is sensitive to ionization, \nwhile the latter is sensitive to density. In the figure are also shown measurements on a small sample of Seyfert galaxies for which we can determine the ionization potential (basically the ratio of ionizing photons over the number of hydrogen atoms) and the gas density. We note that the observed galaxies have average densities ranging from less than 10$^2$ cm$^{-3}$ to less than 10$^4$ cm$^{-3}$ and ionization potential of \n10$^{-2.0}$ $<$ log~U $<$ 10$^{-1.5}$. This is in agreement, for the lower density objects, with conditions of \"coronal emission regions\" in AGNs (Spinoglio \\& Malkan \\cite{sm92}).\n\nAnother example, presented in Fig. 4, is given by the [CII]157$\\mu$m\/[OI]63$\\mu$m ratio as a function of the \n[OIII]88$\\mu$m\/[OI]63$\\mu$m ratio. These low ionization lines are copiously produced in the ISM of galaxies (in photodissociation regions) and the [OIII] line is excited also in HII regions. However we can see from this diagram that while normal galaxies are clustering in a central region that can easily be explained by starburst models, most of the Seyfert galaxies are far from this locus and their rations cannot be reproduced by starburst models. They have much stronger emission of [OI]63$\\mu$m than it would be expected from stellar emission only. To obtain better statistics on this diagram, we will have to wait for the \\textit{Herschel} satellite, that will have the sensitivity to collect far-infrared spectroscopic observations of large samples of local galaxies.\n\n \n\\section{From the local to the distant Universe} \n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig5.eps}\n\\caption{Same as fig.1, but for differerent redshift intervals.\n Different symbols are used for lines from photodissociation regions (squares),\nstellar\/HII region lines (triangles), AGN lines (circles) and\ncoronal lines (stars). The two [OI] lines have been plotted - for\ngraphical reasons - at a ionization potential higher than their\neffective value.}\\end{figure}\n\nMid-IR and far-IR spectroscopy of fine-structure emission lines\ncan be used not only in the local universe but also to measure \nthe excitation conditions in distant cosmological objects.\nFig. 5 shows again, as in Fig.2, the critical density of the lines as a function \nof their ionization potential, not only for the local universe, but \nin three different redshift ranges, one for each frame,\nfor which the rest-frame wavelength of the line is shifted in the\nfar-infrared range. \nIt appears from the figure that although the photodissociation (PDR) \nregime can be probed only in the relatively local universe, because of the long \nwavelengths of the lines tracing this regime, however, the\nstellar emission (e.g. in starburst galaxies) can be probed up to\nhigh z, using many lines in the rest-frame spectral range of 3\n$\\leq$ $\\lambda (\\mu m)$ $\\leq$ 30. Moreover, the ionization from AGN can be\nprobed from the local universe up to redshifs of z$\\leq$5 and the extremely high \nexcitation coronal emission regions are probed by near-IR lines shifted into the\nfar-IR at a redshift of z $\\sim$ 5.\n\nOnce we have proved that we have in the mid-to-far infrared the adequate diagnostic lines, we need\nstill to understand if these lines could be detected by the future space facilities under development or study \nin the future years. Do do so, we will use the observed infrared spectra in local galaxies and let them evolve backwards at earlier cosmological times.\n\n \n\\begin{figure}\n\\includegraphics[width=12cm]{fig6.eps}\n\\caption{Predicted line fluxes as a function of redshift, using as a \nlocal template the prototypical Seyfert type 2 galaxy NGC1068. \nSquares indicate the fluxes of HII region lines, triangles the fluxes \nof lines emitted by AGN, filled circles the fluxes of the \n[OI]$\\mu$m line and open circles the three OH lines \ndetected in emission in NGC1068. The solid lines show how \nthe line fluxes change with redshift adopting luminosity evolution, \nwhile dotted lines without any evolution.}\n\\end{figure}\n\nFor simplicity, and to cover as many transitions possible, we use a \nlocal object which contains both an active nucleus and a starburst, \nthe bright prototype Seyfert 2 galaxy NGC1068.\nThe ISO spectrometers detected in this galaxy many of the lines plotted in Fig. 2 at flux levels\nof 5-200 $\\times$ 10$^{-16}$ W m$^{-2}$ (Alexander {\\em et al.\\\/} \\cite{Ale00}; Spinoglio et al. {\\em et al.\\\/} \\cite{spi05}). \nConsidering this galaxy as a template object, we computed the line\nintensities expected at redshifts ranging from 0.1 to 5. For\nsimplicity, we adopted an Einstein-De Sitter model Universe, with\n$\\Omega_{\\Lambda}$ = $\\Omega_{vac}=0$ and $\\Omega_{M}$= 1,\nH$_{0}$=75 km s$^{-1}$ Mpc$^{-1}$. The luminosity distances have been derived using:\n\\begin{equation}\nd_{L} (z)= (2c\/H_{0})\\cdot [1+z - (1+z)^{1\/2}]\n\\end{equation}\n\nThe results are reported in Fig. 6, where the line intensities are given in W m$^{-2}$, and the expected sensitivities of spectrometers, such ESI (European SPICA Instrument, Swinyard \\cite{swi06}), onboard of the future space observatories, SPICA (Space Infrared Telescope for Cosmology and\nAstrophysics, ISAS-JAXA) (Nakagawa \\cite{Nag04}; Onaka \\& Nakagawa \\cite{ON05}) and FIRI (Far-Infrared Interferometer) are also shown.\n\nWe have assumed that the line luminosities scale as the bolometric luminosity and we have chosen two cases: \\\\\nA) a luminosity evolution proportional to the (z+1)$^{2}$, consistent with the {\\it Spitzer}\nresults at least up to redshift z=2 (P\\'{e}rez-Gonz\\'{a}lez {\\em et al.\\\/} \\cite{Per05}) ;\\\\\nB) no luminosity evolution.\\\\\n\nBecause the star formation process in galaxies was much more enhanced at z=1-2 than today, we consider reasonable to adopt the model with strong evolution at least for the stellar\/HII region lines and, to be conservative, the \"no\nevolution\" one for the AGN lines.\\\\\nWe note that the dependence on different cosmological models is not very strong.\nThe popular model with $\\Omega_{M}$= 0.27, $\\Omega_{vac}$=0.73,\nH$_{0}$=71 km s$^{-1}$ Mpc$^{-1}$ shows greater dilutions,\nincreasing with z, by factors of 1.5 for z=0.5 to 2.5 for z=5. In\nthis case the line intensities of Fig. 6 would decrease by these factors.\n \nWe conclude that a relatively low luminosity object like NGC1068, with an infrared luminosity of \n2 $\\times$ 10$^{11}$ L$_{\\odot}$, \nwill be detected up to a redshift of z=5 by the cooled 3.5m mirror of the SPICA satellite in a few bright and important diagnostic lines, such as the [OI]63$\\mu$m, the [OIII]52$\\mu$m, [OIV]26$\\mu$m, assuming luminosity evolution in the lines.\n\nThe fainter AGN lines (like [NeVI]7.6$\\mu$m) and the molecular lines of OH, will be detected by SPICA in such an object at z $\\sim$ 0.5. For detecting the fainter lines up to z $\\sim$ 5, we will have to wait for larger collecting area space telescopes, as the FIRI project, foreseen beyond the next decade.\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Maximizing $p$-Mean Welfare}\n\nAddressing fair-division instances with identical subadditive valuations, this section presents an efficient algorithm for computing a constant-factor approximation to the $p$-mean welfare objective, uniformly for all $p \\in (-\\infty, 1]$. \n\nThe algorithm consists of two phases, Algorithm~\\ref{Alg} (\\textsc{Alg}) and Algorithm~\\ref{AlgSub} (\\textsc{AlgLow}). In the first phase, ``high-value'' goods are assigned as singletons--we use the approximation algorithm of Feige~\\cite{feige2009maximizing} to obtain an estimate of the optimal $1$-mean welfare and deem a good to be of high value if its valuation is at least a constant (specifically, $\\nicefrac{1}{3.53}$) times this estimate. Intuitively, the estimate provides a useful benchmark, since the optimal $1$-mean welfare upper bounds the optimal $p$-mean welfare for all $p \\in (-\\infty, 1]$ (Proposition~\\ref{Monotonicity}); this bound essentially follows from the generalized mean inequality~\\cite{bullen1988mathematics} which asserts that, for all $p \\in (-\\infty, 1]$, the $p$-mean welfare of any allocation $\\mathcal{A}$ is at most its $1$-mean welfare, ${\\textrm M}_p(\\mathcal{A}) \\leq {\\textrm M}_1(\\mathcal{A})$. \n\nTherefore, in phase one of the algorithm, we sort the goods in non-increasing order by value and iteratively select goods, which by themselves provide a value comparable to that of the optimal $p$-mean welfare. In each iteration, the selected good is assigned as a singleton to an agent and this agent-good pair is removed from consideration. Note that such an update leads to a new fair-division instance with one less good and one less agent, as well as a potentially different optimal $1$-mean welfare. The key technical issue here is that the change in the optimal $1$-mean welfare (and, hence, its estimate obtained via Feige's algorithm) can be non-monotonic. Nonetheless, via an inductive argument, we show that the welfare contribution of the goods assigned (as singletons) in the first phase is sufficiently large (Lemma~\\ref{Induction_argument}). \n\nThe first phase terminates when we obtain an instance $\\mathcal{J}$ wherein each good is of value no more than a constant times its optimal $1$-mean welfare. The second phase (\\textsc{AlgLow}) is designed to address such a fair-division instance. In particular, we show that, in the absence of high-value goods, we can efficiently find an allocation $\\mathcal{B}=(B_i)_i$ such that each bundle $B_i$ is of value at least constant times the optimal $p$-mean welfare of $\\mathcal{J}$. To obtain the allocation $\\mathcal{B}$, we first compute (via Feige's approximation algorithm) an allocation $\\mathcal{S}=(S_j)_j$ that provides a $2$-approximation to the optimal $1$-mean welfare of $\\mathcal{J}$. Subsequently, we show that the subsets $S_j$s, that have appropriately high value, can be partitioned to form the desired bundles $B_i$s, which constitute the allocation $\\mathcal{B}$. \n\nMultiple technical lemmas (in Sections~\\ref{Supporting_Lemmas} and~\\ref{section:stitching-lemma}) are required to show that the two phases in combination lead to the desired $p$-welfare bound. It is also relevant to note that, while the above-mentioned ideas hold at a high level, the formal guarantees are obtained by separately analyzing different ranges of the exponent parameter $p$.\n\n\n\n\n\\floatname{algorithm}{Algorithm}\n\\begin{algorithm}[ht]\n \\caption{\\textsc{Alg}} \\label{Alg\n \\textbf{Input:} A fair-division instance $\\mathcal{I}= \\langle [m],[n],v \\rangle$ with demand oracle access to the subadditive valuation function $v$. \\\\\n \\textbf{Output:} An allocation $\\mathcal{A} = (A_1, A_2, \\ldots, A_n)$ \n \\begin{algorithmic}[1]\n \\STATE Initialize the set of agents ${U}=[n]$, the set of goods $G=[m]$, and bundle $A_i= \\emptyset $ for all $i\\in {U}$\n \\STATE Index all the goods in non-increasing order of value $v(g_1) \\geq v(g_2) \\geq \\ldots \\geq v(g_m)$\\label{Ordered_Goods}\n \\STATE Set $\\mathcal{I}^0= \\I{G}{{U}}$ and initialize $t=1$ \\COMMENT{Recall that ${\\rm F}(\\mathcal{I}) = {\\rm M}_1 (\\mathcal{S})$, where $\\mathcal{S}$ denotes the allocation obtained by executing Feige's algorithm~\\cite{feige2009maximizing} on instance $\\mathcal{I}$.}\n \\WHILE { $v(g_t) \\geq \\frac{1}{3.53} \\ {\\rm F}(\\mathcal{I}^{t-1})$ } \\label{Threshold}\n \\STATE Allocate $A_t \\leftarrow \\{ g_t \\}$ and update $G \\leftarrow G\\setminus \\{g_t\\}$ along with ${U} \\leftarrow {U} \\setminus \\{ t \\}$\n \\STATE Set $\\mathcal{I}^{t}=\\I{G}{{U}}$ and update $t \\leftarrow t+1$\n \\ENDWHILE \\label{step:end-while}\n \\STATE Set $(A_{t}, A_{t+1},\\ldots, A_n) = \\textsc{AlgLow}(G, {U} ,v)$ \\COMMENT{This step corresponds to the second phase of the algorithm which assigns bundles to the remaining $| {U}| = n-t +1$ agents. Also, note that, in the current instance $\\mathcal{J} \\coloneqq \\langle G, {U}, v \\rangle$, for every good $g \\in G$ we have $v(g) < \\frac{1}{3.53} {\\rm F} (\\mathcal{J})$.}\n \\RETURN allocation $\\mathcal{A} = (A_1,A_2,...,A_n).$\n \n \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\floatname{algorithm}{Algorithm}\n\\begin{algorithm}[h]\n \\caption{\\textsc{AlgLow} } \\label{AlgSub}\n \\textbf{Input:} A fair-division instance $\\mathcal{J} = \\langle G, U,v \\rangle$ with demand oracle access to the subadditive valuation function $v$. \\\\\\textbf{Output:} An allocation $\\mathcal{B} = (B_1, B_2, \\ldots, B_{|U|})$ \n \\footnotesize\n \\begin{algorithmic} [1]\n \\STATE Execute Feige's approximation algorithm~\\cite{feige2009maximizing} on the given instance $\\mathcal{J}$ to compute allocation $\\mathcal{S} = (S_1, S_2, \\ldots, S_{|U|})$. \n \\COMMENT{Note that allocation $\\mathcal{S}$ provides a $2$-approximation to the optimal $1$-mean welfare of $\\mathcal{J}$, ${\\rm M}_1 (\\mathcal{S}) = {\\rm F}(\\mathcal{J}) \\geq \\frac{1}{2} {\\rm M}_1 \\left(\\mathcal{A}^*(\\mathcal{J}, 1) \\right)$}\n \\STATE Index the bundles such that $v(S_1) \\geq \\ldots \\geq v(S_{|U|})$ and initialize $i=a = 1$ along with $B_\\ell = \\emptyset$ for $1 \\leq \\ell \\leq |U|$ \\\\ \\COMMENT{Lemma~\\ref{Low_valued} shows that the following loop runs to completion} \n \\WHILE {agent index $a < |U|$}\n \\STATE Consider an arbitrary good $g \\in S_i$ \n \\IF {$v( B_a \\cup \\{ g\\}) < \\frac{1}{3} {\\rm F} (\\mathcal{J}) $ } \n \\STATE Update $B_a \\leftarrow B_a \\cup \\{ g \\}$ and $S_i \\leftarrow S_i \\setminus \\{ g\\}$ \\COMMENT{Here good $g$ is assigned to bundle $B_a$ to increase its value}\n \\ELSE\n \\STATE Update $a \\leftarrow a + 1$ \\COMMENT{This update is performed when sufficient value has been accumulated in a bundle}\n \\ENDIF\n \\IF{$v(S_i) < \\frac{1}{3} {\\rm F} (\\mathcal{J}) $}\n \\STATE Update $i \\leftarrow i + 1$ \\COMMENT{Once the value of $S_i$ drops below $\\frac{1}{3} {\\rm F}(\\mathcal{J})$ we consider the next bundle in $\\mathcal{S}$}\n \\ENDIF\n\\ENDWHILE\n\\STATE $B_{|U|} \\leftarrow B_{|U|} \\cup \\left( G \\setminus (\\bigcup \\limits _{a=1}^{|U|-1} B_a )\\right)$ \\COMMENT{Assign the remaining elements to $B_{|U|}$} \n \n \n \n \n \n \\RETURN partition $\\mathcal{B} = (B_1,\\ldots ,B_{|U|}).$\n \n \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\nThe following theorem constitutes the main result of the current work. It asserts that Algorithm~\\ref{Alg} (\\textsc{Alg}) achieves a constant-factor approximation ratio for the $p$-mean welfare maximization problem.\n\n\n\\newtheorem{thm}{Theorem}[section]\n\\newcommand{Main Theorem }{Main Theorem }\n\n\\newtheorem*{genericthm*}{Main Theorem }\n\\newenvironment{namedthm*}[1]\n {\\renewcommand{Main Theorem }{#1}%\n \\begin{genericthm*}}\n {\\end{genericthm*}}\n \n\n \n\\begin{theorem} [Main Result] \\label{MainTheorem}\nLet $\\mathcal{I} = \\langle [m], [n], v \\rangle$ be a fair-division instance wherein all the agents have an identical, subadditive valuation function $v$. Given demand oracle access to $v$, \\textsc{Alg} computes in polynomial time an allocation $\\mathcal{A}$ that, for all $p \\in (-\\infty, 1]$, provides a $40$-approximation to the optimal $p$-mean welfare, i.e., ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} \\ {\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p))$, for all $p \\in (-\\infty, 1]$; here, $\\mathcal{A}^*(\\mathcal{I},p)$ is the $p$-optimal allocation in $\\mathcal{I}$. \n\\end{theorem}\n\n \n We first consider an instance $\\mathcal{J}$ wherein all the goods are of value a constant times less than ${\\rm F}(\\mathcal{J})$ and prove that, for such an instance, \\textsc{AlgLow} finds an allocation in which the value of every bundle is comparable to the optimal average social welfare of $\\mathcal{J}$. In Section \\ref{subsection:p-inf-half}, we use this fact and supporting lemmas from Sections~\\ref{Supporting_Lemmas} and~\\ref{section:stitching-lemma} to prove Theorem~\\ref{MainTheorem} for $p\\in (-\\infty,0.4)$. Finally, in Section \\ref{subsection:p-half-one}, we prove the main result for $p\\in [0.4,1].$ \n \nWe start with the following observation to upper bound the optimal $p$-mean welfare in terms of the optimal $1$-mean welfare. \n\\begin{Proposition}\\label{Monotonicity}\nLet $\\mathcal{I}$ be a fair-division instance in which all the agents have an identical, subadditive valuation $v$. Then, for each $p\\in(-\\infty,1],$ the optimal $1$-mean welfare is at least as large as the optimal $p$-mean welfare:\n\\begin{align*}\n{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{I},1)) \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)) \\ \\ \\text{for every } \\; p \\in \\left(- \\infty , 1\\right]\n\\end{align*} \n\\end{Proposition}\n\\begin{proof}\nThe generalized mean inequality (see, e.g., ~\\cite{bullen1988mathematics}) applied to the allocation $\\mathcal{A}^{*}(\\mathcal{I},p)$, gives us ${\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)) \\leq {\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},p))$, for all $p \\in (-\\infty, 1]$. By definition, the allocation $\\mathcal{A}^*(\\mathcal{I}, 1)$ maximizes the $1$-mean welfare, ${\\rm M}_1(\\cdot)$, and, hence, the claim follows ${\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},1)) \\geq {\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},p)) \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$. \n\\end{proof}\n\n\n\\section{Approximation Guarantee for \\textsc{AlgLow}}\n\\label{Approx_for_AlgSub}\nThis section addresses the second phase of the algorithm (\\textsc{AlgLow}) that---by the processing performed in the while-loop of \\textsc{Alg}---solely needs to consider fair-division instances $ \\mathcal{J}=\\I{G}{U}$ wherein all the goods $g \\in G$ satisfy $v(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})$, i.e., the goods are of ``low value.'' The following lemma establishes that, for such instances, \\textsc{AlgLow} finds bundles each with value comparable to the optimal $1$-mean welfare (and, hence, comparable to the optimal $p$-mean welfare) of $\\mathcal{J}$. \n\nRecall that here $G$ is a subset of the original set of goods $[m]$, $U$ is a subset of the $[n]$ agents, and ${\\rm F}(\\mathcal{J})$ denotes the $1$-mean welfare (average social welfare) of the allocation computed by Feige's approximation algorithm for instance $\\mathcal{J}$.\n\n\n\n \n\\begin{Lemma}\\label{Low_valued}\nLet $\\mathcal{J} = \\I{G}{U}$ be a fair-division instance in which all the agents have an identical subadditive valuation function $v$, and every good $g\\in G$ satisfies \n$v(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})$. Then, in the demand oracle model, the algorithm \\textsc{AlgLow} efficiently computes an allocation $\\mathcal{B}=(B_1,\\ldots , B_{|U|})$ with the property that, for all $i \\in \\{1, \\ldots, |U|\\}$,\n\\begin{align*}\nv(B_i)\\geq \\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1)) \\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p)).\n\\end{align*}\n\\end{Lemma}\n\\begin{proof}\nFor input instance $\\mathcal{J}$, Feige's algorithm returns an allocation $\\mathcal{S}=(S_1,S_2,\\ldots,S_{|U|})$ with near-optimal average social welfare:\n\\begin{align*}\n\\frac{1}{|U|}\\summi{1}{|U|}{S} & = {\\rm F}(\\mathcal{J}) \\geq \\frac{1}{2}{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))\n\\end{align*}\n\nGiven allocation $\\mathcal{S}=(S_1,S_2,\\ldots,S_{|U|})$, we show that one can partition $S_i$s to form $|U|$ bundles such that the value of each bundle is at least $\\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))$. Note that, by the assumption in the lemma, for each $g \\in G$ we have \n\\begin{align}\\label{2}\nv(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})\n\\end{align} \n\nLet $H \\coloneqq \\left\\{ i \\in \\{1, 2, \\ldots, |U|\\} \\mid v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\right\\}$ denote the subsets in $\\mathcal{S}$ with value at least $\\frac{1}{3} {\\rm F}(\\mathcal{J})$. These are the subsets we split to form bundles $B_i's$ that form the output allocation $\\mathcal{B}$.\n\nSince $v$ is subadditive, we have the following lower bound on the cumulative value of the subsets in $H$\n\\begin{align}\\label{3}\n\\sum \\limits _{i \\in H} v(S_i) \\geq \\sum \\limits _{j=1}^{|U|} v(S_j) - \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\cdot |U| = \\frac{2}{3}\\ |U| \\cdot {\\rm F}(\\mathcal{J})\n\\end{align}\n\\newtheorem{Claim}{Claim}\n\\begin{Claim}\nLet $\\mathcal{S} = (S_1,\\ldots,S_{|U|})$ be the allocation computed by Feige's algorithm for input instance $\\mathcal{J}.$ Then, every $S_i$, with the property that $v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J})$, can be partitioned into at least $k=\\left ( \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})}-1\\right )$ subsets $T^1_i,\\ldots , T^k_i$ such that $v(T^j_i)\\geq \\frac{1}{20}{\\rm F}(\\mathcal{J})$ for each $T^j_i$.\n\\end{Claim}\n\\begin{proof}\nInitialize $T_i^1$ to be the empty set. Then, we keep transferring goods---in any order and one at a time---from $S_i$ to $T_i^1$ till the value of $T_i^1$ goes over $\\frac{1}{3}{\\rm F}(\\mathcal{J}).$ Returning the last such good back into $S_i$, the populated set $T_i^1$ satisfies \n\\begin{align*}\nv(T_i^1) & \\geq \\frac{1}{3}{\\rm F}(\\mathcal{J}) - \\frac{1}{3.53}{\\rm F}(\\mathcal{J}) \\tag{using (\\ref{2}) and the subadditivity of $v$} \\nonumber \\\\ & \\geq \\frac{1}{20}{\\rm F}(\\mathcal{J}) \n\\end{align*}\nNote that, by construction, $v(T_i^1) \\leq \\frac{1}{3}{\\rm F}(\\mathcal{J})$. Therefore, using the subadditivity of $v$, we get $v(S_i\\setminus T_i^1) \\geq v(S_i)-\\frac{1}{3}{\\rm F}(\\mathcal{J})$.\n\nWe can repeat the above process to obtain subsets $T_i^1,\\ldots T_i^k$ and stop when $v\\left(S_i\\setminus ( T_i^1 \\cup \\ldots \\cup T_i^k) \\right) \\leq \\frac{1}{3}{\\rm F}(\\mathcal{J}).$ Given that, for each $T_i^j$, we remove a subset of value atmost $\\frac{1}{3} {\\rm F}(\\mathcal{I})$ from $S_i$, the subadditivity of $v$ gives us $v \\left(S_i\\setminus ( T_i^1 \\cup \\ldots \\cup T_i^k) \\right) \\geq v(S_i) - \\frac{k{\\rm F}(\\mathcal{J})}{3}$. Hence, the following lower bound holds $k\\geq \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1$.\nIn other words, we can extract at least $\\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1$ bundles, each of value no less than $\\frac{1}{20}{\\rm F}(\\mathcal{J})$, from $S_i$.\n\\end{proof}\nWe now apply the same procedure to every subset in $H \\coloneqq \\left\\{ i \\in \\{1, 2, \\ldots, |U| \\} \\mid v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\right\\}$ to obtain $k^{'} = \\sum \\limits _{i=1}^{|H|} \\left ( \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1 \\right ) $ bundles, each of value at least $\\frac{1}{20}{\\rm F}(\\mathcal{J})$. Using this equation and inequality (\\ref{3}), we get\n\\begin{align*}\nk^{'} & \\geq \\frac{2|U|{\\rm F}(\\mathcal{J})}{{\\rm F}(\\mathcal{J})} - |H| \\geq 2|U| - |U| \\tag{Since $|H|<|U|$} \\\\\n& = |U| \\hspace{3.3cm}\n\\end{align*}\n\nIn conclusion, one can construct at least $|U|$ bundles of value at least $\\frac{1}{20}{\\rm F}(\\mathcal{J}) \\geq \\frac{1}{40}{\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1))$. Note that this observation implies that \\textsc{AlgLow} successfully finds $|U|$ bundles each of value at least $\\frac{1}{40} {\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1))$. \n\nProposition~\\ref{Monotonicity} gives us ${\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1)) \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p))$ and, hence, the stated claim follows. \n\\end{proof}\n\n\\section{Proof of Numeric Inequality from Lemma \\ref{Good_Transfer}} \\label{app 3}\nThis section establishes the numeric inequalities used in Sections \\ref{subsection:p-infty-zero-good-transfer} and \\ref{subsection:p-zero-half-good-transfer}. Specifically, for $p\\in (-\\infty,0)$\n\\begin{align}\n \\left( \\frac{1}{2} - \\frac{1}{40}\\right )^p + \\left ( \\frac{1}{2}\\right )^p &\\leq 1 + \\left ( \\frac{2}{11.33} \\right )^p \\label{ineq_neg}\n\\end{align}\nAlso, for $p\\in (0,0.4)$, we have \n\\begin{align}\n\\left( \\frac{1}{2} - \\frac{1}{40}\\right )^p + \\left ( \\frac{1}{2}\\right )^p &\\geq 1 + \\left ( \\frac{2}{11.33} \\right )^p \\label{ineq_pos} \n\\end{align}\n\n\nWrite $a$ to denote $\\left( \\frac{1}{2} - \\frac{1}{40}\\right ) $, $b$ to denote $\\left ( \\frac{1}{2}\\right )$ and $c$ to denote $\\left ( \\frac{2}{11.33} \\right ).$ Consider the following differentiable function $f(p)\\coloneqq a^p+b^p-1-c^p$. In order to obtain bounds (\\ref{ineq_neg}) and (\\ref{ineq_pos}), it suffices to prove that $f(p)>0$ for all $p\\in (0,0.4)$ and $f(p)\\leq 0$ for all $p\\in(-\\infty,0)$. We first show a useful property of the function $f(p)$ that will help us in deriving these inequalities.\n\\begin{center}\n \\includegraphics[scale=0.3]{graph1.png}\n \\captionof{figure}{Plot of the function $f(p)$ }\n \\label{fig:sample_figure}\n\\end{center}\n\n\n\n\\begin{Claim}\\label{extreme_points_are_maxima}\nEvery extreme point of the function $f(p)=a^p+b^p-1-c^p$ is a local maximum; here, $a=\\left( \\frac{1}{2} - \\frac{1}{40}\\right ) $, $b=\\left ( \\frac{1}{2}\\right )$ and $c=\\left ( \\frac{2}{11.33} \\right ).$\n\\end{Claim}\n\n\\begin{proof}\nWe consider the first and second derivatives of the smooth function $f$:\n\\begin{align}\nf'(p)= a^p\\log (a) + b^p\\log (b) - c^p\\log (c) \\label{$f'$}\\\\ \\text{and } f''(p) = a^p(\\log (a))^2 + b^p(\\log (b))^2 - c^p(\\log (c))^2 \\label{$f''$}\n\\end{align} \nLet $\\overline{p}$ be any point that satisfies $f'(\\overline{p})=0. $ That is, $c^{\\overline{p}}\\log (c)= a^{\\overline{p}}\\log (a) + b^{\\overline{p}}\\log (b) $. Instantiating equation (\\ref{$f''$}) with $p=\\overline{p}$ and substituting the previous expression for $c^{\\overline{p}}\\log (c)$, we get \n\\begin{center}\n$f''(\\overline{p})=a^{\\overline{p}}\\log (a)(\\log (a)-\\log (c))+b^{\\overline{p}}\\log (b)(\\log (b)-\\log (c))$\n\\end{center}\nIn the above equation, note that the terms $(\\log (a)-\\log (c))$ and $(\\log (b)-\\log (c))$ are positive, since $\\frac{a}{c} >1$ and $\\frac{b}{c} >1$, while the terms $\\log (a)$ and $\\log (b)$ are negative since $a,b<1.$ Therefore, $f''(\\overline{p})<0$, which means $\\overline{p}$ is a local maximum. We have now shown that every extreme point of $f$ is a local maximum.\n\n\\end{proof}\nNote that $f(0.4)>0$ and $f(0.41)<0$, hence, the intermediate value theorem implies that there exists a point $r \\in (0.4, 0.41)$ such that $f(r) = 0$. We will show that $f$ is nonnegative over the interval $[0,r]$ and, hence, $f(p) \\geq 0$ for all $p \\in (0, 0.4)$. \n\nGiven that $f(0)=0$ and $f(r) = 0$, we can apply the mean value theorem to $f$ in the range $[0,r]$ to conclude that there is a point $p_0 \\in [0,r]$ satisfying $f'(p_0)=0$. Using Claim \\ref{extreme_points_are_maxima}, we will next show that $f(p)$ has a unique extreme point, which is a local maximum, in the range $[0,r]$. \n\n\\begin{Claim}\\label{unique_maximum}\nLet $p^* \\in [0,r]$ be a point that satisfies $f'(p^*)=0$, then it is the unique extreme point of $f(\\cdot)$ and is a local maximum.\n\\end{Claim}\n\\begin{proof}\nSince $f'(p^*)=0$, $p^*$ is an extreme point. Additionally, from Claim \\ref{extreme_points_are_maxima}, $p^*$ is a local maximum. \n\nAssume, towards a contradiction, that there is another local maximum $p_1 \\in \\mathbb{R}$ and, without loss of generality, assume that $p_1>p^*.$ We will show that this implies the existence of a local minimum between $p^*$ and $p_1$, hence contradicting Claim \\ref{extreme_points_are_maxima}.\n\nConsider the point $p_{min}$ defined as $p_{min}=\\text{arginf} \\left\\{ \\ f(p) \\mid p \\in [p^*, p_1] \\right\\}$. Note that $p_{min}\\neq p^*,p_1$ because $p^*$ and $p_1$ are local maxima, which means their function values are greater than those of the points in an $\\varepsilon$-neighborhood around them. Therefore, applying Fermat's theorem for stationary points, we get $f'(p_{min})=0$. Since $p_{min}$ is also the infimum of $p\\in[p^*,p_1],$ it is a local minimum between $p^*$ and $p_1.$ Therefore, by way of contradiction, we obtain the stated claim. \n\\end{proof}\n\\noindent \\textit{Proof of inequality (\\ref{ineq_pos}):} We now prove that $f(p)\\geq0$ for $p \\in [0, 0.4].$\nAssume, for the sake of contradiction, that there exists is a point $q \\in [0, 0.4]$ such that $f(q)<0$.\n\nThe facts that $f(0) = 0$, $f'(0) >0$ and $f(q)<0$, along with the mean value theorem, imply that there exists a point $\\widehat{p} \\in [0, q)$ such that $f'(\\widehat{p})=0$. Additionally, applying mean value theorem with inequalities $f(0.4)>0$, $f'(0.4)<0$, and $f(q)<0$, we get a different point $\\overline{p}\\in (q, 0.4]$ such that $f'(\\overline{p})=0$. Existence of two distinct extreme points contradicts Claim \\ref{unique_maximum} and, hence, establishes that $f(p)\\geq 0$ for all $p \\in [0,0.4]$. \\\\\n\n\\noindent \\textit{Proof of inequality (\\ref{ineq_neg}):} We split the proof of this inequality depending on the range of $p$:\\\\\n\n\\noindent\nCase 1: $p\\in (-\\infty,-1]$. Note that $a>\\frac{1}{2.2}$, $b=\\frac{1}{2}$, $c<\\frac{1}{5}$, and $p$ is negative.\nSubstituting these bounds for $a,b$ and $c$ in the expression for $f(p)$, we get $f(p)<\\left(\\frac{1}{2.2}\\right)^p+\\left(\\frac{1}{2}\\right)^p-1-\\left(\\frac{1}{5}\\right)^p.$ Equivalently, we have $f(p)<(2.2)^{|p|}+2^{|p|}-1-5^{|p|}.$ Since $5^{|p|}\\geq(2.2)^{|p|}+2^{|p|}$ for $p\\leq-1$, we conclude that $f(p)<0$ for $p\\in(-\\infty,-1].$\\\\\n\\noindent\nCase 2: $p\\in (-1,0)$. Assume, for the sake of contradiction, that there is a $\\widetilde{p} \\in(-1,0)$ such that $f(\\widetilde{p})>0$.\nNote that $f(-1) <0$. Hence, by applying intermediate value theorem to $-1$ and $\\widetilde{p}$, we have a point $\\widehat{p} \\in (-1, \\widetilde{p})$ such that $f(\\widehat{p})=0$. Additionally, mean value theorem with $0$ and $\\widehat{p}$ implies that there exists a point $\\overline{p}\\in [\\widehat{p},0)$ such that $f'(\\overline{p})=0$. Since this contradicts Claim \\ref{unique_maximum}, we have $f(p) \\leq 0$ for all $p\\in(-1,0)$.\n\nThese two cases imply that inequality (\\ref{ineq_neg}) holds for all $p\\in (-\\infty,0).$\n\n\n\\section{APX-Hardness of Maximizing $p$-Mean Welfare} \\label{APX_Hardness}\nIn this section, we prove that the problem of computing a $p$-optimal allocation is {\\rm APX}-hard, for all $p\\in (-\\infty,1]$, in the demand oracle model. \nThat is, we show that there exists a constant $c>1$ such that it is {\\rm NP}-hard to approximate the optimal $p$-mean welfare within a factor of $c$, even if we are given access to demand queries.\n\n\\begin{theorem}\\label{APX_hardness_theorem}\nGiven a fair division instance $\\mathcal{I}=\\I{[m]}{[n]}$, wherein the agents have identical, subadditive valuations, the $p$-mean welfare maximization problem is {\\rm APX}-Hard for all $p\\in (-\\infty,1]$, in the demand oracle model.\n \\end{theorem}\n\nWe prove this hardness result by developing a gap-preserving reduction from the Gap-3DM problem. An instance $\\mathcal{C}$ of this problem consists of three disjoint sets $X$, $Y$, and $Z$, of cardinality $q$ each, along with a collection of $3$-uniform hyperedges $E \\subset X \\times Y \\times Z$. \nThe goal is to find a matching (i.e., a subset of pairwise disjoint hyperedges) $M \\subset E$ of maximum cardinality. \n\nFormally, Gap-3DM is the gap version of $3$-dimensional matching and it entails distinguishing between the following types of instances: \\\\\n(i) {\\rm YES}-instance: There is a perfect matching (a matching of size $q$) in the given instance $\\mathcal{C}$.\\\\\n(ii) {\\rm NO}-instance: All matchings in $\\mathcal{C}$ are of size at most $\\alpha q$, with $\\alpha <1$.\\\\\nThe Gap-3DM problem is known to be {\\rm NP}-hard, for an absolute constant $\\alpha <1$~\\cite{ostrovsky2014s}.\\footnote{Note that the given instances in this problem are promised to be either {\\rm YES} or {\\rm NO} instances.} \n\nTheorem \\ref{APX_hardness_theorem} follows from a gap-preserving reduction. In particular, we will prove that in the {\\rm YES} case (i.e., when \nthe given instance of Gap-3DM has a perfect matching) the reduced instance of the welfare problem admits an allocation with $p$-mean welfare at least $3$. In the {\\rm NO} case, the optimal $p$-mean welfare is less than $3 \\ c (\\alpha)$, where $c (\\alpha) <1$ is a constant (that depends of $\\alpha$). \n\nTherefore, given a $\\frac{1}{c(\\alpha)}$-approximation algorithm for $p$-mean welfare maximization, one could distinguish between the {\\rm YES} and {\\rm NO} instances of Gap-3DM. Hence, the hardness of Gap-3DM implies that it is {\\rm NP}-hard to approximate the optimal $p$-mean welfare with a constant factor of $\\nicefrac{1}{c(\\alpha)}$. \n\n\\subsection{Proof of Theorem \\ref{APX_hardness_theorem}} \n\nGiven an instance $\\mathcal{C}$ of Gap-3DM with $3$-uniform hyperedges $E=\\{E_1, \\ldots, E_T\\}$ over size-$q$ sets $X$, $Y$, and $Z$, we construct an instance $\\mathcal{I}$ of $p$-mean welfare maximization with $q$ agents and $3q$ goods, one for each vertex $X \\cup Y \\cup Z$. All the agents share a common XOS valuation function $v$: define $v(S) \\coloneqq \\max \\limits _{1\\leq i\\leq T} \\left\\{|S \\cap E_i| \\right\\}$ for each subset of goods $S\\subset X\\cup Y\\cup Z$.\n\nNote that the value of any subset $S$ is upper bounded by 3, since each $E_i$ is a 3-uniform hyperedge. We now prove that this reduction is gap preserving. \n\nWhen the input instance $\\mathcal{C}$ is a {\\rm YES} instance, there is a matching of size $q$. We assign the three goods corresponding to each edge in the matching to a distinct agent. In this allocation each agent gets a bundle of value $3$. Therefore, the optimal $p$-mean welfare in this case is at least $3$. Conversely, when $\\mathcal{C}$ is a {\\rm NO} instance, every matching is of size at most $\\alpha q$. We claim that in this case, the optimal $p$-mean welfare in the reduced instance $\\mathcal{I}$ is upper bounded by $3 \\ c(\\alpha)$, where $c(\\alpha) =\\frac{2+\\alpha}{3} <1$. \n\nLet $\\mathcal{A}^*(\\mathcal{I},p)=(A^*_1,\\ldots, A^*_q)$ denote a $p$-optimal allocation in instance $\\mathcal{I}$. Here, in any allocation, an agent's value for her bundle is either $0$, $1$, $2$, or $3$. We partition the bundle in the optimal allocation into two collections\n$\\mathcal{H} \\coloneqq \\{A^*_i \\mid v(A^*_i)=3\\} \\text{ and } \\overline{\\mathcal{H}} \\coloneqq \\{A^*_i \\mid v(A^*_i)\\leq 2\\}$.\nEvery bundle $A^*_i$ in $\\mathcal{H}$ contains at least one hyperedge $E_j$. Since the bundles in $\\mathcal{H}$ are disjoint, the hyperedges contained in different bundles are nonintersecting. Hence, such hyperedges form a matching in $\\mathcal{C}$ of size at least $|\\mathcal{H}|.$ Recall that $\\mathcal{C}$ is a {\\rm NO} instance, i.e., every matching in $\\mathcal{C}$ of size at most $\\alpha q.$ Therefore, there are at most $\\alpha q$ bundles in $\\mathcal{A}^*(\\mathcal{I},p)$ of value $3$, $|\\mathcal{H}| \\leq \\alpha q$. Using this inequality we can upper bound the optimal $p$-mean welfare in instance $\\mathcal{I}$ as follows. \n\\begin{Claim}\n${\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p))\\leq 2+ \\alpha = 3 \\left( \\frac{2+\\alpha}{3}\\right) $\n\\end{Claim}\n\\begin{proof}\nRecall that $\\mathcal{H}$ and $\\overline{\\mathcal{H}}$ form a partition of the $q$ bundles in $\\mathcal{A}^*(\\mathcal{I},p)$. Let $|\\mathcal{H}|=\\overline{\\alpha} q$ for some $\\overline{\\alpha}\\leq \\alpha$. Then, $\\sum \\limits _{A^*_i \\in \\overline{\\mathcal{H}}} v(A^*_i)\\leq 2(1-\\overline{\\alpha})q$. Therefore, \n\\begin{align*}\n{\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p)) &\\leq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{I},p)) \\tag{via the generalized mean inequality}\\\\\n& \\leq \\frac{1}{q}\\left( 3\\overline{\\alpha} q + 2(1-\\overline{\\alpha})q \\right) \\tag{averaging over the values of bundles in $\\mathcal{H}$ and $\\overline{\\mathcal{H}}$}\\\\\n&\\leq 2+\\alpha \\tag{since $\\overline{\\alpha}\\leq \\alpha$}\n\\end{align*}\n\\end{proof}\nHence, we have a polynomial-time reduction from the Gap-3DM to the $p$-mean welfare maximization problem such that:\\\\\n(i) When Gap-3DM instance $\\mathcal{C}$ is a {\\rm YES} instance, the optimal $p$-mean welfare is at least $3$.\\\\\n(ii) When $\\mathcal{C}$ is a {\\rm NO} instance, the optimal $p$-mean welfare is at most $3\\ c(\\alpha)$, for a constant $c(\\alpha) <1$. \n\nAs mentioned previously, such a gap-preserving reduction establishes the {\\rm APX}-hardness of $p$-mean welfare maximization. \n\nFinally, note that we can efficiently simulate the demand oracle for the valuation function $v$ in the constructed instance $\\mathcal{I}$. For any additive function $f$, the response to a demand query---with prices $p_j$s associated with the goods---is simply the subset of goods $g$ that satisfy $f(g) - p_g \\geq 0$. The XOS valuation function $v$ in the reduction is obtained by considering a maximum over $T$ additive functions. The parameter $T$ is the number of edges in the given Gap-3DM instance and, hence, is polynomially bounded. Therefore, we can efficiently simulate the demand oracle for $v$ by explicitly optimizing over all the $T$ additive functions. Overall, we get that the {\\rm APX}-hardness holds in the demand oracle model and the theorem follows. \n\\section{Conclusion and Future Work}\nThis work studies the problem of allocating indivisible goods among agents that share a common subadditive valuation. We show that, for such settings, one can always (and in polynomial-time) find a single allocation that simultaneously approximates a range of generalized-mean welfares, to within a constant factor of the optimal. \n\n\nFor ease of presentation, we focussed on the case in which the agents' valuations are exactly identical. Nonetheless, it can be shown that the developed results are somewhat robust: if, say, the agents' valuations are point-wise and multiplicatively close to each other, then again one can obtain meaningful approximation guarantees. Here, an interesting direction of future work is to address settings in which we have a fixed number of distinct valuation functions across all the agents. A nontrivial improvement on the developed approximation guarantee will also be interesting. \n\n\n\n\n\n\\section{Introduction}\n\nA significant body of recent work, in algorithmic game theory, has been directed towards the study of fair and efficient allocation of indivisible goods among agents; see, e.g.,~\\cite{endriss2017trends} and~\\cite{brandt2016handbook}. This thread of research has led to the development of multiple algorithms and platforms (e.g., {Spliddit}~\\cite{goldman2015spliddit}) which, in particular, address settings wherein discrete resources (that cannot be fractionally allocated) need to be partitioned among multiple agents. Contributing to this line of work, the current paper studies discrete fair division from a welfarist perspective. \n\nWe specifically address the problem of finding allocations (of indivisible goods) that (approximately) maximize the \\emph{generalized means} of the agents' valuations. Formally, for exponent parameter $p \\in \\mathbb{R}$, the $p${th} generalized mean, of $n$ nonnegative values $\\{v_i\\}_{i=1}^n$, is defined as ${\\rm M}_p(v_1, \\ldots, v_n) \\coloneqq \\left( \\frac{1}{n} \\sum_i v_i^p \\right)^{\\frac{1}{p}}$. Parameterized by $p$, this family of functions includes well-studied fairness and efficiency objectives, such as average social welfare ($p=1$), Nash social welfare ($p \\to 0$), and egalitarian welfare ($p \\to -\\infty$). In fact, generalized means---with the exponent parameter $p$ in the range $(-\\infty, 1]$---admit a fundamental axiomatic characterization: up to monotonic transformations, generalized means (with $p \\in (-\\infty, 1]$) exactly constitute \\emph{the} family of welfare functions that satisfy the \\emph{Pigou-Dalton transfer principle} and a few other key axioms~\\cite{moulin2004fair}.\\footnote{Note that generalized means are ordinally equivalent to CES (constant elasticity of substitution) functions.} Hence, by way of developing a single approximation algorithm for maximizing generalized means, the current work provides a unified treatment of multiple fairness and efficiency measures. \n\nWith generalized mean as our objective, we focus on fair-division instances in which the agents have a common \\emph{subadditive} (i.e., complement free) valuation. Formally, a set function $v$, defined over a set of indivisible goods $[m]$, is a said to be subadditive iff, for all subsets $A$ and $B$ of $[m]$, we have $v(A \\cup B) \\leq v(A) + v(B)$. This class of functions includes many other well-studied valuation families, namely \\emph{XOS}, \\emph{submodular}, and \\emph{additive} valuations.\\footnote{Recall that a submodular function $f$ is defined by a diminishing returns property: $f(A + e) - f(A) \\geq f(B + e) - f(B)$, for all subsets $A \\subseteq B$ and $e \\notin B$.} These function classes have been used extensively in computer science and mathematical economics to represent agents' valuations. Of particular relevance here are results that (in the context of combinatorial auctions) address the problem of maximizing social welfare under submodular, XOS, and, more generally, subadditive valuations~\\cite{nisan2007algorithmic}. \n\n\nThe focus on a common valuation function across the agents provides a technically interesting and applicable subclass of fair-division problems--as a stylized application, consider a setting in which the agents' values represent money, i.e., for every agent, the value of each subset (of the goods) is equal to the subset's monetary worth. Here, one encounters subadditivity when considering goods that are substitutes of each other. Also, from a technical standpoint, we note that the problem of maximizing social welfare is {\\rm APX}-hard even under identical submodular~\\cite{khot2008inapproximability} and subadditive valuations~\\cite{dobzinski2005approximation}. Appendix \\ref{APX_Hardness} extends this hardness result to all $p \\in (-\\infty, 1]$.\\\\\n\n\n\\noindent\n{\\bf Our Results:} Addressing fair-division instances with identical subadditive valuations, we develop an efficient constant-factor approximation algorithm for the generalized-mean objective (Theorem~\\ref{MainTheorem}). Specifically, our algorithm computes an allocation (of the indivisible goods among the agents), $\\mathcal{A}$, with the property that its generalized-mean welfare, ${\\rm M}_p (\\mathcal{A})$, is at least ${1}\/{40}$ times the optimal $p$-mean welfare, for all $p \\in (-\\infty, 1]$. This result in fact implies an interesting existential guarantee as well: if in a fair-division instance the agents' valuations are identical and subadditive, then there exists a single allocation that uniformly approximates the optimal $p$-mean welfare for all $p \\in (-\\infty, 1]$. \n\nThe tradeoff between fairness and economic efficiency is an important consideration in fair division literature.\\footnote{For example, consider the work on price of fairness~\\cite{bertsimas2011price,bei2019price}} The relevance of the above-mentioned existential guarantee is substantiated by the fact that this result reasonably mitigates the fairness-efficiency tradeoff in the current context; it shows that for identical subadditive valuations there exists a single allocation which is near optimal with respect to efficiency objectives (in particular, social welfare) as well as fairness measures (e.g., egalitarian welfare). Note that such an allocation cannot be simply obtained by selecting an arbitrary partition that (approximately) maximizes social welfare: under identical additive valuations, all the allocations have the same social welfare, even the ones with egalitarian welfare equal to zero. One can also construct instances, with identical subadditive valuations, wherein particular allocations have optimal egalitarian welfare, but subpar social welfare. \n\nEven specific instantiations of our algorithmic guarantee provide novel results: while the problem of maximizing Nash social welfare, among $n$ agents, admits an $\\mathcal{O}(n \\log n)$-approximation under nonidentical submodular valuations~\\cite{garg2020approximating}, the current work provides a novel (constant-factor) approximation guarantee for maximizing Nash social welfare when the agents share a common subadditive (and, hence, submodular) valuation.\\footnote{Under nonidentical additive valuations, there exists a polynomial-time $1.45$-approximation algorithm for maximizing Nash social welfare~\\cite{barman2018finding}. Furthermore, under identical additive valuations, maximizing Nash social welfare admits a polynomial-time approximation scheme~\\cite{nguyen2014minimizing,barman2018greedy}.} Analogously, the instantiation of our result for egalitarian welfare is interesting in and of itself.\n\nGiven that the valuations considered in this work express combinatorial preferences, a naive representation of such set functions would require exponential (in the number of goods) values, one for each subset of the goods. Hence, to primarily focus on the underlying computational aspects and not on the representation details, much of prior work assumes that the valuations are provided via oracles that can only answer particular type of queries. The most basic oracle considered in literature answers \\emph{value queries}: given a subset of the indivisible goods, the value oracle returns the value of this subset. In this value oracle model, the work of Vondr\\'{a}k~\\cite{vondrak2008optimal} considers submodular valuations and provides an efficient $\\frac{e}{e-1}$-approximation algorithm for maximizing social welfare. Using this method as a subroutine and, hence, completely in the value oracle model, our algorithm achieves the above-mentioned approximation guarantee for identical submodular valuations. \n\n\nAnother well-studied oracle addresses \\emph{demand queries}. Specifically, such an oracle, when queried with an assignment of prices $p_1, \\ldots, p_m \\in \\mathbb{R}$ to the $m$ goods, returns $\\max_{S \\subseteq [m]} \\left( v(S) - \\sum_{j \\in S} p_j \\right)$, for the underlying valuation function $v$.\\footnote{Observe that a value query can be simulated via polynomially many demand queries. Though, the converse is not true~\\cite{nisan2007algorithmic}.} Demand oracles have been often utilized in prior work for addressing social welfare maximization in the context of subadditive and XOS valuations~\\cite{nisan2007algorithmic}. In particular, the work of Fiege~\\cite{feige2009maximizing} shows that, under subadditive valuations and assuming oracle access to {demand queries},\\footnote{This result holds even if the agents have distinct, but subadditive, valuations.} the social welfare maximization problem admits an efficient $2$-approximation algorithm. Demand queries are unavoidable in the subadditive case: one can directly extend the result of Dobzinski et al.~\\cite{DobzinskiNS10} to show that, even under identical (subadditive) valuations, any sub-linear (in $n$) approximation of the optimal social welfare requires exponentially many value queries. At the same time, we note that our algorithm requires demand oracle access \\emph{only} to implement the $2$-approximation algorithm of Fiege~\\cite{feige2009maximizing} as a subroutine. Beyond this, we can work with the value oracle. \\\\\n\n\n\n\\noindent\n{\\bf Related Work:} Multiple algorithmic and hardness results have been developed to address welfare maximization in the context of indivisible goods\/discrete resources. Though, in contrast to the present paper, prior work in this direction has primarily addressed one welfare function at a time. \n\nAs mentioned previously, maximizing social welfare and Nash social welfare (see, e.g., \\cite{cole2018approximating} and references therein) has been actively studied in algorithmic game theory. Egalitarian welfare has also been addressed in prior work--this welfare maximization problem is also referred to as the max-min allocation problem (or the Santa Claus problem); see, e.g.,~\\cite{DBLP:journals\/corr\/AnnamalaiKS14}. Specifically, for maximizing egalitarian welfare under additive and nonidentical valuations, the result of Chakrabarty et al.~\\cite{chakrabarty2009allocating} provides an $\\widetilde{\\mathcal{O}}(n^{\\varepsilon})$-approximation algorithm that runs in time $\\mathcal{O}(n^\\frac{1}{\\varepsilon})$; here $n$ denotes the number of agents and $\\varepsilon >0$. Furthermore, under nonidentical submodular valuations, the problem of maximizing egalitarian welfare is known to admit a polynomial-time $\\widetilde{\\mathcal{O}}(n^{1\/4} m^{1\/2})$-approximation algorithm~\\cite{goemans2009approximating}; here $m$ is the number of goods. In contrast to these sublinear approximations, this paper shows that, if the agents' valuations are identical, then even under subadditive valuations the problem of maximizing egalitarian welfare admits a constant-factor approximation guarantee. \n\n\n\n\n\\section{Notation and Preliminaries}\n\nAn instance of a fair-division problem corresponds to a tuple $\\langle [m], [n], v \\rangle$, where $[m]= \\left\\{1,2,\\ldots, m \\right\\}$ denotes the set of $m \\in \\mathbb{N}$ indivisible {goods} that have to be allocated (partitioned) among the set of $n \\in \\mathbb{N}$ agents, $[n]=\\{1, 2, \\ldots, n\\}$. \nHere, $v: 2^{[m]} \\mapsto \\mathbb{R}_+$ represents the (identical) valuation function of the agents;\\footnote{Recall that this work addresses fair-division instances in which all the agents have a common valuation function.} specifically, $v(S) \\in \\mathbb{R}_+$ is the value that each agent $i \\in [n]$ has for a subset of goods $S \\subseteq [m]$. \n\nWe will assume throughout that the valuation function $v$ is (i) normalized: $v(\\emptyset) = 0$, (ii) monotone: $v(A) \\leq v(B)$ for all $A \\subseteq B \\subseteq [m]$, and (iii) {subadditive}: $v(A \\cup B) \\leq v(A) + v(B)$ for all subsets $A, B \\subseteq [m]$. \n\n\nWrite $\\Pi_n([m])$ to denote the collection of all $n$ partitions of the indivisible goods $[m]$. We use the term \\textit{allocation} to refer to an $n$-partition $\\mathcal{A} = \\allo{A}{}{1} \\in \\Pi_n([m])$ of the $m$ goods. Here, $A_i$ denotes the subset of goods allocated to agent $i \\in [n]$ and will be referred to as a \\emph{bundle}.\n\n\nGeneralized (H\\\"{o}lder) means, ${\\rm M}_p$, constitute a family of functions that capture multiple fairness and efficiency measures. Formally, for an exponent parameter $p \\in \\mathbb{R}$, the $p${th} generalized mean of $n$ nonnegative numbers $x_1,\\ldots , x_n \\in \\mathbb{R}_+$ is defined as $\\Mp{x}{1} \\coloneqq \\left( \\frac{1}{n} \\sum \\limits _{i=1}^n x_i^p \\right )^\\frac{1}{p}$.\n\nNote that, when $p=1$, ${\\rm M}_p$ reduces to the arithmetic mean. Also, as $p$ tends to zero, ${\\rm M}_p$, in the limit, is equal to the geometric mean and $\\lim_{p \\rightarrow -\\infty} \\Mp{x}{1} = \\min\\{x_1, x_2, \\ldots, x_n\\}$. Hence, following standard convention, we will write ${\\rm M}_0(x_1, \\ldots, x_n) = \\left(\\prod_{i=1}^n x_i \\right)^{1\/n}$ and ${\\rm M}_{-\\infty}(x_1, \\ldots, x_n) = \\min_i x_i $.\n\nConsidering generalized means as a parameterized collection of welfare objectives, we define the \\emph{$p$-mean welfare}, ${\\rm M}_p(\\mathcal{A})$, of an allocation $\\mathcal{A}=(A_1, A_2, \\ldots, A_n)$ as \\begin{align}\n{\\rm M}_p(\\mathcal{A}) & \\coloneqq {\\rm M}_p\\left( v(A_1), \\ldots, v(A_n) \\right) = \\left( \\frac{1}{n} \\sum_{i=1}^n v (A_i)^p \\right)^{1\/p} \\label{eq:generalized-mean}\n\\end{align}\nHere, $v$ is the (common) valuation function of the agents. Indeed, with $p$ equal to one, zero, and $-\\infty$, the $p$-mean welfare, respectively, corresponds to (average) social welfare, Nash social welfare, and egalitarian welfare. \n\n\nGiven a fair-division instance $\\mathcal{I}=\\langle [m], [n], v \\rangle$ and $p \\in (-\\infty, 1]$, ideally, we would like to find an allocation $\\mathcal{A} = (A_1, \\ldots, A_n)$ with as large an ${\\rm M}_p(\\mathcal{A})$ value as possible, i.e., maximize the $p$-mean welfare. An allocation that achieves this goal will be referred to as a \\emph{$p$-optimal allocation} and denoted by $\\mathcal{A}^*(\\mathcal{I}, p)=(A^*_1(\\mathcal{I}, p), A^*_2(\\mathcal{I}, p), \\ldots, A^*_n(\\mathcal{I}, p))$. \n\n\n\nWe note that, under identical, subadditive valuations, finding a $p$-optimal allocation is {\\rm APX}-hard, for any $p \\in (-\\infty, 1]$ (Appendix~\\ref{APX_Hardness}). Hence, the current work considers approximation guarantees. In particular, for fair-division instances $\\mathcal{I}$ in which the agents have a common subadditive valuation, we develop a polynomial-time algorithm that computes a single allocation $\\mathcal{A}$ with the property that ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p({\\mathcal{A}^{*}}(\\mathcal{I}, p))$ for all $p \\in (-\\infty, 1]$. That is, the developed algorithm achieves an approximation ratio of $40$ uniformly for all $p \\in (-\\infty, 1]$. \n\n\nThe work of Fiege~\\cite{feige2009maximizing} shows that, for subadditive valuations, the social-welfare maximization problem (equivalently, the problem of maximizing ${\\rm M}_1(\\cdot)$) admits an efficient $2$-approximation algorithm, assuming oracle access to {demand queries}. In particular, such an oracle, when queried with an assignment of prices $p_1, \\ldots, p_m \\in \\mathbb{R}$ to the $m$ goods, returns $\\max_{S \\subseteq [m]} \\left( v(S) - \\sum_{j \\in S} p_j \\right)$. Our algorithm requires demand oracle access {only} to implement the $2$-approximation algorithm of Fiege~\\cite{feige2009maximizing} as a subroutine. Beyond this, we can work with the basic value oracle, which when queried with a subset of goods $S \\subseteq [m]$, returns $v(S)$.\n\nIn fact, if the underlying valuation is submodular, then one can invoke the result of Vondr\\'{a}k~\\cite{vondrak2008optimal} (instead of using the approximation algorithm by Feige~\\cite{feige2009maximizing}) and efficiently obtain a $\\frac{e}{e-1}$-approximation for the social-welfare maximization problem in the value oracle model. Hence, under a submodular valuation, our algorithm can be implemented entirely in the standard value oracle model. \n\nFor a fair-division instance $\\mathcal{I}$, write ${\\rm F}(\\mathcal{I})$ to denote the $1$-mean welfare ${\\rm M}_1$ (i.e., the average social welfare) of the allocation computed by the approximation algorithm of Feige~\\cite{feige2009maximizing}. The approximation guarantee established in~\\cite{feige2009maximizing} ensures that---for any instance $\\mathcal{I}$ with a subadditive valuation---we have ${\\rm F}(\\mathcal{I}) \\geq \\frac{1}{2} {\\textrm M}_1(\\mathcal{A}^{*}(\\mathcal{I}, 1))$. Here, $\\mathcal{A}^{*}(\\mathcal{I}, 1)$ denotes a $1$-optimal allocation, i.e., it maximizes the (average) social welfare in $\\mathcal{I}$. \n\n\\section{Proof of Theorem \\ref{MainTheorem} for $p \\in [0.4,1]$} \\label{subsection:p-half-one}\n\n\n\nFor instance $\\mathcal{I}$, let \\textsc{Alg} assign the $k$ highest-valued goods as singletons in its while-loop. Specifically, write $\\widehat{G}=\\left\\{g_1,\\ldots, g_k\\right\\}$ to denote the $k$ goods that are assigned in {while}-loop of \\textsc{Alg}. Instance $\\mathcal{J} = \\langle [m] \\setminus \\{g_1, \\ldots, g_k\\}, [n] \\setminus [k], v \\rangle$ is passed as input to \\textsc{AlgLow}, which returns allocation $\\mathcal{B}=(B_{k+1},\\ldots,B_n)$. Recall that $\\mathcal{B}$ satisfies Lemma~\\ref{Low_valued}. Finally, let $\\mathcal{A}=(\\{g_1\\},\\ldots,\\{g_k\\},B_{k+1},\\ldots , B_{n})$ denote the allocation returned by \\textsc{Alg}.\nAlso, as before, let $\\mathcal{A}^*(\\mathcal{I},p) = (A^*_1(\\mathcal{I},p),\\ldots, A^*_n(\\mathcal{I},p))$ denote the $p$-optimal allocation of $\\mathcal{I}$. \n\n \nWe will prove the following bound for $p \\in [0.4,1]$ and, hence, establish the stated approximation guarantee \n\\begin{align}\n\\left( \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p + \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p \\right)^{\\frac{1}{p}} & \\geq \\frac{1}{40} \\left(\\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p))^p \\right)^{\\frac{1}{p}}\n\\end{align}\n\nWrite $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ to denote the allocation (specifically, an $n$-partition) obtained by removing the goods $\\widehat{G} =\\{g_1,\\ldots,g_k\\}$ from the bundles in $\\mathcal{A}^*(\\mathcal{I},p)$, i.e., $ \\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G} \\coloneqq \\left( A^*_i(\\mathcal{I},p) \\setminus \\widehat{G} \\right)_{i=1}^n$.\nSubadditivity of $v$ ensures that, for all $i \\in [n]$, the bundle $A^{*}_i(\\mathcal{I},p)$ satisfies \n\\begin{align*}\nv(A^{*}_i(\\mathcal{I},p)) & \\leq v \\left( A^{*}_i (\\mathcal{I},p) \\setminus \\widehat{G}\\right) + \\sum \\limits_{ g \\in \\widehat{G} \\cap A^{*}_i(\\mathcal{I},p)} v(g). \n\\end{align*}\n\nSince $p>0$, exponentiating the previous inequality by $p$ gives us \n\\begin{align}\n \\left(v(A^{*}_i(\\mathcal{I},p)) \\right)^p & \\leq \\left(v \\left( A^{*}_i (\\mathcal{I},p) \\setminus \\widehat{G}\\right) + \\sum \\limits_{ g \\in \\widehat{G} \\cap A^{*}_i(\\mathcal{I},p)} v(g) \\right)^p \\nonumber \\\\\n & \\leq \\left(v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G}) \\right)^p + \\sum \\limits_{g\\in \\widehat{G}\\cap A^{*}_i(\\mathcal{I},p)} v(g)^p \\label{ineq:p-exp}\n\\end{align}\n\nThe last inequality follows from the fact that $(x+y)^p \\le x^p+y^p$, for all $p \\in [0.4, 1]$ and $x, y \\in \\mathbb{R}_+$.\n\nAveraging equation (\\ref{ineq:p-exp}) over $i \\in [n]$ leads to \n \\begin{align}\\label{17}\n \\frac{1}{n} \\summi{1}{k}{g}^p + \\frac{1}{n}\\sum\\limits_{i=1}^{n}{v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G}))^p} \n \\geq \\frac{1}{n}\\sum\\limits_{i=1}^{n}{v(A^{*}_i(\\mathcal{I},p))^p}\n \\end{align}\n\nTo show that the $k$ goods in $\\widehat{G}$ (which are allocated as singletons) substantially contribute towards $p$-mean welfare of the computed allocation $\\mathcal{A}$, we will next establish the following lower bound for all $1\\leq t \\leq k$\n\\begin{align*}\nv(g_t)^p & \\geq \\frac{1}{(7.06)^p}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) ^p \\right)\n\\end{align*}\n\nRecall that $\\mathcal{I}^t$ denotes the fair-division instance $\\I{[m]\\setminus \\{ g_1,\\ldots, g_t\\}}{[n]\\setminus \\{1,\\ldots t\\}}$, for $1\\leq t\\leq k$. Write $\\mathcal{A}^*(\\mathcal{I}^t,p)=(A^*_{t+1} (\\mathcal{I}^t,p),\\ldots,A^*_{n}(\\mathcal{I}^t,p))$ to denote a $p$-optimal allocation of instance $\\mathcal{I}^t$.\n\nThe selection criterion of the while-loop in \\textsc{Alg} and the fact that Feige's algorithm achieves an approximation ratio of $2$ ensure \n\\begin{align}\n v(g_t) &\\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{7.06}\\left ( \\frac{1}{n-t+1} \\sum\\limits_{j=t}^{n}v(A^{*}_j(\\mathcal{I}^{t-1},1)) \\right) \\label{ineq:bound-g-t-low}\n\\end{align}\n\nIndex the $n$ bundles in allocation $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ in non-increasing order of value $ v(A^*_1(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq v(A^*_2(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq \\ldots \\geq v(A^*_n(\\mathcal{I},p)\\setminus \\widehat{G})$ and note that the arithmetic mean of the values of the first $n-t+1$ bundles is at least as large as the overall arithmetic mean\n\\begin{align}\n\\frac{1}{n-t+1}\\sum \\limits _{j=1}^{n-t+1}v(A^*_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq \\frac{1}{n}\\sum \\limits _{j=1}^{n}v(A^*_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\label{ineq:top-mean}\n\\end{align}\n\nGiven that allocation $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ constitutes an $n$-partition of the set of goods $[m] \\setminus \\widehat{G}$ and allocation $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 1) = (A^*_{t} (\\mathcal{I}^{t-1},1),\\ldots, A^*_{n}(\\mathcal{I}^{t-1},1))$ is an $(n-t+1)$-partition of $[m]\\setminus \\{g_1, \\ldots, g_{t-1}\\} \\supseteq [m] \\setminus \\widehat{G}$, we have the following containment $\\bigcup \\limits _{j=1}^{n-t+1}\\left(A_j^*(\\mathcal{I},p)\\setminus \\widehat{G}\\right ) \\subseteq \\bigcup \\limits _{j=t}^{n} \\left(\\ A^*_j(\\mathcal{I}^{t-1},1)\\right)$. Furthermore, by definition, allocation $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 1)$ achieves the maximum possible average social welfare among all $(n-t+1)$ partitions of $[m] \\setminus \\{g_1, \\ldots, g_t\\}$. Therefore, we have \n\\begin{align}\n \\frac{1}{n-t+1}\\sum\\limits_{j=t}^{n} v(A^{*}_j(\\mathcal{I}^{t-1},1)) & \\geq \\frac{1}{n-t+1}\\sum\\limits_{j=1}^{n-t+1} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\label{ineq:opt-sub-problem}\n\\end{align}\n\nEquations (\\ref{ineq:bound-g-t-low}) and (\\ref{ineq:opt-sub-problem}) lead to \n \\begin{align*}\n v(g_t) & \\geq \\frac{1}{7.06}\\left( \\frac{1}{n-t+1}\\sum\\limits_{j=1}^{n-t+1} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\right)\\\\\n &\\geq \\frac{1}{7.06}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\right) \\tag{using inequality (\\ref{ineq:top-mean})} \\\\ \n &\\geq \\frac{1}{7.06}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right) ^{\\frac{1}{p}} \\tag{via the generalized mean inequality}\n\\end{align*}\nExponentiating both sides of the previous inequality by $p$ gives us the desired lower bound\n\\begin{align}\\label{19}\n v(g_t)^p \\ge \\frac{1}{(7.06)^p}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) ^p \\right)\n \\end{align} \n \nEquation (\\ref{19}) enables us to bound the $p$-welfare contribution of the goods $\\widehat{G}=\\{g_1, \\ldots, g_k\\}$ assigned as singletons \n\\begin{align}\\label{20}\n \\frac{2}{n} \\sum\\limits_{i=1}^kv(g_i)^p \\geq \\frac{1}{n} \\sum\\limits_{i=1}^kv(g_i)^p\n + \\frac{k}{n} \\ \\frac{1}{(7.06)^p} \\left(\\frac{1}{n} \\sum \\limits_{i=1}^n v ( A^{*}_i (\\mathcal{I},p)\\setminus \\widehat{G})^p\\right)\n\\end{align}\n\nRecall that \\textsc{AlgLow}---with input instance $\\mathcal{J}=\\mathcal{I}^k$---returns allocation $\\mathcal{B}=(B_{k+1}, \\ldots B_n)$. Next we lower bound the values of these bundles $B_{k+1}, \\ldots, B_n$. \n\\begin{align*}\nv(B_j) & \\geq \\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1)) \\tag{via Lemma~\\ref{Low_valued}} \\\\\n& = \\frac{1}{40 }\\left( \\frac{1}{n-k} \\sum\\limits_{i=k+1}^{n} v(A^{*}_i(\\mathcal{J},1)) \\right) \\tag{by defintion, $\\mathcal{A}^{*}(\\mathcal{J},1) = \\left( \\ A^{*}_i(\\mathcal{J},1) \\ \\right)_{i=k+1}^n $} \\\\\n& \\geq \\frac{1}{40 }\\left( \\frac{1}{n-k}\\sum\\limits_{j=1}^{n-k} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\right)\\tag{instantiating inequality (\\ref{ineq:opt-sub-problem}) for $\\mathcal{J} = \\mathcal{I}^k$} \\\\\n&\\geq \\frac{1}{40}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\right) \\tag{using inequality (\\ref{ineq:top-mean})} \\\\ \n &\\geq \\frac{1}{40}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right) ^{\\frac{1}{p}} \\tag{via the generalized mean inequality}\n\\end{align*}\n\n\n\n\nExponentiating by $p$ and summing over all $j \\in \\{k+1,\\ldots n\\}$, we have\n\\begin{align} \\label{21}\n \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p \\ge \\frac{n-k}{n} \\frac{1}{(40) ^p}\\left(\\frac{1}{n}\n \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right)\n\\end{align}\n\nCombining inequalities (\\ref{20}) and (\\ref{21}) gives us\n\\begin{align*}\n \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p + \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p & \\geq\n \\frac{1}{2n}\\ \\sum\\limits_{i=1}^k v(g_i)^p +\n \\frac{k}{2n} \\cdot \\frac{1}{(7.06)^p} \\left (\\frac{1}{n}\\sum\\limits_{j=1}^{n}v ( A^{*}_i (\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \\\\ & \\ \\ \\ \\ \\ +\n \\frac{n-k}{n}\\frac{1}{(40) ^p}\\left(\\frac{1}{n}\\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \n \\end{align*}\n\nNote that $2 \\times (7.06)^p \\leq (40) ^p$ for all $p \\in [0.4,1]$, hence, the previous inequality simplifies to \n \\begin{align*}\n \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p + \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p & \\geq\n \\frac{1}{2n}\\sum\\limits_{i=1}^{k}v(g_i)^p +\n \\frac{1}{(40) ^p}\\left(\\frac{1}{n}\\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \\\\\n & \\geq \\frac{1}{(40) ^p} \\left(\\frac{1}{n} \\sum\\limits_{i=1}^{k}v(g_i)^p +\n \\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right) \\tag{since $(40)^p > 2$ for $p \\in [0.4,1]$} \\\\\n & \\geq \\frac{1}{(40) ^p}\\left(\\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p))^p \\right) \\tag{using inequality (\\ref{17})}\n \\end{align*}\n\nTaking the $p${th} root (with $p>0$) on both sides of the last inequality gives us the desired result for the computed allocation $\\mathcal{A} = (\\{g_1\\},\\ldots,\\{g_k\\},B_{k+1},\\ldots , B_{n})$\n\\begin{align*}\n{\\rm M}_p(\\mathcal{A})\\ge \\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)).\n\\end{align*}\n\nThis completes the proof of Theorem \\ref{MainTheorem} for all $p \\in (-\\infty,1]$.\n\n\\section{Proof of Theorem \\ref{MainTheorem} for $p \\in (-\\infty,0.4)$}\n\\label{subsection:p-inf-half}\n\n\nAs before, we will write $\\mathcal{I}=\\I{[m]}{[n]}$ to denote the given fair-division instance. Also, write $\\mathcal{J} = \\mathcal{I}^k = \\I{[m]\\setminus \\{g_1,\\ldots,g_k\\}}{[n]\\setminus [k]}$ to denote the instance obtained at the termination of the while loop in \\textsc{Alg}. That is, instance $\\mathcal{J}$ is obtained after allocating the $k$ highest-valued goods as singletons to different agents and $\\mathcal{J}$ is passed on as an input to \\textsc{AlgLow}. \n\nAlso recall that $\\mathcal{A}^*(\\mathcal{I},p)=(A^*_1(\\mathcal{I},p),\\ldots, A^*_n(\\mathcal{I},p))$ and $\\mathcal{A}^*(\\mathcal{J}, p)= \\left( A^*_{k+1}(\\mathcal{J},p),\\ldots, A^*_n(\\mathcal{J},p) \\right)$ denote the $p$-mean optimal allocations of instances $\\mathcal{I}$ and $\\mathcal{J}$, respectively.\n\nSo far, we have established two results \\\\\n\\noindent\n(i) Lemma~\\ref{Induction_argument}: The allocation $(\\{g_1\\},\\ldots ,\\{g_k\\}, A^*_{k+1}(\\mathcal{J},p), \\ldots, A^*_{n}(\\mathcal{J},p) )$ achieves welfare comparable to the optimal $p$-mean welfare (i.e., comparable to ${\\rm M}_p \\left(\\mathcal{A}^*(\\mathcal{I},p) \\right)$), for $p \\in (-\\infty, 0.4)$. \\\\\n\\noindent\n(ii) Lemma~\\ref{Low_valued}: For the instance $\\mathcal{J}$ and any $p\\in (-\\infty, 0.4]$, \\textsc{AlgLow} computes an allocation $\\mathcal{B}= (B_{k+1},\\ldots,B_{n})$ such that, for all $j \\in \\{k+1, \\ldots, n\\}$,\\footnote{For notational convenience, we index the bundles in allocation $\\mathcal{B}$ from $k+1$ to $n$.}\n\\begin{align}\\label{16}\nv(B_j)\\geq\\frac{1}{40}{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))\\geq\\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p)) \n\\end{align}\n\nThe allocation returned by \\textsc{Alg} for input instance $\\mathcal{I}$ is $\\mathcal{A}=(\\{g_1\\},\\ldots,\\{g_k\\}, B_{k+1}, \\ldots ,B_n)$. We will prove Theorem~\\ref{MainTheorem}, for $p \\in (-\\infty,0.4)$, by showing that Lemma~\\ref{Induction_argument} and Lemma~\\ref{Low_valued}, in conjunction, imply that the $p$-mean welfare of $\\mathcal{A}$ is a constant times the optimal $p$-mean optimal; specifically, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, for $p\\in(-\\infty,0.4)$.\n\nWe split the proof of this inequality into three parts, depending on the range of the exponent parameter $p$.\n\n\\noindent\n\\textbf{Case 1:} $p \\in (-\\infty, 0)$. Since in this case $p$ is negative, to obtain the desired inequality, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, it suffices to show that\n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(B_j\\right)^p\\leq \\frac{1}{(40) ^p}\\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align*}\n\nExponentiating both sides of equation (\\ref{16}) by $p <0$ and summing over $j \\in \\{k+1, \\ldots, n\\}$ lead to \n\\begin{align}\n\\sum \\limits _{j=k+1}^nv(B_j)^p\\leq\\frac{1}{(40) ^p}\\sum \\limits _{j=k+1}^{n}v(A^{*}_j(\\mathcal{J},p))^p \n\\end{align}\n\nWe add $\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p$ to both sides of the previous equation and apply Lemma~\\ref{Induction_argument} (with $p \\in (-\\infty, 0)$) to obtain the desired inequality \n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits _{j=k+1}^n v(B_i)^p \\leq \\frac{1}{\\left( 40 \\right)^p} \\left( (40)^p\\sum \\limits _{i=1}^k v(g_i)^p + \\sum \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},p))^p \\right)\n\\leq \\frac{1}{40 ^p}\\left( \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\\right)\\hspace{3.6cm}\n\\end{align*}\n\n\n\\noindent {\\bf Case 2:} $p \\in (0,0.4)$.\nNote that in this case $p>0$. Hence, to obtain the desired inequality, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, it suffices to show that\n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(B_j\\right)^p\\geq \\frac{1}{(40) ^p}\\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align*}\n\nExponentiating both sides of equation (\\ref{16}) by $p >0$ and summing over $j \\in \\{k+1, \\ldots, n\\}$ lead to \n\n\\begin{align*}\n\\sum \\limits _{j=k+1}^nv(B_i)^p & \\geq \\frac{1}{(40) ^p}\\sum \\limits _{j=k+1}^{n}v(A^{*}_j(\\mathcal{J},p))^p.\n\\end{align*}\n\nWe add $\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p$ to both sides of the previous equation and apply Lemma~\\ref{Induction_argument} (with $p \\in (0,0.4)$) to obtain the desired inequality \n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits _{j=k+1}^n v(B_j)^p &\\geq \\frac{1}{\\left( 40\\right)^p} \\left( (40)^p \\sum \\limits _{i=1}^k v(g_i)^p + \\sum \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},p))^p \\right)\n\\geq \\frac{1}{(40) ^p}\\left( \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\\right)\\hspace{3.6cm}\n\\end{align*}\n\n\n\n\n\\noindent {\\bf Case 3:} $p=0$. In this case, ${\\rm M}_0(\\mathcal{A}) = \\left( \\prod \\limits _{i=1}^kv(g_i) \\prod \\limits _{j=k+1}^{n} v(B_j)\\right)^\\frac{1}{n}.$\nMultiplying inequality (\\ref{16}) over all $j \\in \\{k+1,\\ldots, n\\}$ gives us \n\\begin{align*}\n\\prod \\limits _{j=k+1}^{n} v(B_i) \\geq \\frac{1}{(40)^{n-k}} \\left( {\\rm M}_0 \\left(\\mathcal{A}^{*}(\\mathcal{J}, 0) \\right) \\right)^{n-k} & = \\frac{1}{(40)^{n-k}}\\prod \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},0))\n\\end{align*}\n\nNext we multiply both sides of this inequality by $\\prod_{i=1}^k v\\left(g_i\\right)$ and obtain \n\\begin{align*} \n\\prod \\limits _{i=1}^k v(g_i) \\prod \\limits _{j=k+1}^n v(B_i)\\geq \\frac{1}{(40) ^{n-k}}\\prod \\limits_{i=1}^k v(g_i) &\\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{J},0) ) \\geq \\frac{1}{(40) ^n}({\\rm M}_0(\\mathcal{A}^{*}(\\mathcal{I}, 0)))^n \\tag{via Lemma~\\ref{Induction_argument} with $p=0$}\n\\end{align*}\nTaking the $n${th} root on both sides, we obtain the desired result ${\\rm M}_0(\\mathcal{A}) \\geq \\frac{1}{40} {\\rm M}_0(\\mathcal{A}^{*}(\\mathcal{I},0))$. \\\\\n\n\nTheorem \\ref{MainTheorem} now stands proved for $p \\in (-\\infty,0.4)$. \n\n\\section*{Acknowledgements}\nSiddharth Barman gratefully acknowledges the support of a Ramanujan Fellowship (SERB - {SB\/S2\/RJN-128\/2015}) and a Pratiksha Trust Young Investigator Award.\n\n\\bibliographystyle{alpha}\n\n\\section{Structural Lemma} \\label{Supporting_Lemmas}\nThe following lemma provides a structural property of $p$-optimal allocations $\\mathcal{A}^*(\\mathcal{I}, p)$, for $p \\in (-\\infty,0.4)$. It states that the only way allocation $\\mathcal{A}^*(\\mathcal{I}, p)$ has a bundle $A^*_i(\\mathcal{I}, p)$ of notably high value is through a single good $g \\in A^*_i(\\mathcal{I}, p)$ that by itself has high value. \n\n\n\\begin{Lemma}\\label{Good_Transfer}\nLet $\\mathcal{L}= \\I{[M]}{[N]}$ be a fair-division instance wherein all the $N \\in \\mathbb{N}$ agents have an identical, subadditive valuation $v$ over the set of $M\\in\\mathbb{N}$ goods. In addition, let $\\mathcal{A}^{*}(\\mathcal{L},p) = \\{ A^{*}_{1}(\\mathcal{L},p), \\cdots , A^{*}_{N}(\\mathcal{L},p)\\}$ be a $p$-mean optimal allocation in $\\mathcal{L}$, for any $p \\in \\left( -\\infty ,0.4 \\right)$. \n\nIf for any bundle $A^{*}_i(\\mathcal{L},p)$, with $i \\in [N]$, we have $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L})$, then there exists a good $g\\in A_i^*(\\mathcal{L},p)$ with the property that that $v(g)\\geq \\frac{1}{40} v(A_i^*(\\mathcal{L},p)).$\n\\end{Lemma}\n\nThe proof of the above lemma is divided into three parts (Sections~\\ref{subsection:p-infty-zero-good-transfer}, \\ref{subsection:p-zero-half-good-transfer}, and~\\ref{subsection:p-zero-good-transfer}) depending on the range of the exponent parameter $p.$ \n\\subsection{Proof of Lemma \\ref{Good_Transfer} for $p \\in (-\\infty,0)$}\n\\label{subsection:p-infty-zero-good-transfer}\n\nAssume, towards a contradiction, that $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L}) $, for some $i \\in [N] $, and $v(g) \\leq \\frac{1}{40} v(A^{*}_i(\\mathcal{L},p))$ for all $g \\in A^{*}_i(\\mathcal{L},p).$ Recall that the $1$-mean welfare of the allocation returned by Feige's algorithm satisfies ${\\rm F}(\\mathcal{L}) \\geq \\frac{1}{2}{\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1))$. Pick a bundle $A^{*}_j(\\mathcal{L},p)$ (in $\\mathcal{A}^*(\\mathcal{L}, p)$) with the property that $v(A^{*}_j(\\mathcal{L},p)) \\leq 2{\\rm F}(\\mathcal{L})$. Such a bundle exists, since $2{\\rm F}(\\mathcal{L})\\geq {\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1)) \\geq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{L},p))$. \n\nDefine a partition---$A'_i$ and $A'_j$---of $A^*_i$ as follows: \\\\\n\\noindent\n(i) Initialize $A'_j$ to be the empty set. Then, we keep transferring goods from $A_i^*(\\mathcal{L},p)$ to $A'_j$ (one at a time and in an arbitrary order) and stop as soon as the value of $A'_j$ exceeds $\\frac{v(A^*_i(\\mathcal{L},p))}{2}$.\\\\\n(ii) Denote the remaining set of goods as $A'_i \\coloneqq A^*_i(\\mathcal{L},p) \\setminus A'_j$. Note that, by construction, $v(A'_j)\\geq \\frac{v(A^{*}_i(\\mathcal{L},p))}{2}$ and \n$v(A'_i)\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right ) v(A^*_i(\\mathcal{L},p))$. The last inequality follows from the fact that $v$ is subadditive and the assumption that goods in $A^*_i(\\mathcal{L},p)$ are of value at most $\\frac{1}{40}v(A^*_i(\\mathcal{L},p))$. \\\\\n(iii) Write $\\mathcal{B}= \\{B_1, \\ldots, B_N\\}$ to denote the allocation obtained by replacing the bundles $A^{*}_i(\\mathcal{L},p)$ and $A^{*}_j(\\mathcal{L},p)$ in $\\mathcal{A}^*(\\mathcal{L},p)$ by $A'_i$ and $A^{*}_j(\\mathcal{L},p) \\cup A'_j$, respectively: $B_i = A'_i$ and $B_j = A^*_j(\\mathcal{L},p) \\cup A'_j$ along with $B_\\ell = A^*_\\ell (\\mathcal{L},p)$ for all $\\ell \\in [N] \\setminus \\{i, j \\}$. \n\nWe will show that $\\mathcal{B}$ has $p$-mean welfare strictly greater than that of the $p$-optimal allocation ${\\mathcal{A}^{*}}(\\mathcal{L},p)$. Hence, by way of contradiction, the desired result follows.\n\nRecall that the current case addresses exponent parameters that are negative, $p \\in (-\\infty, 0)$. Hence, the following inequality implies that the $p$-mean welfare of $\\mathcal{B}$ is strictly greater than that of $\\mathcal{A}^*(\\mathcal{L},p)$:\n\\begin{align}\nv(A'_i)^p+v(A'_j)^p< v(A^{*}_i(\\mathcal{L},p))^p+v(A^{*}_j(\\mathcal{L},p))^p \\label{ineq:desired-neg-p}\n\\end{align}\n\nHowever, for negative $p$, the lower bounds on the values of $A'_i$ and $A'_j$ gives us\n\\begin{align*}\nv(A'_i)^p+v(A'_j)^p\\leq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p v(A^{*}_i(\\mathcal{L},p))^p + \\left (\\frac{1}{2} \\right )^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nIn addition, using the bounds $v(A^*_j(\\mathcal{L},p)) \\leq 2 {\\rm F}( \\mathcal{L}) < \\frac{2}{11.33} v(A^*_i(\\mathcal{L},p))$, we get \n\\begin{align*}\nv(A^{*}_i(\\mathcal{L},p))^p+ v(A^{*}_j(\\mathcal{L},p))^p & > v(A^{*}_i(\\mathcal{L},p))^p+ \\left( \\frac{2}{11.33} \\right)^p v(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nTherefore, the desired equation (\\ref{ineq:desired-neg-p}) follows from the following numeric inequality, which is established in Appendix \\ref{app 3}.\n\\begin{align*}\n \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p + \\left (\\frac{1}{2} \\right )^p &\\leq 1+ \\left( \\frac{2}{11.33} \\right)^p \\quad \\text{ for all } p\\in (-\\infty,0).\n\\end{align*}\nThis establishes Lemma \\ref{Good_Transfer} for $p \\in (-\\infty,0)$.\n\n\\subsection{Proof of Lemma \\ref{Good_Transfer} for $p \\in (0, 0.4)$}\n\\label{subsection:p-zero-half-good-transfer}\nAssume, towards a contradiction, that $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L}) $, for some $i \\in [N] $, and $v(g) \\leq \\frac{1}{40} v(A^{*}_i(\\mathcal{L},p))$ for all $g \\in A^{*}_i(\\mathcal{L},p).$ Recall that the $1$-mean welfare of the allocation returned by Feige's algorithm satisfies ${\\rm F}(\\mathcal{L}) \\geq \\frac{1}{2}{\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1))$. Pick a bundle $A^{*}_j(\\mathcal{L},p)$ (in $\\mathcal{A}^*(\\mathcal{L}, p)$) with the property that $v(A^{*}_j(\\mathcal{L},p)) \\leq 2{\\rm F}(\\mathcal{L})$. Such a bundle exists, since $2{\\rm F}(\\mathcal{L})\\geq {\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1)) \\geq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{L},p))$. \n\nDefine a partition---$A'_i$ and $A'_j$---of $A^*_i(\\mathcal{L},p)$ as follows: \\\\\n\\noindent\n(i) Initialize $A'_j$ to be the empty set. Then, we keep transferring goods from $A_i^*(\\mathcal{L},p)$ to $A'_j$ (one at a time and in an arbitrary order) and stop as soon as the value of $A'_j$ exceeds $\\frac{v(A^*_i(\\mathcal{L},p))}{2}$.\\\\\n(ii) Denote the remaining set of goods as $A'_i \\coloneqq A^*_i(\\mathcal{L},p) \\setminus A'_j$. Note that, by construction, $v(A'_j)\\geq \\frac{v(A^{*}_i(\\mathcal{L},p))}{2}$ and \n$v(A'_i)\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right ) v(A^*_i(\\mathcal{L},p))$. The last inequality follows from the fact that $v$ is subadditive and the assumption that goods in $A^*_i(\\mathcal{L},p)$ are of value at most $\\frac{1}{40}v(A^*_i(\\mathcal{L},p))$. \\\\\n(iii) Write $\\mathcal{B}= \\{B_1, \\ldots, B_N\\}$ to denote the allocation obtained by replacing the bundles $A^{*}_i(\\mathcal{L},p)$ and $A^{*}_j(\\mathcal{L},p)$ in $\\mathcal{A}^*(\\mathcal{L},p)$ by $A'_i$ and $A^{*}_j(\\mathcal{L},p) \\cup A'_j$, respectively: $B_i = A'_i$ and $B_j = A^*_j(\\mathcal{L},p) \\cup A'_j$ along with $B_\\ell = A^*_\\ell (\\mathcal{L},p)$ for all $\\ell \\in [N] \\setminus \\{i, j \\}$. \n\nWe will show that $\\mathcal{B}$ has $p$-mean welfare strictly greater than that of the $p$-optimal allocation ${\\mathcal{A}^{*}}(\\mathcal{L},p)$. Hence, by way of contradiction, the desired result follows.\n\nNotice that the construction of $A'_i$, $A'_j$ and $\\mathcal{B}$ hold for $p=0$ as well. We will use these sets in the next section to prove an analogous result for Nash social welfare. \n\nThe current case addresses exponent parameters that are positive $p \\in (0, 0.4)$. Hence, the following inequality implies that the $p$-mean welfare of $\\mathcal{B}$ is strictly greater than that of $\\mathcal{A}^*(\\mathcal{L},p)$\n\\begin{align}\nv(A'_i)^p+v(A'_j)^p & > v(A^{*}_i(\\mathcal{L},p))^p+v(A^{*}_j(\\mathcal{L},p))^p \\label{ineq:desired-pos-p}\n\\end{align}\n\nHowever, for positive $p$, the lower bounds on the values of $A'_i$ and $A'_j$ gives us\n\\begin{align*}\nv(A'_i)^p+v(A'_j)^p & \\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p v(A^{*}_i(\\mathcal{L},p))^p + \\left (\\frac{1}{2} \\right )^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nIn addition, using the bounds $v(A^*_j(\\mathcal{L},p)) \\leq 2 {\\rm F}( \\mathcal{L}) < \\frac{2}{11.33} v(A^*_i(\\mathcal{L},p))$, we get \n\\begin{align*}\nv(A^{*}_i(\\mathcal{L},p))^p+ v(A^{*}_j(\\mathcal{L},p))^p & < v(A^{*}_i(\\mathcal{L},p))^p + \\left( \\frac{2}{11.33} \\right)^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nTherefore, the desired equation (\\ref{ineq:desired-pos-p}) follows from the following numeric inequality, which is established in Appendix \\ref{app 3}.\n\\begin{align*}\n \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p + \\left (\\frac{1}{2} \\right )^p &\\geq 1+ \\left( \\frac{2}{11.33} \\right)^p \\hbox{ for } p\\in (0,0.4).\n\\end{align*}\n\nThis establishes Lemma \\ref{Good_Transfer} for $p \\in (0, 0.4)$.\n\n\n\\subsection{Proof of Lemma \\ref{Good_Transfer} for Nash Social Welfare ($p=0$)}\n\\label{subsection:p-zero-good-transfer}\n\n\nRecall the sets $A'_i$, $A'_j$, and the allocation $\\mathcal{B}$ defined in Section \\ref{subsection:p-zero-half-good-transfer}. We will show that $\\mathcal{B}$ has Nash welfare strictly greater than that of ${\\mathcal{A}^{*}}(\\mathcal{L}, 0)$, and hence, by way of contradiction, establish the desired result.\n\n In order to obtain ${\\rm M}_0(\\mathcal{B})>{\\rm M}_0(\\mathcal{A}^*(\\mathcal{L},0))$, it suffices to prove that $v(A'_i)v(A'_j)>v\\left( A^{*}_i(\\mathcal{L},0)\\right)v( A^{*}_j(\\mathcal{L},0)).$\nThe lower bounds we obtained on the values of $A'_i$ and $A'_j$ gives us \n\\begin{align*}\nv(A'_i)v(A'_j)&\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )v\\left( A^{*}_i (\\mathcal{L},0)\\right) \\frac{v\\left( A^{*}_i (\\mathcal{L},0)\\right)}{2}\n\\geq 0.2 \\ v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2 \n\\end{align*}\n\nIn addition, we have \n\\begin{align*}\nv\\left( A^{*}_i (\\mathcal{L},0)\\right) v\\left( A^{*}_j (\\mathcal{L},0)\\right) & <\\frac{2}{11.33} \\ v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2\n<0.18 \\ v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2 \n\\end{align*}\nTherefore, the lemma holds for $p=0$ as well. \n\n\n\n\\section{Combination Lemma} \\label{section:stitching-lemma}\nThe following lemma shows that the goods assigned as singletons in the while-loop of $\\textsc{Alg}$ (Algorithm~\\ref{Alg}), along with a $p$-optimal allocation of instance $\\mathcal{J}$ that remains at the termination of the loop, lead to a $p$-mean welfare that is comparable to the optimal, for all $p \\in (-\\infty, 0.4)$.\n\nAs shown previously in Lemma~\\ref{Low_valued}, $\\textsc{AlgLow}$---with instance $\\mathcal{J}$ as input---achieves a constant-factor approximation for the $p$-mean welfare objective. Hence, the lemma established in this section will enable us to combine the welfare guarantees of the goods assigned in the while-loop of \\textsc{Alg} and the allocation computed by \\textsc{AlgLow} to obtain the desired approximation result for $p \\in (-\\infty, 0.4)$. \n\nSpecifically, given a fair-division instance $\\mathcal{I} = \\langle [m], [n], v \\rangle$ as input, let $\\{g_1, \\ldots, g_k\\}$ denote the set of goods that get assigned as singletons in the while-loop of $\\textsc{Alg}$ (Algorithm~\\ref{Alg}). Furthermore, for $1 \\leq t \\leq k$, let $\\mathcal{I}^t$ denote the instance obtained at the end of the $t$th iteration of this while-loop. Since in the first $t$ iterations $\\textsc{Alg}$ assigns goods $\\{g_1, \\ldots, g_t\\}$ to the first $t$ agents as singletons, we have $\\mathcal{I}^t = \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$. In particular, $\\mathcal{J} \\coloneqq \\mathcal{I}^k$ is the instance that remains after the termination of the while-loop in $\\textsc{Alg}$ and this instance is passed on to $\\textsc{AlgLow}$ as input.\n\nInstance $\\mathcal{J}$ consists of $n-k$ agents and, hence, in $\\mathcal{J}$, any $p$-optimal allocation $\\mathcal{A}^*(\\mathcal{J},p)$ contains $(n-k)$ bundles. For notational convenience, we will index these bundles from $k+1$ to $n$, i.e., $\\mathcal{A}^*(\\mathcal{J},p)= \\left( A^*_{k+1} (\\mathcal{J},p), \\ldots, A^*_{n} (\\mathcal{J},p) \\right)$.\n\n\n\\begin{Lemma}\\label{Induction_argument}\nGiven a fair-division instance $\\mathcal{I} = \\langle [m], [n], v \\rangle$ with an identical subadditive valuation $v$, let $\\{g_1, \\ldots, g_k\\}$ denote the set of goods that get assigned as singletons in the while-loop of $\\textsc{Alg}$ and let $\\mathcal{J}=\\langle [m] \\setminus \\{g_1, \\ldots, g_k \\}, [n] \\setminus [k], v \\rangle$ be the instance that remains after the termination of this loop. In addition, let $\\mathcal{A}^*(\\mathcal{I},p)=\\left(A^*_1 (\\mathcal{I},p), \\ldots, A^*_n(\\mathcal{I},p) \\right)$ and $\\mathcal{A}^*(\\mathcal{J},p) = \\left( A^*_{k+1} (\\mathcal{J},p), \\ldots, A^*_{n} (\\mathcal{J},p) \\right)$ denote $p$-optimal allocations of instances $\\mathcal{I}$ and $\\mathcal{J}$, respectively. Then, with constant $\\alpha = 40$, \n\\begin{itemize}\n\\item For $p \\in (-\\infty, 0)$, we have $\\alpha^p \\sum \\limits_{i=1}^k v(g_i)^p \\ + \\ \\sum \\limits_{j=k+1}^{n} v (A^{*}_j(\\mathcal{J},p) )^p \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p ))^p$. \n\n\\item For $p \\in (0,0.4)$, we have $\\alpha^p \\sum \\limits_{i=1}^k v \\left( g_i\\right)^p \\ + \\ \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{J},p)\\right)^p \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p ))^p$.\n\\item For $p=0$, we have $\\alpha^k \\prod \\limits_{i=1}^k v(g_i) \\ \\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{J},p) ) \\geq \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},p))$. \n \n\\end{itemize}\n\\end{Lemma}\nWe will prove Lemma~\\ref{Induction_argument} by considering different ranges of the exponent parameter $p$ separately. However, in all of the ranges, the desired inequality is obtained by inducting on the number of iterations of the while-loop in \\textsc{Alg}. \n\n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for $p\\in(-\\infty,0)$}\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $p$-mean optimal, $\\mathcal{A}^*(\\mathcal{I}^t,p)=\\left( A^*_{t+1}(\\mathcal{I}^{t},p),\\ldots , A^*_n(\\mathcal{I}^t,p)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\\begin{align}\\label{7}\n(40) ^p\\sum \\limits_{i=1}^t v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^t,p)\\right)^p & \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align}\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{7}) are equal to each other. Therefore, the base case holds.\n\n\\noindent\n\\emph{Induction Step:} We establish inequality (\\ref{7}) for $t$, assuming that it holds for $t-1$. \n\n Consider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},p)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is a $p$-optimal allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p) =\\left( A^*_{t}(\\mathcal{I}^{t-1},p),\\ldots , A^*_n(\\mathcal{I}^{t-1},p) \\right)$. \n \n We lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},p)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},p)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},p))$; since, $3.53 \\times 11.33 \\leq 40$. \\\\\n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},p)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the highest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},p )) \\label{8}\n\\end{align}\nNote that inequality (\\ref{8}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. Furthermore, since the current case addresses negative $p \\in (-\\infty, 0)$, we have $ (40)^p \\ v(g_{t})^p\\leq v(A^*_{t}(\\mathcal{I}^{t-1},p ))^p$. \n\nWe add $(40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p $ to both sides of the previous inequality to obtain\n\n\\begin{align}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\leq (40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p +v(A^*_{t}(\\mathcal{I}^{t-1},p ))^p +\\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p \\label{ineq:interim}\n\\end{align}\n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},p )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1},p)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1},p)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t},p)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},p) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},p)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is the $p$-optimal allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align*}\n\\left(\\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p\\right)^\\frac{1}{p} & \\leq \\left( \\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j( \\mathcal{I}^{t},p)\\right)^p \\right)^\\frac{1}{p}.\n\\end{align*}\n\nExponentiating both sides by $p$ (which in the current case is negative) and multiplying by $n-t$, gives us $ \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\geq \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p .$ Therefore, inequality (\\ref{ineq:interim}) extends to\n\n\\begin{align*}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\leq 40 ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\\\\n& \\leq \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align*}\nThe last inequality follows from the induction hypothesis. Setting $t=k$ gives us the desired inequality for $p\\in(-\\infty,0)$\n\\begin{align}\\label{9}\n( 40 )^p\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^k,p)\\right)^p & \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align}\nTherefore, Lemma \\ref{Induction_argument} holds for $p\\in (-\\infty,0)$.\n\n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for $p\\in (0,0.4)$}\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $p$-mean optimal, $\\mathcal{A}^*(\\mathcal{I}^t,p)=\\left( A^*_{t+1}(\\mathcal{I}^{t},p),\\ldots , A^*_n(\\mathcal{I}^t,p)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\\begin{align} \n(40) ^p\\sum \\limits_{i=1}^t v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^t,p)\\right)^p & \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \\label{ineq:ind}\n\\end{align}\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{7}) are equal to each other. Therefore, the base case holds.\n\n\\noindent\n\\emph{Induction Step:} We establish inequality (\\ref{ineq:ind}) for $t$, assuming that it holds for $t-1$. \n\nConsider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},p)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is a $p$-optimal allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p) =\\left( A^*_{t}(\\mathcal{I}^{t-1},p),\\ldots , A^*_n(\\mathcal{I}^{t-1},p) \\right)$. \\\\\n\nWe lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},p)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},p)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},p))$; since, $3.53 \\times 11.33 \\leq 40$. \n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},p)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the highest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},p )) \\label{ineq:g-t-val}\n\\end{align}\n\nNote that inequality (\\ref{ineq:g-t-val}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. Furthermore, since the current case addresses positive $p\\in (0,0.4)$, we have $ (40)^p \\ v(g_{t})^p \\geq v(A^*_{t}(\\mathcal{I}^{t-1},p ))^p$. \n\nWe add $(40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p $ to both sides of the previous inequality to obtain\n\n\\begin{align}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\geq (40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p +v(A^*_{t}(\\mathcal{I}^{t-1},p ))^p +\\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p \\label{ineq:interim-positive}\n\\end{align}\n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},p )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1},p)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1},p)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t},p)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},p) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},p)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is the $p$-optimal allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align*}\n\\left(\\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p\\right)^\\frac{1}{p} & \\leq \\left( \\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j( \\mathcal{I}^{t},p)\\right)^p \\right)^\\frac{1}{p}.\n\\end{align*}\n\nExponentiating both sides by $p$ (which in the current case is positive) and multiplying by $n-t$, gives us $ \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\leq \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p .$ Therefore, via inequality (\\ref{ineq:interim-positive}), we obtain\n\\begin{align*}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\geq 40 ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\\\\n& \\geq \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align*}\nThe last inequality follows from the induction hypothesis. Setting $t=k$ gives us the desired inequality for $p \\in (0, 0.4)$\n\\begin{align} \n( 40 )^p\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^k,p)\\right)^p & \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align}\nTherefore, Lemma \\ref{Induction_argument} holds for $p \\in (0, 0.4)$. \n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for Nash Social Welfare ($p=0$)}\n\\label{subsection:p-zero}\n\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $0$-mean optimal (i.e., Nash optimal), $\\mathcal{A}^*(\\mathcal{I}^t, 0)=\\left( A^*_{t+1}(\\mathcal{I}^{t},0),\\ldots , A^*_n(\\mathcal{I}^t,0)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\n\\begin{align}\\label{13}\n \\prod \\limits_{i=1}^t v(g_i) \\ \\prod \\limits_{j=t+1}^{n} v(A^*_j (\\mathcal{I}^t,0) ) \\geq \\left( \\frac{1}{40}\\right)^t \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0)).\n\\end{align}\n\n\\noindent\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{13}) are equal to each other. Therefore, the base case holds. \\\\\n\n\n\\noindent\n\\emph{Induction Step:} We establish inequality (\\ref{13}) for $t$, assuming that it holds for $t-1$. \n\n Consider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},0)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0)$ is a $0$-optimal (Nash optimal) allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0) =\\left( A^*_{t}(\\mathcal{I}^{t-1},0),\\ldots , A^*_n(\\mathcal{I}^{t-1},0) \\right)$. \\\\\n \n We lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},0)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},0)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},0))$; since, $3.53 \\times 11.33 \\leq 40$. \n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},0)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the largest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},0 )) \\label{ineq:val-g-t-0}\n\\end{align}\n\n\nHere, inequality (\\ref{ineq:val-g-t-0}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. \n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},0)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},0 )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1}, 0)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 0)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t}, 0)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},0) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},0)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$ is the $0$-optimal (Nash optimal) allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align}\n\\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right ) > \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t-1},0)\\right ) \\label{ineq:prod-induction}\n\\end{align}\n\nMultiplying by $v(g_1) \\ v(g_2) \\ldots v(g_{t-1}) \\ v(g_{t})$ on both sides of the previous inequality and using equation (\\ref{ineq:val-g-t-0}), we get\n\\begin{align*}\nv(g_1) \\ v(g_2) \\ldots v(g_{t-1}) \\ v(g_{t}) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right) & \\geq \\frac{1}{40}v\\left(g_1\\right)\\ldots v\\left(g_{t-1}\\right) \\ v\\left(A^{*}_{t}(\\mathcal{I}^{t-1},0)\\right) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right) \\\\\n& \\geq \\frac{1}{40}v\\left(g_1\\right)\\ldots v\\left(g_{t-1}\\right) \\ v\\left(A^{*}_{t}(\\mathcal{I}^{t-1},0)\\right) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t-1},0)\\right) \\tag{via inequality (\\ref{ineq:prod-induction})}\\\\\n &\\geq \\left(\\frac{1}{40}\\right)^{t} \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0)) \\tag{using the induction hypothesis}\n\\end{align*}\n\n\nSetting $t=k$ gives us the desired inequality for $p =0$\n\\begin{align*}\n(40)^k \\prod \\limits_{i=1}^k v(g_i) \\ \\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{I}^k,0) ) \\geq \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0)).\n\\end{align*}\nTherefore, Lemma \\ref{Induction_argument} holds for $p =0$ as well. \\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn \\cite{FFT16}, a special class of group actions on CAT(0) cube complexes is defined by the behaviour of halfspaces under the action (see Definition \\ref{RAAGlike} in this paper). Since they generalise right-angled Artin groups (RAAGs) acting on the associated RAAG-complex (the universal cover of the associated Salvetti complex), they are called \\emph{RAAG-like} actions.\\par\nThe authors define quasimorphisms on all groups acting non-transversally \\footnote{A group acts \\emph{non-transversally}, if orbits of halfspaces are nested. In particular, RAAG-like actions are non-transverse.} on CAT(0) cube complexes, generalising Fujiwara--Epstein counting quasimorphisms on free groups (introduced in \\cite{EF97}). The quasimorphisms in \\cite{FFT16} are shown to have defect of at most $6$ and, hence, their homogenisation at most $12$. This is done using the median property of CAT(0) spaces.\\par\nThe authors then prove \\emph{effectiveness} of these quasimorphisms, that is, for every element $g$ acting hyperbolically on the CAT(0) cube complex, one of their homogeneous quasimorphisms $\\overline{\\phi}$ satisfies $\\overline{\\phi}(g)\\geq 1$. For this, using Haglunds combinatorial axis (\\cite{Hag07}), they isolate a subcomplex with respect to $g$ called `essential set' and construct a $g$-equivariant embedding of this into a Euclidean space. They then use a rather intricate series of technical lemmas to prove that $\\overline{\\phi}(g)\\geq 1$.\\par\nThe purpose of this paper is to show that these technical arguments can be avoided and replaced by a short proof that also uses the full properties of RAAG-like actions.\\par\nFinally, we conclude as in \\cite{FFT16} that hyperbolic elements of RAAG-like actions have stable commutator length at least $1\/24$ using the Bavard Duality. In particular, RAAGs have a stable commutator length gap of $1\/24$. Note that, using a different class of quasimorphisms, Heuer \\cite{Heu19} has already proved a better (and optimal) bound of $1\/2$ for the stable commutator length in RAAGs.\n\n\\section{Premliminaries}\n\\subsection{Halfspaces in CAT(0) cube complexes}\nLet $X$ be a CAT(0) cube complex. Denote by $\\mathcal{H}(X)$ the set of half-spaces and by $\\overline{\\Phi}$ the complement of a halfspace $\\Phi$. Two halfspaces $\\Phi,\\Psi \\in \\mathcal{H}$ are said to be \\emph{nested}, if either $\\Phi\\subseteq \\Psi$, $\\overline{\\Phi} \\subseteq \\Psi$, $\\Phi\\subseteq \\overline{\\Psi}$ or $\\overline{\\Phi}\\subseteq \\overline\\Psi$. Otherwise, they are \\emph{transverse}. Two distinct $\\Phi,\\Psi \\in \\mathcal{H}$ are \\emph{tightly nested}, if they are nested and there is no $\\Phi'\\in \\mathcal{H}$ with $\\Phi\\subsetneq \\Phi' \\subsetneq \\Psi$ or $\\overline{\\Phi}\\subsetneq \\Phi' \\subsetneq \\Phi$ etc..\\par\nFor every oriented edge $E$ in $X$, there is exactly one $\\Phi\\in\\mathcal{H}$ such that the beginning vertex of $E$ is in $\\Phi$ and the end vertex is in $\\overline{\\Phi}$. We say that $\\Phi$ and $E$ are \\emph{dual} to each other.\\par\n\nGiven $x,y\\in \\mathcal{H}$ the \\emph{interval} between $x$ and $y$ is \n$$[x,y]:= \\left\\{\\Phi\\in\\mathcal{H}: x\\notin\\Phi \\textrm{ and } y\\in\\Phi \\right\\}.$$\nGiven vertices $x,y,z\\in X$, there is a unique vertex $m(x,y,z)$ called \\emph{median} with the property $[a,b]=[a,m]\\cup[m,b]$ for any distinct pair $a,b$ in ${x,y,z}$ (see Preliminaries in \\cite{FFT16} for a simple proof).\n\n\n\\subsection{Haglund's combinatorial axis}\nLet $X$ be a CAT(0) cube complex. We can introduce a metric called \\emph{combinatorial distance} $d^c$ on the set of vertices $X^{(0)}$, by defining $d^c(x,y)$ as the minimal number of edges in an edge path from $x$ to $y$. A \\emph{combinatorial geodesic} is an optimal (with respect to $d^c$) oriented edge path.\\par\n\nThe \\emph{translation distance} of an automorphism $g$ of $X$ is the natural number\n$$\\delta(g)=\\min_{x\\in X^{(0)}} d^c(x,gx).$$\nAn automorphism $g$ of a CAT(0) cube complex $X$ is \\emph{hyperbolic}, if it fixes no vertex of $X$, i.e. $\\delta(g)>0$. Otherwise, $g$ is called \\emph{elliptic}. A combinatorial axis is an infinite combinatorial geodesic on which $g$ acts as a shift. According to Haglund in \\cite{Hag07}, if $g$ is a hyperbolic automorphism all of whose powers act without inversion (that is, there are no $\\Phi\\in\\mathcal{H}$ and $n\\in\\mathbb{Z}$ with $g^n\\Phi=\\overline{\\Phi}$), then every vertex in $X^{(0)}$ on which $g$ attains its translation distance is contained in some combinatorial axis.\\par\n\nAs in \\cite{FFT16}, let $A^+_g$ denote all halfspaces dual to an oriented edge in some combinatorial axis (indeed, a halfspace dual to some combinatorial axis is also dual to all other ones according to \\cite{Hag07}). Clearly, $A^+_g = \\bigcup_{n\\in \\mathbb{N}} [g^no,g^{n+1}o]$ for any vertex $o$ where $g$ attains its translation distance. An important fact is that for $\\Phi,\\Psi \\in A^+_g$ either $\\Phi\\subseteq \\Psi$, $\\Phi\\supseteq \\Psi$ or $\\Phi$ and $\\Psi$ are transverse, which follows because a combinatorial geodesic may never leave a halfspace after entering.\n\n\n\\subsection{The Bavard Duality}\nGiven a group $G$, the \\emph{commutator length} $\\textrm{cl}$ is a function $\\textrm{cl}: [G,G] \\rightarrow \\mathbb{N}$, where $[G,G]$ is the commutator subgroup. For $g\\in[G,G]$, it is defined as the minimal number of commutators whose product is $g$. The \\emph{stable commutator length} is the well defined limit \n$$\\textrm{scl}(g)= \\lim_{n\\to \\infty} \\frac{\\textrm{cl}(g^n)}{n}.$$\\par\nBounds of $\\textrm{scl}$ can be estimated using \\emph{homogeneous quasimorphisms} and the \\emph{Bavard Duality}. A quasimorphism is a function $\\phi: G\\rightarrow \\mathbb{R}$ with a bounded \\emph{defect} \n$$D(\\phi):= \\sup_{g,h\\in G} |\\phi(gh)-\\phi(g)-\\phi(h)|.$$ \nThe quasimorphism $\\phi$ is called homogeneous, if $\\phi(g^n)= n\\phi(g)$ for all $g$ and $n\\in \\mathbb{Z}$. Denote by $Q(G)$ the space of homogeneous quasimorphisms on $G$. Every quasimorphism yields a homogeneous quasimorphism called \\emph{homogenisation} $\\overline{\\phi}(g):= \\lim_{n\\to\\infty} \\frac{\\phi(g^n)}{n}$ with the following property:\n\\begin{mylem}\nLet $\\phi$ be a quasimorphism. Then its homogenisation satisfies $D(\\overline{\\phi})\\leq 2D(\\phi)$.\n\\end{mylem}\nThe Bavard Duality states that:\n\\begin{mythm}[\\cite{Bav91}]\\label{bavard}\nFor any $g\\in[G,G]$\n$$\\mathrm{scl}(g)= \\sup_{\\overline{\\phi}\\in Q(G)} \\frac{\\overline{\\phi}(g)}{2D(\\overline{\\phi}))}.$$\n\\end{mythm}\nTherefore, to prove that $\\textrm{scl}(g)$ is bounded from below by some constant, it suffices to find a homogeneous quasimorphism which has low enough defect (we say it is \\emph{efficient}) and at the same time does not vanish on $g$ (that is, it is \\emph{effective}).\\footnote{See \\cite[Chapter 2]{Cal09} for more detail and proofs on quasimorphisms and scl.}\n\n\\section{RAAG-like actions}\nLet us reproduce the definition of RAAG-like actions given in \\cite[Chapter 7]{FFT16}:\n\\begin{mydef}\\label{RAAGlike}\nLet $G$ be a group acting on a CAT(0) cube complex $X$ with halfspaces $\\mathcal{H}(X)$. The action is called \\emph{RAAG-like} if the following are satisfied:\n\\begin{enumerate}[(i)]\n\\item There are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $h\\overline{\\Phi} = \\Phi$ (``no inversions'')\n\\item there are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi$ and $h\\Phi$ transverse (``non-transverse''),\n\\item there are no tightly nested $\\Phi,\\Phi'\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi$ and $h\\Phi'$ transverse,\n\\item there are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi\\subset h\\overline{\\Phi}$ tightly.\n\\end{enumerate}\nA group is called RAAG-like, if it has a faithful RAAG-like action on some CAT(0) cube complex.\n\\end{mydef}\n\\begin{myrem}\nIf $G$ acts on $X$ freely, then RAAG-likeness of the action is equivalent to $X\/G$ being a \\emph{A-special} (often simply called \\emph{special}) cube complex in the sense of Haglund and Wise \\cite[Definition 3.2]{HW08}. In particular, we have the following correspondences: \n\\begin{enumerate}[(i)]\n\\item corresponds to all hyperplanes in $X\/G$ being two-sided, \n\\item corresponds to no hyperplane in $X\/G$ intersecting itself,\n\\item corresponds to no pair of hyperplanes in $X\/G$ inter-osculating and \n\\item corresponds to no pair of hyperplanes in $X\/G$ directly self-osculating.\n\\end{enumerate}\nHence $G$ is the fundamental group of an A-special cube complex and conversely the fundamental group of an A-special cube complex acts RAAG-like and freely on its universal cover. In particular, RAAGs are RAAG-like, as they are the fundamental group of an A-special cube complex (their Salvetti complex).\n\\end{myrem}\n\\begin{mylem}\\label{RAAGhyperbolic}\nEvery non-trivial element of a RAAG-like action is hyperbolic.\n\\end{mylem}\n\\begin{proof}\nSuppose $h\\in G$ is elliptic, i.e. $h$ has at least one fixed vertex, and acts non-trivially. If for some fixed vertex of $h$ in $X$, every incident edge is fixed, then all neighbouring vertices of $v$ are also fixed. Therefore, there must be some fixed vertex $v$ with an incident edge which is not fixed, or else every single vertex of $X$ would be fixed. Let $E$ be adjacent to $v$ and mapped to some other edge $F$ adjacent to $v$. If $E$ and $F$ bound a square, then $h$ is transverse, as the halfspace $\\Phi$ dual to $E$ is transverse to $h\\Phi$, the halfspace dual to $F$. If they do not bound a square, then $\\Phi$ and $h\\Phi$ are tightly nested if they are not transverse.\n\\end{proof}\n\n\\section{The quasimorphisms and their defect}\nFrom now on, let $G$ be a group with a non-transverse (not necessarily RAAG-like) action on a CAT(0) cube complex $X$.\\par\nWe recall the quasimorphisms defined in \\cite[Chapter 4]{FFT16} and, for completeness, the proof that their defect is bounded by $12$.\n\n\\begin{mydef}\nA \\emph{segment} is a series of half-spaces $\\gamma= \\left\\{\\Phi_0,..., \\Phi_r\\right\\}$ such that $\\Phi_i\\supsetneq \\Phi_{i+1}$ \\emph{tightly} for $0\\leq i\\gamma'$, $\\gamma'>\\gamma$, $\\overline{\\gamma}> \\gamma'$ and $\\gamma'> \\overline{\\gamma}$ respectively in these cases.\n\\end{myrem}\n\\begin{myrem}\nIf $S$ is a set of non-overlapping segments, then for any $\\gamma_1, \\gamma_2\\in S$ either $\\gamma_1> \\gamma_2$ or $\\gamma_2> \\gamma_1$. Thus, if $S$ is finite, it must contain a maximal segment that contains every other segment in $S$, and a minimal segment that is contained by every other segment in $S$, respectively.\n\\end{myrem}\n\\begin{mydef}\nGiven a segment $\\gamma$, let $G\\gamma = \\{g\\gamma: g\\in G\\}$ denote the set of \\emph{copies of $\\gamma$}. The function $c_{\\gamma}: X^2\\rightarrow \\mathbb{R}$ is defined sucht that $c_{\\gamma}(x,y)$ is the cardinality of the largest non-overlapping subset of $G\\gamma$ in $[x,y]$.\\par\nFurthermore, define $\\omega_{\\gamma}:X^2\\rightarrow \\mathbb{R}$ by $\\omega_{\\gamma}(x,y):= c_{\\gamma}(x,y) - c_{\\bar{\\gamma}}(x,y)$.\n\\end{mydef}\n\\begin{myrem}\n$\\omega_{\\gamma}(\\cdot,\\cdot)$ is $G$-invariant, i.e. $\\omega_{\\gamma}(x,y)= \\omega_{\\gamma}(gx,gy)$ for any $x,y\\in X$ and $g\\in G$, since any non-overlapping subset of $G\\gamma$ in $[x,y]$ can be pushed by $g$ to one in $[gx,gy]$, and vice versa.\\par\nFurthermore, $\\omega_{\\gamma}(\\cdot,\\cdot)$ is antisymmetric, since if $g\\gamma\\in [x,y]$, then $g\\overline{\\gamma}\\in [y,x]$, and vice versa.\n\\end{myrem}\nThe following lemmas show that $\\omega_{\\gamma}(o,go)$ as a function of $g$ (where $o$ is any vertex of $X$) is a quasimorphism.\n\\begin{mylem}\nFor $x,m,y\\in X$ with $m=m(x,m,z)$,\n$$\\left| \\omega_{\\gamma}(x,y)-\\omega_{\\gamma}(x,m) -\\omega_{\\gamma}(m,y)\\right| < 2$$\nholds.\n\\end{mylem}\n\\begin{proof}\nLet us first prove $c_{\\gamma}(x,y)\\geq c_{\\gamma}(x,m)+ c_{\\gamma}(m,y)-1$. Let $S_1$ and $S_2$ be maximal non-overlapping sets of copies of $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r\\right\\}$ in $[x,m]$ and $[m,y]$, respectively. Let $g\\gamma$ be the minimal element of $S_1$. We have $a\\gamma> b\\gamma$ for any $a\\gamma\\in S_1\\setminus\\{g\\gamma\\}$ and $b\\gamma\\in S_2$, because for any $a\\Phi_k \\in a\\gamma$ and $b\\Phi_l \\in b\\gamma$ we have $a\\Phi_k\\subsetneq g\\Phi_0 \\subsetneq b\\Phi_0 \\subsetneq b\\Phi_l$. Thus, $S_1\\setminus\\{g\\gamma\\}$ and $S_2$ do not overlap, whence the inequality follows.\\par\nLet us now prove $c_{\\gamma}(x,z)\\leq c_{\\gamma}(x,m)+ c_{\\gamma}(m,y) +1$. Let $S$ be a maximal set of copies of $\\gamma$ in $[x,m]$. There can be at most one copy $g\\gamma\\in S$ containing halfspaces $\\Phi$ and $\\Psi$ such that $y\\in\\Phi$ and $y\\notin \\Psi$ since all other copies of $\\gamma$ in $S$ either elementwise contain $\\Phi$ or are contained elementwise in $\\Psi$. The remaining $|S|-1$ copies can be assigned to sets $S_1$ and $S_2$ contained in $[x,y]$ and $[m,y]$, respectively, which proves the inequality.\\par\nThis proves $|c_{\\gamma}(x,y)- c_{\\gamma}(x,m)- c_{\\gamma}(m,y)| <1$ and therefore the lemma, as\n\\begin{equation*}\n\\begin{split}\n&\\left|\\omega_{\\gamma}(x,y)-\\omega_{\\gamma}(x,m) -\\omega_{\\gamma}(m,y)\\right|\\\\\n &\\leq \\left|c_{\\gamma}(x,y)-c_{\\gamma}(x,m) -c_{\\gamma}(m,y)\\right|\n+ \\left|c_{\\bar{\\gamma}}(x,y)-c_{\\bar{\\gamma}}(x,m) -c_{\\bar{\\gamma}}(m,y) \\right|\\\\\n&\\leq 2\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\begin{mylem}\\label{omega}\nFor any $x,y,z\\in X$\n$$\\left|\\omega_{\\gamma}(x,y)+\\omega_{\\gamma}(y,z) + \\omega_{\\gamma}(z,x)\\right| \\leq 6$$\nholds.\n\\end{mylem}\n\\begin{proof}\nLet $m$ be the median of $x,y,z$. By the last lemma, $\\left| \\omega_{\\gamma}(x,y)- \\omega_{\\gamma}(x,m)- \\omega_{\\gamma}(m,y)\\right|< 2$ holds, and analogous inequalities after replacing $x$ or $y$ by $z$. Therefore, \n\\begin{equation*}\n\\begin{split}\n\\left|\\omega_{\\gamma}(x,y)+\\omega_{\\gamma}(y,z) + \\omega_{\\gamma}(z,x)\\right| \\leq& |\\omega_{\\gamma}(x,m)+\\omega_{\\gamma}(m,y)\n\t\t\t+\\omega_{\\gamma}(y,m)\\\\\n\t\t&+\\omega_{\\gamma}(m,z)\n\t\t\t+\\omega_{\\gamma}(z,m)+\\omega_{\\gamma}(m,x)| + 6\\\\\n=& 6,\n\\end{split}\n\\end{equation*}\nwhere antisymmetry of $\\omega_{\\gamma}$ was used on the last line.\n\\end{proof}\n\n\\begin{mylem}\\label{defect}\nGiven a segment $\\gamma$ and a vertex $x_0\\in X$, the map $\\phi_{\\gamma}:G\\rightarrow \\mathbb{R}$ given by $\\phi_{\\gamma}(g)= \\omega_{\\gamma}(x_0,gx_0)$ is a quasimorphism with defect bounded by $6$.\\par\nAs a consequence, its homogenisation $\\overline{\\phi}_{\\gamma}$ has defect bounded by $12$.\n\\end{mylem}\n\\begin{proof}\n\\begin{equation*}\n\\begin{split}\n|\\delta\\phi_{\\gamma}(g,h)| &= |\\phi_{\\gamma}(gh)-\\phi_{\\gamma}(g)-\t\\phi_{\\gamma}(h)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)-\\omega_{\\gamma}(x_0,gx_0)-\t\\omega_{\\gamma}(x_0,hx_0)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)-\\omega_{\\gamma}(x_0,gx_0)-\t\\omega_{\\gamma}(gx_0,ghx_0)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)+\\omega_{\\gamma}(gx_0,x_0)+\t\\omega_{\\gamma}(ghx_0,gx_0)|\\\\\n\t&\\leq 6,\n\\end{split}\n\\end{equation*}\nwhere $\\omega_{\\gamma}(x_0,hx_0)= \\omega_{\\gamma}(gx_0, ghx_0)$, antisymmetry of $\\omega_{\\gamma}$ and Lemma \\ref{omega} were used in this order.\n\\end{proof}\n\n\n\n\\section{Effectiveness}\nFrom now on let $G$ be a group with a RAAG-like action on $X$. Let $g\\in G$ be a hyperbolic element and let $o\\in X$ denote a vertex where $g$ attains its translation distance.\\par\nThe aim is to find a segment $\\gamma$ in $[o,go]$ such that for $m\\in\\mathbb{N}$ the interval $[g^mo,g^{m+1}o]$ contains at least one copy of $\\gamma$ and no copies of $\\overline{\\gamma}$. This will guarantee $\\overline{\\phi_{\\gamma}}(g)\\geq 1$.\\par\nThe following are the segments we need:\n\\begin{mydef}\nA segment $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ in $[o,go]$ is called $g$-nested if $\\gamma> g\\gamma$. It is a maximal $g$-nested segment, if it is not contained in any other $g$-nested segment in $[o,go]$\n\\end{mydef}\n\\begin{myrem}\nA maximal $g$-nested segment always exists, since a single halfspace inn $[o,go]$ is a $g$-nested segment by non-transversality.\n\\end{myrem}\nThe $g$-nestedness is to guarantee, that $[g^mo,g^{m+1}o]$ contains a copy of $\\gamma$ for every $m\\in\\mathbb{N}$, while the maximality will be crucial to ensure that no copies of $\\overline{\\gamma}$ occur in these intervals.\\par\nHere is a useful characterisation of maximality:\n\\begin{mylem}\\label{maxnestchar}\nLet $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ be maximal $g$-nested in $[o,go]$. Then:\n\\begin{enumerate}[(i)]\n\\item Any $\\Psi\\in [o,go]$ with $\\Psi \\supsetneq \\Phi_0$ is transverse to $g^{-1}\\Phi_r$.\n\\item Any $\\Psi\\in [o,go]$ with $\\Phi_r \\supsetneq \\Psi$ is transverse to $g\\Phi_0$.\n\\end{enumerate}\n\\end{mylem}\n\\begin{proof}\nLet $\\Psi\\in [o,go]$ with $\\Psi \\supsetneq \\Phi_0$ and suppose by contradiction that $\\Psi$ is not transverse to $g^{-1}\\Phi_r$. Since $o\\notin\\Psi$, we have $g^{-1}\\Phi_r \\supsetneq \\Psi$ as $\\Psi\\supseteq g^{-1}\\Phi_r$ would imply $o\\notin g^{-1}\\Phi_r$. Let $\\Psi'$ be a halfspace with $\\Psi'\\supsetneq \\Phi_0$ tightly and $\\Psi\\supseteq\\Psi'$. Clearly, $g^{-1}\\Phi_r \\supsetneq \\Psi'$. Therefore, $\\{g^{-1}\\Psi'\\}\\cup g^{-1}\\gamma > \\{\\Psi'\\}\\cup \\gamma$. Applying $g$ yields $\\{\\Psi'\\}\\cup \\gamma> \\{g\\Psi'\\}\\cup g\\gamma$ which means $\\{\\Psi'\\}\\cup \\gamma$ is $g$-nested and thereby $\\gamma$ not maximal $g$-nested.\\par\nThe proof of the second part is symmetric.\n\\end{proof}\nThe following lemma, overlooked in \\cite{FFT16}, will be the key:\n\\begin{mylem}\\label{lesserorgreater}\nLet $\\alpha = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ be a segment in $A^+_g$ and $h\\in G$ such that $h\\overline{\\alpha}\\subset A^+_g$. Then either $h\\overline{\\alpha}> \\alpha$ or $\\alpha> h\\overline{\\alpha}$.\n\\end{mylem}\n\\begin{proof}\nNote that for $0\\leq k\\leq r$, exactly one of $\\Phi_k\\supsetneq h\\overline{\\Phi}_k$ or $h\\overline{\\Phi}_k \\supsetneq \\Phi_k$ must hold, because they are nested ($h$ is non-transverse) and equality would amount to an inversion ($\\Phi_k\\supsetneq h\\Phi_k$ and $h\\Phi_k \\supsetneq \\Phi_k$ are impossible, as $\\alpha,h\\overline{\\alpha}\\subset A^+_g$).\\par\nWe may assume $0 \\gamma$ or (2) $\\gamma> h\\overline{\\gamma}$.\\par\nCase (1): If $h\\overline{\\gamma}>g^{-1}\\gamma$, then $o\\notin h\\overline{\\Phi}_r$ clearly cannot hold. If $g^{-1}\\gamma> h\\overline{\\gamma}$, then $h\\overline{\\Phi}_r$ is in $[o,go]$ and contains $\\Phi_0$, but is not transverse to $g^{-1}\\Phi_r$, in contradiction to Lemma \\ref{maxnestchar}.\\par\t\nCase (2): If $g\\gamma> h\\overline{\\gamma}$, then $go\\in h\\overline{\\Phi}_r$ clearly cannot hold. If $h\\overline{\\gamma}>g\\gamma$, then $h\\overline{\\Phi}_r$ is in $[o,go]$ and contained in $\\Phi_r$, but is not transverse to $g\\Phi_0$, in contradiction to Lemma \\ref{maxnestchar}.\n\\end{proof}\nTying things together gives us the following:\n\\begin{mythm}\\label{effective}\nLet $\\gamma$ be maximal $g$-nested in $[o,go]$. Then $\\overline{\\phi}_{\\gamma}(g) \\geq 1$.\n\\end{mythm}\n\\begin{proof}\nLet $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$. For $n>0$, $\\left\\{ g^0\\gamma,...,g^n\\gamma \\right\\}$ is non-overlapping and in $[o,go]$. Therefore, $c_{\\gamma}(g^n)\\geq n$.\\par\nOn the other hand, if $h\\overline{\\gamma}\\subset [o,g^no]$, then $g^{-m}h\\overline{\\Phi}_r\\in [o,go]$ (and $g^{-m}h\\overline{\\gamma}\\subset A^+$) for some $m\\in\\mathbb{N}$. But this contradicts Lemma \\ref{almostdone}. Hence, $c_{\\overline{\\gamma}}(g^n)= 0$.\\par\nNow \n\\begin{equation*}\n\\overline{\\phi}_{\\gamma}(g)= \\lim_{n\\rightarrow \\infty} \\frac{\\phi_{\\gamma}(g^n)}{n}= \\lim_{n\\rightarrow \\infty} \\frac{\\omega_{\\gamma}(o,g^no)}{n}\\geq \\frac{n}{n} = 1\n\\end{equation*}\n\\end{proof}\nAn application of the Bavard Duality yields the main result: \n\\begin{mycor}\nLet $G$ be a group with a RAAG-like action on a CAT(0) cube complex. Then any element acting non-trivially has $\\mathrm{scl}(g)\\geq \\frac{1}{24}$. In particular RAAG-like groups have a stable commutator length gap of $1\/24$.\n\\end{mycor}\n\\begin{proof}\nBy Lemma \\ref{RAAGhyperbolic} every element of $G$ is hyperbolic. By the Theorem \\ref{effective} there is a quasimorphism $\\overline{\\phi}_{\\gamma}$ with $\\overline{\\phi}_{\\gamma}(g) \\geq 1$ that has defect $\\leq 12$ by Lemma \\ref{defect}. By the Bavard Duality (Theorem \\ref{bavard}) the corollary follows.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDeep discriminative models (\\emph{e.g.}~deep regression forests, deep neural decision forests) have recently been applied to many computer vision problems with remarkable success.\nThey compute the input to output mapping for regression or classification by virtue of deep neural networks~\\cite{Kontschieder_2015_ICCV,he2016deep,Simonyan2015,shen_deep_2018,chendeepage,chen_using_2017}.\nIn general, DDMs probably perform better when large amounts of effective training data (less noisy and balanced) is available.\nHowever, such ideal data is hard to collect, especially when large amounts of labels are required.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{Figure1.pdf}\n\t\\caption{The motivation of considering underrepresented examples in DRFs. \\textbf{(a):} The histogram shows the number of face images at different ages, and the average entropy curve represents the predictive uncertainty. We observe the high entropy values correspond to \\emph{underrepresented samples}. \\textbf{(b):} The histogram of the selected face images at pace 1 in SPL. \\textbf{(c):} The proposed new self-paced learning paradigm: easy and underrepresented samples first.}\n\t\\label{Figure1}\n\\end{figure}\n\nComputer vision literatures are filled with scenarios in which we are required to learn DDMs, not only robust to confusing and noisy examples, but also capable to conquer imbalanced data problem~\\cite{zeng2019soft,ren2018learning,cui2019class,khan2019striking,Kortylewski_2019_CVPR_Workshops}.\nOne typical approach is to learn discriminative features through rather deep neural networks, and feed them into a \\emph{cost-sensitive} discriminative function, often with regularization~\\cite{Kai2018Deep}.\nThe other typical approach reweights training samples according to their cost values~\\cite{cui2019class,khan2019striking} or gradient directions~\\cite{ren2018learning} (\\emph{i.e.}~meta learning).\nThese strategies are unlike our human beings, who lean things gradually---start with easy concepts and build up to complex ones, and can exclude extremely hard ones.\nMore importantly, we have a sense of \\emph{uncertainty} for some samples (\\emph{e.g.}~seldom seen) and progressively improve our capability to recognize them.\nThus, the main challenge towards realistic discrimination lies how to mimic our human discrimination system might work.\n\n\n\nThis line of thinking makes us resort to self-paced learning---a gradual learning regime inspired by the manner of humans~\\cite{Kumar2010Self}.\nIn fact, there are rare studies on the problem of self-paced DDMs.\nThen, a natural question arises: \\emph{can the self-paced regime lead DDMs to achieve more robust and less biased solutions? }\n\nMotivated by this, we propose a new self-paced learning paradigm for DDMs, which tackles the fundamental ranking and selecting problem in SPL from a new perspective: fairness.\nTo the best of our knowledge, this is the first work considering \\emph{ranking fairness} in SPL.\nSpecifically, we focus on deep regression forests (DRFs), a typical discriminative method, and propose self-paced deep regression forests with consideration on underrepresented examples (SPUDRFs).\nFirst, by virtue of SPL, our model distinguishes confusing and noisy examples from regular ones, and emphasizes more on ``good'' examples to obtain robust solutions.\nSecond, our method considers underrepresented examples, which may incur neglect in SPL since visual data is often imbalanced, renderring less bisaed solutions.\nThird, we build up a new self-paced learning paradigm: ranking samples on the basis of both likelihood and entropy (predictive uncertainty), as shown in Fig.~\\ref{Figure1}, which could be easily combined with a variety of DDMs.\n\nFor verification, we apply the SPUDRFs framework on two computer vision problems: (\\romannumeral1) facial age estimation, and (\\romannumeral2) head pose estimation.\nExtensive experimental results demonstrate the efficacy of our proposed new self-paced paradigm for DDMs.\nMoreover, on both aforementioned problems, SPUDRFs almost achieve the state-of-the-art performances.\n\n\n\\section{Related Work}\n\nThis section reviews the deep discriminative methods for facial age estimation and head pose estimation, and SPL methods.\n\n\\noindent\\textbf{Facial Age Estimation.}\nDDM based facial age estimation methods, for example~\\cite{niu_ordinal_2016,chen_using_2017,gao_age_2018,shen_deep_2018,li2019bridgenet}, employ DNNs to precisely model the mapping from image to age.\nOrdinal-based approaches~\\cite{niu_ordinal_2016,chen_using_2017} resort to a set of sequential binary queries---each query refers to a comparison with a predefined age, to exploit the inter-relationship (ordinal information) among age labels.\nImproved deep label distribution learning (DLDL-v2)~\\cite{gao_age_2018} explores the underlying age distribution patterns to effectively accommodates age ambiguity.\nBesides, deep regression forests (DRFs)~\\cite{shen_deep_2018} connect random forests to deep neural networks and achieve promising results.\nBridgeNet~\\cite{li2019bridgenet} uses local regressors to partition the data space and gating networks to provide continuity-aware weights.\nThe final age estimation result is the mixture of the weighted regression results.\nOverall, these DDM based approaches have enhanced age estimation performance largely; however, they plausibly ignore one problem: the interference arising from confusing and noisy examples---facial images with PIE (\\emph{i.e.}~pose, illumination and expression) variation, occlusion, misalignment and so forth.\n\n\n\\noindent\\textbf{Head Pose Estimation.}\nFor head pose estimation, Riegler~\\cite{riegler2013hough} \\emph{et al.}~utilized convolutional neural networks (CNNs) to learn patch features of facial images and achieved better performance.\nIn~\\cite{huang2018mixture}, Huang \\emph{et al.}~adopted multi-layer perceptron (MLP) networks for head pose estimation and proposed multi-modal deep regression networks to fuse RGB and depth information.\nIn~\\cite{wang2019deep}, Wang \\emph{et al.}~proposed a deep coarse-to-fine network for head pose estimation.\nIn~\\cite{ruiz2018fine}, Ruiz \\emph{et al.}~used a large synthetically expanded head pose dataset to train rather deep multi-loss CNNs for head pose estimation and gained satisfied accuracy.\nIn~\\cite{kuhnke2019deep}, Kuhnke \\emph{et al.}~proposed domain adaptation for head pose estimation, assuming shared and continuous label spaces.\nDespite seeing much success, these methods seldom consider the potential problems caused by imbalanced and noisy training data, which may exactly exist in visual problems.\n\n\\noindent\\textbf{Self-Paced Learning.}\nThe SPL is a gradual learning paradigm, which builds on the intuition that, rather than considering all training samples simultaneously, the algorithm should be presented with the training data from easy to difficult, which facilitates learning~\\cite{Kumar2010Self,meng_theoretical_2017}.\nVariants of SPL methods have been proposed recently with varying degrees of success.\nFor example, in~\\cite{jiang2015self}, Zhao \\emph{et al.}~generalized the conventional binary (hard) weighting scheme for SPL to a more effective real valued (soft) weighting manner.\nIn~\\cite{ma2017self}, Ma \\emph{et al.}~proposed self-paced co-training which applies self-paced learning to multi-view or multi-modality problems.\nIn~\\cite{han2017self}, Han \\emph{et al.}~made some efforts on mixture of regressions with SPL strategy, to avoid poorly conditioned linear sub-regressors.\nIn~\\cite{Ren2017RoSR,Ren2020SAMVC}, Ren \\emph{et al.}~introduced soft weighting schemes of SPL to reduce the negative influence of outliers and noisy samples.\nIn fact, the majority of these mentioned methods can be cast as the combination of SPL and shallow classifiers, where SVM and logistic regressors are usually involved.\nIn computer vision, due to the remarkable performance of DNNs, some authors have realized SPL may guide DDMs to achieve more robust solutions recently.\nIn~\\cite{ijcai2017}, Li \\emph{et al.}~sought to enhance the learning robustness of CNNs with SPL, and proposed SP-CNNs.\nHowever, \\cite{ijcai2017} omits one important problem in the discriminative model: the imbalance of training data.\nIn contrast to SP-CNNs, our SPUDRFs model has three advantages: (i) it emphasizes ranking fairness (\\emph{i.e.}~considering underrepresented examples) in SPL, and hence tends to achieve less biased solutions; (ii) its learning regime is fundamental and can be easily combined with other DDMs, especially the ones with predictive uncertainty; (iii) it creatively explores how SPL can integrate with DMMs with a probabilistic interpretation.\n\n\n\nOur work is inspired by the existing works~\\cite{jiang2014self,yang2019self} which take the class diversity in the sample selection of SPL into consideration.\nJiang \\emph{et al.}~\\cite{jiang2014self} encouraged the class diversity in sample selection at the early paces of self-paced training.\nYang \\emph{et al.}~\\cite{yang2019self} defined a metric, named complexity of image category, to measure sample number and recognition difficult jointly, and adopted this measure for sample selection in SPL.\nIn fact, the aforementioned two methods realize the lack of class diversity in SPL's sample selection may achieve biased solutions since visual data is often imbalanced.\nBut what causes lack of class diversity is exactly the ranking unfairness as underrepresented examples may often have large loss (particular in DDMs).\nNot only that, \\cite{yang2019self,jiang2014self} are only suitable for classification, but not regression (with continuous and high dimensional output).\nIn this paper, we will go further along this direction, aiming to tackle the fundamental problem in SPL: ranking unfairness.\n\n\n\\section{Preliminaries}\n\nIn this section, we review the basic concepts of deep regression forests (DRFs)~\\cite{shen_deep_2018}.\n\n\n\n\\noindent \\textbf{Deep Regression Tree.} DRFs usually consist of a number of deep regression trees.\nA deep regression tree, given input-output pairs $\\left\\{\\mathbf{x}_i, y_i\\right\\}_{n=1}^N$, where $\\mathbf{x}_i\\in\\mathbb{R}^{D_x}$ and $y_i\\in\\mathbb{R}$, models the mapping from input to output through DNNs coupled with a regression tree.\nA regression tree $\\mathcal{T}$ consists of split nodes $\\mathcal{N}$ and leaf nodes $\\mathcal{L}$~\\cite{shen_deep_2018}.\nMore specifically, each split node $n \\in \\mathcal{N}$ possesses a split to determine whether input $\\mathbf{x}_i$ goes to the left or right subtree; each leaf node $\\ell \\in \\mathcal{L}$ corresponds to a Gaussian distribution $p_{\\ell}(y_i)$ with mean $\\mu_l$ and variance $\\sigma^2_l$.\n\n\n\n\\noindent \\textbf{Split Node.}\nSplit node has a split function, $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta}) : \\mathbf{x}_i \\rightarrow[0,1]$, which is parameterized by $\\bm{\\Theta}$---the parameters of DNNs.\nConventionally, the split function is formulated as $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})=\\sigma\\left(\\mathbf{f}_{\\varphi(n)}(\\mathbf{x}_i ; \\bm{\\Theta})\\right)$, where $\\sigma(\\cdot)$ is the sigmoid function, $\\varphi(\\cdot)$ is an index function to specify the $\\varphi(n)$-th element of $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ in correspondence with a split node $n$, and $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ denotes the learned deep features.\nAn example to illustrate the sketch chart of the DRFs is shown in Fig.~\\ref{Figure1}, where $\\varphi_1$ and $\\varphi_2$ are two index functions for two trees.\nThe probability that $\\mathbf{x}_i$ falls into the leaf node $\\ell$ is given by:\n\n\\begin{equation}\n\\label{Eq.1}\n\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}=\\prod_{n \\in \\mathcal{N}} s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})^{[\\ell \\in \\mathcal{L}_{n_l}]}\\left(1-s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})\\right)^{\\left[\\ell \\in \\mathcal{L}_{n_r}\\right]},\n\\end{equation}\nwhere $[\\mathcal{H}]$ denotes an indicator function conditioned on the argument $\\mathcal{H}$. In addition, $\\mathcal{L}_{n_l}$ and $\\mathcal{L}_{n_r}$ correspond to the sets of leaf nodes owned by the subtrees $\\mathcal{T}_{n_l}$ and $\\mathcal{T}_{n_r}$ rooted at the left and right children ${n}_{l}$ and ${n}_{r}$ of node $n$, respectively.\n\n\n\n\\noindent \\textbf{Leaf Node.} For tree $\\mathcal{T}$, given $\\mathbf{x}_i$, each leaf node $\\ell \\in \\mathcal{L}$ defines a predictive distribution over $y_i$, denoted by $p_{\\ell}(y_i)$.\nTo be specific, $p_{\\ell}(y_i)$ is assumed to be a Gaussian distribution: $\\mathcal{N}\\left(y_i|\\mu_l, \\sigma^2_l\\right)$.\nThus, considering all leaf nodes, the final distribution of $y_i$ conditioned on $\\mathbf{x}_i$ is averaged by the probability of reaching each leaf:\n\\begin{equation}\n\\label{Eq.2}\np_{\\mathcal{T}}(y_i | \\mathbf{x}_i ; \\bm{\\Theta}, \\bm{\\pi})=\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i),\n\\end{equation}\nwhere $\\bm{\\Theta}$ and $\\bm{\\pi}$ represent the parameters of DNNs and the distribution parameters $\\left\\{\\mu_l,\\sigma^2_l\\right\\}$, respectively.\nIt can be viewed as a mixture distribution, where $\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}$ denotes mixing coefficients and $ p_{\\ell}(y_i)$ denotes the Gaussian distributions associated with the $\\ell^{th}$ leaf node.\nNote that $\\bm{\\pi}$ varies along with tree $\\mathcal{T}_k$, and thus we rewrite it as $\\bm{\\pi}_k$ below.\n\n\n\n\\noindent \\textbf{Forests of Regression Trees.}\nSince a forest comprises a set of deep regression trees $\\mathcal{F}=\\left\\{\\mathcal{T}_1,...,\\mathcal{T}_k\\right\\}$, the predictive output distribution, given $\\mathbf{x}_i$, is obtained by averaging over all trees:\n\\begin{equation}\n\\label{Eq.3}\np_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\n=\n\\frac{1}{K}\\sum_{k=1}^K p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i, \\bm{\\Theta}, \\bm{\\pi}_k\\right),\n\\end{equation}\nwhere $K$ is the number of trees and $\\bm{\\Pi}=\\left\\{\\bm{\\pi}_1,...,\\bm{\\pi}_K\\right\\}$.\n$p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ can be viewed as the likelihood that the $i^{th}$ sample has output $y_i$.\n\n\n\n\n\\section{Self-Paced DRFs with Consideration on Underrepresented Examples}\nThe problems in training DDMs for visual tasks arise from: (\\romannumeral1) the noisy and confusing examples, and (\\romannumeral2) the imbalance of training data.\nIntuitively inspired by the gradual learning manner of humans, we resort to self-paced learning and explore whether the DDMs, by virtue of SPL, tend to achieve more robust solutions.\nPerhaps not easily, in existing SPL, we observe ranking unfairness, as shown in Fig.~\\ref{Figure1}.\nMotivated by this observation, we propose SPUDRFs, which starts learning with easy yet underrepresented examples, and build up to complex ones.\nSuch a paradigm avoids overlooking the ``minority'' of training samples, leading to less biased solutions.\n\n\n\n\\subsection{Underrepresented Examples}\n\\label{Uncertainty}\nUnderrepresented examples mean ``minority'', as which the examples with similar or the same labels are scarce.\nUnsurprisingly, we observe that they may incur unfairness treatment in the early paces of SPL (see Fig.~\\ref{Figure1}(b)), due to imbalanced data distribution.\nThe underrepresented level could be measured by predictive uncertainty.\nGiven the sample $\\mathbf{x}_i$, its predictive uncertainty is formulated as the entropy of its predictive output distribution $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$:\n\\begin{equation}\n\\label{Eq.4}\nH\\left [p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\\right] = \\frac{1}{K}\\sum^K_{k=1}H\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right],\n\\end{equation}\nwhere $H\\left[ \\cdot\\right ]$ denotes entropy, and the entropy corresponds to the $k^{th}$ tree is:\n\\begin{equation}\n\\label{Eq.5}\nH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right] = -\\int p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\ln p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right) dy_i,\n\\end{equation}\nThe large the entropy is, the more uncertain the prediction should be, \\emph{i.e.}, the more underrepresented the sample is.\nConsidering underrepresented samples can be interpreted as adequately utilizing the ``information'' inherent in such examples in SPL training.\n\n\nAs previously discussed, $p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i; \\bm{\\Theta}, \\bm{\\pi}_k\\right)$ is a mixture distribution, taking the form $\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i)$, where $\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}$ denotes mixing coefficients and $p_{\\ell}(y_i)$ denotes the Gaussian distribution associated with the $\\ell\n^{th}$ leaf node.\nIn Eq.~(\\ref{Eq.5}), the integral of mixture of Gaussians is non-trivial. Monte Carlo sampling provides a way to calculate it, but incurs large computational cost~\\cite{huber2008entropy}.\nHere, we use the lower bound of this integral to approximate its true value:\n\\begin{equation}\n\\label{Eq.6}\nH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right]\\approx\\frac{1}{2}\\sum_{\\ell\\in\\mathcal{L}}\\omega_\\ell(\\mathbf{x}_i|\\mathbf{\\Theta})\\left[\\ln \\left(2\\pi \\sigma_\\ell^2\\right)+1\\right].\n\\end{equation}\nThe underrepresented examples are often scarce, and have not been treated fairly, resulting in large prediction uncertainty (\\emph{i.e.}~entropy).\n\n\\subsection{ Objective Function}\n\nRather than considering all the samples simultaneously, our proposed SPUDRFs are presented with the training data in a meaningful order, that is, easy and underrepresented examples first.\nSpecifically, we define a latent variable $v_i$ that indicates whether the $i^{th}$ sample is selected $(v_i = 1)$ or not $(v_i = 0)$ depending on how easy and underrepresented it is for training.\nOur objective is to jointly maximize the log likelihood with\nrespect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$, and learn the latent selecting variables $\\mathbf{v}=\\left(v_1,...,v_N\\right)^T$.\nWe prefer to select the underrepresented examples, which probably have higher predictive uncertainty (\\emph{i.e.}~entropy), particularly in the early paces.\nIt builds on the intuition that the underrepresented examples may incur neglect since they are the ``minority'' in training data.\nTherefore, we maximize a self-paced term regularized likelihood function, meanwhile considering predictive uncertainty,\n\\begin{equation}\n\\label{Eq.7}\n\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i \\right \\} + \\lambda\\sum_{i=1}^N v_i ,\n\\end{equation}\nwhere $\\lambda$ is a parameter controlling the learning pace, $\\lambda>0$, $\\gamma$ is the parameter imposing on entropy, and $H_i$ denotes the predictive uncertainty of the $i^{th}$ sample, as previously discussed in Sec.~\\ref{Uncertainty}.\nWhen $\\gamma$ decays to 0, the objective function is equivalent to the log likelihood function with respect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nEq.~(\\ref{Eq.7}) indicates each sample is weighted by $v_i$, and whether $\\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i>-\\lambda$ determines\nthe $i^{th}$ sample is selected or not.\nThat is, the sample with high likelihood value or high predictive uncertainty may be selected.\nThe optimal $v_i^*$ is:\n\\begin{align}\n\\label{Eq.8}\nv_i^* = \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i > -\\lambda$}\\\\\n0 & \\textrm{otherwise}\n\\end{array} \\right.,\n\\end{align}\nwhere $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ is written as $ p_{\\mathcal{F}i}$ for simplicity.\n\nOne might argue the noisy and hard examples tend to have high predictive uncertainty also, rendering being selected in the early paces.\nIn fact, from Eq.~(\\ref{Eq.8}), we observe whether one sample is selected is determined by both its predictive uncertainty and the log likelihood of being predicted correctly.\nThe noisy and hard examples probably have relatively large loss \\emph{i.e.}~low log likelihood, avoiding being selected at the very start.\n\nIteratively increasing $\\lambda$ and decreasing $\\gamma$, samples are dynamically involved in the training of DRFs, starting with easy and underrepresented examples and ending up with all samples.\nNote every time we retrain DRFs, that is, maximizing Eq.~(\\ref{Eq.7}), our model is initialized to the result of the last iteration.\nAs such, our model is initialized progressively by the result of the previous pace---adaptively calibrated by ``good'' examples.\nThis also means we place more emphasis on easy and underrepresented examples rather than confusing and noisy ones.\nThus, SPUDRFs are prone to have more robust and less biased solutions since we adequately consider the underrepresented examples.\n\n\\noindent \\textbf{Mixture Weighting.}\nIn the previous section, we adopt a hard weighting scheme to assign data points to paces, in which one sample is either selected $(v_i=1)$ or not $(v_i=0)$.\nSuch a weighting scheme appears to be less accurate as it omits the importance of samples.\nHence, we adopt a mixture weighting scheme~\\cite{jiang2014easy}, where the selected samples are weighted by its importance, ling in the range $0\\leq v_i \\leq 1$.\nThe objective function with mixture weighting is defined as:\n\\begin{equation}\n\\label{Eq.9}\n\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i\\right \\} + \\zeta \\sum_{i=1}^N \\log\\left(v_i + \\zeta\/\\lambda\\right) ,\n\\end{equation}\nwhere $\\zeta$ is a parameter controlling the learning pace.\nWe set $\\zeta=\\left(\\frac{1}{\\lambda'}-\\frac{1}{\\lambda}\\right)^{-1}$, and $\\lambda>\\lambda'>0$ to construct a reasonable soft weighting formulation.\nThe self-paced regularizer in Eq.~(\\ref{Eq.9}) is convex with respect to $v\\in\\left[0,1\\right]$.\nThen, setting the partial gradient of Eq.~(\\ref{Eq.9}) with respect to $v_i$ as zero will lead to the following:\n\\begin{equation}\n\\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i + \\frac{\\zeta}{v_i + \\zeta\/\\lambda} = 0.\n\\end{equation}\nThen, the optimal solution of $v_i$ is given by:\n\\begin{align}\nv_i^* = \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\geq -\\lambda' $}\\\\\n0 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\leq -\\lambda $}\\\\\n\\frac{-\\zeta}{\\log p_{\\mathcal{F}i} + \\gamma H_i} - \\zeta\/\\lambda & \\textrm{otherwise}\n\\end{array} \\right.\n\\label{Eq.11}\n\\end{align}\nIf either the log likelihood or the predictive uncertainty is too large, $v^*_i$ equals to 1.\nIn addition, if the likelihood and the predictive uncertainty are both too small, $v^*_i$ equals to 0.\nExcept the above two situations, the soft weighting calculation (\\emph{i.e.}, the last line of Eq.~(\\ref{Eq.11})) is adopted.\n\n\\noindent \\textbf{Curriculum Reconstruction.}\nThe underrepresented examples play an important role in our SPUDRFs algorithm.\nAs previously mentioned, the proposed new self-paced regime coupled with a mixture weighting scheme emphasizes more on underrepresented examples, rendering better solutions.\nSince the intrinsic reason that causes predictive uncertainty is plausibly the imbalanced training data, we further re-balance data distribution via a curriculum reconstruction strategy.\nMore specifically, we distinguish the underrepresented examples (whose $H_i$ is lager than $\\beta$) from regular ones at each pace, and augment them into the training data.\n\n\n\n\\subsection{Optimization}\n\\label{Learning}\nWe propose a two-step alternative search strategy (ASS) algorithm to solve SPUDRFs: (\\romannumeral1) update $\\mathbf{v}$ for sample selection with fixed $\\bm{\\Theta}$ and $\\bm{\\Pi}$, and (\\romannumeral2) update $\\bm{\\Theta}$ and $\\bm{\\Pi}$ with current fixed sample weights $\\mathbf{v}$.\n\n\\noindent\\textbf{Optimizing $\\bm{\\Theta}$ and $\\bm{\\Pi}$.}\nThe parameters $\\left\\{\\bm{\\Theta},\\bm{\\Pi}\\right\\}$ and weights $\\mathbf{v}$ are optimized alternatively.\nWith fixed $\\mathbf{v}$, our DRFs is learned by alternatively updating $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nIn \\cite{shen_deep_2018}, the parameters $\\bm{\\Theta}$ for split nodes (\\emph{i.e.}~parameters for VGG) are updated through gradient descent since the loss is differentiable with respect to $\\bm{\\Theta}$.\nWhile the parameters $\\bm{\\Pi}$ for leaf nodes are updated by virtue of variational bounding~\\cite{shen_deep_2018} when fixing $\\bm{\\Theta}$.\n\n\\noindent\\textbf{Optimizing $\\mathbf{v}$.}\nAs previously discussed, $v_i$ is a binary variable or real variable ranged in $\\left[0, 1\\right]$.\nIt indicates how to weight the $i^{th}$ sample during training.\nThe parameter $\\lambda$ could be initialized to obtain 50\\% samples to train the model, and is then progressively increased to involve 10\\% more data in each pace.\nThe parameter $\\gamma$ could be initialized empirically and is progressively decayed to zero.\nThe training stops when all the samples are selected, at $\\gamma=0$.\nAlong with increasing $\\lambda$ and decreasing $\\gamma$, DRFs are trained to be more ``mature''.\nThis learning process is like how our human beings learn one thing from easy and uncertain to complex.\n\n\n\n\n\\section{Experimental Results}\n\\subsection{Tasks and Benchmark Datasets}\n\\noindent \\textbf{Age Estimation.}\nThe Morph \\uppercase\\expandafter{\\romannumeral2~\\cite{ricanek2006morph}} dataset contains 55,134 unique face images of 13618 individuals with unbalanced gender and ethnicity distributions, and is the most popular publicly available real age dataset.\nThe FG-NET~\\cite{panis2016overview} dataset includes 1,002 color or gray images of 82 people with each subject almost accompanied by more than 10 photos at different ages.\nSince all images were taken in a totally uncontrolled environment, there exists a large deviation on lighting, pose and expression (\\emph{i.e.}~PIE) of faces inside the dataset.\n\n\n\\noindent \\textbf{Head Pose Estimation.}\nThe BIWI dataset~\\cite{fanelli2013random} contains 20 subjects, of which 10 are male and 6 are female, besides, 4 males have been chosen twice with wearing glasses or not.\nIt includes 15678 images collected by a Kinect sensor device for different persons and head poses\nwith pitch, yaw and roll angles mainly ranging within $\\pm 60^{\\circ}$, $\\pm 75^{\\circ}$ and $\\pm 50^{\\circ}$.\n\n\\subsection{Experimental Setup}\n\n\\noindent\\textbf{Dataset Setting.}\nThe settings of different datasets are given below.\n\\begin{itemize}\n\t\\item \\textbf{Morph \\uppercase\\expandafter{\\romannumeral2}.} Following the recent relevant work~\\cite{shen_deep_2018}, the images in Morph \\uppercase\\expandafter{\\romannumeral2} were divided into two sets: 80\\% for training and the rest 20\\% for testing. The random division was repeated\n\t5 times and the reported performance was averaged over these 5 times. The VGG-Face~\\cite{parkhi2015deep} networks were chosen as the pre-trained model.\n\t\\item \\textbf{FG-NET.} The leave-one-person-out scheme~\\cite{shen_deep_2018} was adopted, where the images of one person were selected for testing and the remains for training. The VGG-16 networks were pre-trained on the IMDB-WIKI~\\cite{rothe2018deep} dataset.\n\t\\item \\textbf{BIWI.} Similarly, 80\\% of the whole data was randomly chosen for training and the rest 20\\% for testing, and this operation was repeated 5 times. Moreover, the VGG-FACE networks were the pre-trained model.\n\\end{itemize}\n\n\n\\noindent\\textbf{Evaluation Metrics.}\nThe first evaluation metric is the mean absolute error (MAE), which is defined as the average absolute error between the ground truth and the predicted output: $\\sum_{i=1}^{N}\\left|\\hat{y}_{i}-y_{i}\\right|\/N$, $\\hat{y_{i}}$ represents the estimated output of the $i^{th}$ sample, and $N$ is the total number of testing images.\nThe other evaluation metric is cumulative score (CS), which denotes the percentage of images sorted in the range of $\\left[y_{i}-L, y_{i}+L\\right]$: $CS(L)=\\sum_{i=1}^{N}\\left[\\vert\\hat{y}_{i}-y_{i}\\vert \\leq L\\right]\/N \\cdot 100 \\%$, where $[ \\cdot ]$ denotes an indicator function and $L$ is the error range.\n\n\\noindent\\textbf{Preprocessing and Data Augmentation.}\nOn the Morph \\uppercase\\expandafter{\\romannumeral2} and FG-NET datasets, MTCNN~\\cite{zhang_joint_2016} was used for joint face detection and alignment.\nFurthermore, following~\\cite{shen_deep_2018}, we augmented training images in three ways: (\\romannumeral1) random cropping (5 times); (\\romannumeral2) adding Gaussian white noise with variance of 0.0001 (2 times); (\\romannumeral3) random horizontal flipping (2 times). The whole number of samples was increased by 20 times after augmentation.\nOn the BIWI dataset, we utilized the depth images for training and did not augment training images.\n\n\n\\noindent\\textbf{Parameters Setting.}\nThe VGG-16~\\cite{Simonyan2015} was employed as the fundamental backbone networks of SPUDRFs.\nThe hyper-parameters of VGG-16 were: training batch size (32 on Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI, 8 on FG-NET), drop out ratio (0.5), max iterations of each pace ($80k$ on Morph \\uppercase\\expandafter{\\romannumeral2}, $20k$ on FG-NET, and $40k$ on BIWI), stochastic gradient descent (SGD), initial learning rate (0.2 on Morph \\uppercase\\expandafter{\\romannumeral2}, 0.1 on BIWI, 0.02 on FG-NET) by reducing the learning rate ($\\times$0.5) per $10k$ iterations. The hyper-parameters of SPUDRFs were: tree number (5), tree depth (6), output unit number of feature learning (128), iterations to update leaf node predictions (20), number of mini-batches used to update leaf node predictions (50).\nIn the first pace, 50\\% samples which are easy or underrepresented were selected for training.\nHere, $\\lambda$ was set to guarantee the first 50\\% samples with large $\\log p_{\\mathcal{F}i} + \\gamma H_i$ values involved.\n$\\lambda'$ was set to ensure 10\\% of selected samples with soft weighting.\n$\\gamma$ was initialized to be 15 on the Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI datasets, and 5 on the FG-NET dataset.\n$\\beta$ was set to select 1180 and 2000 samples as the ones needed to be augmented twice at each pace on the Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI datasets.\nThe number of paces was empirically set to be 10, 3 and 6 on the Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET, and BIWI datasets, and except the first pace, an equal proportion of the rest data was gradually involved at each pace.\n\n\n\n\n\n\\subsection{Validity of Our Proposed Method}\n\\label{valid}\n\\noindent \\textbf{Self-paced Learning Strategy.}\nThe validity of self-paced strategy in training DDMs is mainly demonstrated by the following experiments on the MorphII dataset.\nWe first used all training images in the Morph \\uppercase\\expandafter{\\romannumeral2} datasets to train DRFs so as to rank samples at the beginning pace.\nRetraining proceeded with progressively increasing $\\lambda$ such that every 1\/9 of the rest data was gradually involved at each pace, where $\\gamma$ was decreased to the half of its previous value every time.\nIn the last pace, the value $\\gamma$ was constrainedly set to be 0.\nThe visualization of this process can be found in Fig.~\\ref{SPUDRFs_validation}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{SPUDRFs_validation.pdf}\n\t\\caption{The gradual learning process of SP-DRFs and SPUDRFs. \\textbf{Left:} The typical worst cases at each iteration become more confusing and noisy along\n\t\twith iteratively increasing $\\lambda$ and decreasing $\\gamma$. The two numbers below each image are the real age (left) and predicted age (right). \\textbf{Right:} The MAEs of SP-DRFs and SPUDRFs at each pace descend gradually. The SPUDRFs show its superiority of taking predictive uncertainty into consideration, when compared with SP-DRFs.}\n\t\\label{SPUDRFs_validation}\n\\end{figure}\nFig.~\\ref{SPUDRFs_validation} illustrates the representative face images in each learning pace of SPUDRFs, along with increasing $\\lambda$ and decreasing $\\gamma$.\nThe two numbers below each image are the real age (left) and predicted age (right).\nWe observe that the training images in the latter paces are obviously more confusing and noisy than the ones in the early paces.\nSince our model is initialized by the results of the previous retraining pace, meaning adaptively calibrated by ``good'' examples.\nAs a result, it has improved performance than DRFs, where the MAE is improved from 2.17 to 1.91, and the CS is promoted from 92.79\\% to 93.31\\% (see Fig.~\\ref{morph_experiment}(a)).\n\n\nFig.~\\ref{SPUDRFs_validation} also shows the comparison between SP-DRFs and SPUDRFs on the Morph \\uppercase\\expandafter{\\romannumeral2} datasets.\nThe yellow bar denotes the MAE of SP-DRFs, while the orange bar denotes for SPUDRFs.\nWe find the MAE of SPUDRFs is lower than SP-DRFs at each pace, particularly the last pace ($1.91$ against $2.02$).\nAs we discussed previously, as in Fig.~\\ref{Figure1}, SPUDRFs are prone to reach less biased solutions due to the wider covering range of leaf nodes, owing to considering underrepresented examples.\nThis experiment could be regarded as an ablation study of considering ranking fairness in SPL.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{Uncertainty_efficacy.pdf}\n\t\\caption{The leaf node distribution of SP-DRFs and SPUDRFs in gradual learning process. Three paces, \\emph{i.e.}~pace 1, 3, and 6, are randomly chosen for visualization. For SP-DRFs, the Gaussian means of leaf nodes (the red points in the second row) are concentrated in a small range, incurring seriously biased solutions. For SPUDRFs, the Gaussian means of leaf nodes (the orange points in the third row) distribute widely, leading to much better MAE performance.}\n\t\\label{Uncertainty_efficacy}\n\\end{figure}\n\\noindent\\textbf{Considering Underrepresented Examples.}\n\\label{ExpUnderSamples}\nOn the BIWI dataset, the necessity of considering ranking fairness in SPUDRFs is further demonstrated.\nIn SP-DRFs, DRFs was first trained on the basis of all data, and the samples were ranked and selected for the first pace according to this result.\nSubsequently, every $10\\%$ of the rest samples were progressively involved for retraining.\n$\\lambda$ was progressively increased while $\\gamma$ was progressively decreased until zero.\nIn SP-DRFs, the same self-paced strategy was adopted as in SPUDRFs, but without considering ranking fairness (\\emph{i.e.}~underrepresented examples).\n\n\nFig.~\\ref{Uncertainty_efficacy} visualizes the leaf node distributions of SP-DRFs and SPUDRFs in the progressive learning process.\nThe Gaussian means $\\mu_l$ associated with the 160 leaf nodes, where each 32 leaf nodes are defined for 5 trees, are plotted in each sub-figures.\nThree paces, \\emph{i.e.}~pace 1, 3, and 6, are randomly chosen for visualization.\nOnly pitch and yaw angles are shown for clarity.\nBesides, the distribution of angle labels (\\emph{i.e.}~pitch and yaw) are also shown, where the imbalance problem of data distribution is obvious.\n\nIn Fig.~\\ref{Uncertainty_efficacy}, the comparison results between SP-DRFs and SPUDRFs demonstrate the efficacy of considering ranking fairness in SPL.\nFor SP-DRFs, the Gaussian means of leaf nodes (red points in the second row) are concentrated in a small range, incurring seriously biased solutions.\nThat means the underrepresented examples have been neglected in SPL training.\nThe poor MAEs are the evidence for this, which are even inferior to DRFs (see Fig.~\\ref{biwi_experiment}(a)).\nSPUDRFs rank samples by log likelihood coupled with entropy, and are prone to achieve less biased solutions, as shown in the third rows of Fig.~\\ref{Uncertainty_efficacy}.\nSuch an experiment could be also regard as an ablation study of the proposed ranking algorithm.\n\n\\subsection{Comparison with State-of-the-art Methods}\n\\label{sec:blind}\n\nWe compared our SPUDRFs with other state-of-the-art methods on the Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET and BIWI datasets.\n\\begin{figure}[t]\n\t\\centering \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.82}{\n\t\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tLSVR \\cite{guo_human_2009} & 4.31 & 66.2\\% \\\\\n\t\t\t\tRCCA \\cite{Huerta2014Facial} & 4.25 & 71.2\\% \\\\\n\t\t\t\tOHRank \\cite{Chang2011Ordinal} & 3.82 & N\/A \\\\\n\t\t\t\tOR-CNN \\cite{niu_ordinal_2016} & 3.27 & 73.0\\% \\\\\n\t\t\t\tRanking-CNN \\cite{chen_using_2017} & 2.96 & 85.0\\% \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 2.17 & 91.3\\% \\\\\n\t\t\t\tDLDL-v2 \\cite{gao_age_2018}& 1.97 & N\/A \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.02} & \\textbf{92.79\\%} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{1.91} & \\textbf{93.31\\%} \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t& \\raisebox{-0.83in}{\\includegraphics[width=0.555\\textwidth]{morph_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves of the comparison methods.}\n\t\\label{morph_experiment}\n\\end{figure}\n\n\\noindent\\textbf{Results on Morph \\uppercase\\expandafter{\\romannumeral2}.}\nFig. \\ref{morph_experiment}(a) compares SPUDRFs with other baseline methods: LSVR~\\cite{guo_human_2009}, RCCA \\cite{Huerta2014Facial}, OHRank~\\cite{Chang2011Ordinal}, OR-CNN \\cite{niu_ordinal_2016}, Ranking-CNN \\cite{chen_using_2017}, DRFs~\\cite{shen_deep_2018}, and DLDL-v2~\\cite{gao_age_2018}.\nFirstly, owing to the effective feature learning ability of DNNs, the SPUDRFs method is much superior to the shallow model based approaches, such as LSVR~\\cite{guo_human_2009} and OHRank~\\cite{Chang2011Ordinal}.\nSecondly, duing to the valid self-paced regime, our SPUDRFs outperform other DDMs, and lead to more robust and less biased solutions.\nThirdly, SPUDRFs outperform SP-DRFs on both MAE and CS, and achieve state-of-the-art performance.\nFig.~\\ref{morph_experiment}(b) shows the CS comparison on this dataset.\nWe observe that the CS of SPUDRFs reachs 93.31\\% at error level $L=5$, which is significantly better than DRFs and obtained 2.01\\% increment.\n\\begin{figure}\n\t\\centering \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.82}{\n\t\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tIIS-LDL \\cite{xin_geng_facial_2013} & 5.77 & N\/A \\\\\n\t\t\t\tLARR \\cite{guodong_guo_image-based_2008} & 5.07 & 68.9\\% \\\\\n\t\t\t\tMTWGP \\cite{Yu2010Multi} & 4.83 & 72.3\\% \\\\\n\t\t\t\tDIF \\cite{han_demographic_2015} & 4.80 & 74.3\\% \\\\\n\t\t\t\tOHRank \\cite{Chang2011Ordinal} & 4.48 & 74.4\\% \\\\\n\t\t\t\tCAM \\cite{Luu2013Contourlet} & 4.12 & 73.5\\% \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 3.06 & 83.33\\% \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.84} & \\textbf{84.73\\%} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{2.77} & \\textbf{85.53\\%}\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\n\t\t& \\raisebox{-0.83in}{\\includegraphics[width=0.555\\textwidth]{fgnet_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the FGNET dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves of the comparison methods.}\n\t\\label{fgnet_experiment}\n\\end{figure}\n\n\n\\noindent\\textbf{Results on FG-NET.} Fig.~\\ref{fgnet_experiment}(a) shows the comparison results of SPUDRFs with the state-of-the-art approaches on FG-NET dataset.\nAs can be seen, SPUDRFs reach an MAE of 2.77 years, which reduces the MAE of DRFs by 0.29 years.\nBesides, the CS comparison is shown in Fig.~\\ref{fgnet_experiment}(b), SPUDRFs consistently outperform other recent proposed methods at different error levels, proving that our method is effective in enhancing the robustness of facial age estimation.\n\n\n\\begin{figure}\n\t\\centering \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.78}{\n\t\t\t\\begin{tabular}{@{}l|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tHF \\cite{riegler2013hough} & 4.95 \\\\\n\t\t\t\tSVR \\cite{drucker1997support} & 3.14 \\\\\n\t\t\t\tRRF \\cite{liaw2002classification} & 3.06 \\\\\n\t\t\t\tKPLS \\cite{al2012partial} & 2.88 \\\\\n\t\t\t\tSAE \\cite{hinton2006reducing} & 1.94 \\\\\n\t\t\t\tMoDRN \\cite{huang2018mixture} & 1.62 \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 1.44 \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.08} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{1.18} \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t& \\raisebox{-0.8in}{\\includegraphics[width=0.53\\textwidth]{biwi_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the BIWI dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves the comparison methods.}\n\t\\label{biwi_experiment}\n\\end{figure}\n\n\\noindent\\textbf{Results on BIWI.}\nFig.~\\ref{biwi_experiment}(a) shows the comparison results of our method with several state-of-the-art approaches.\nThe experimental results reveal the proposed SPUDRFs method achieves the best performance with an MAE of 1.18, which is state-of-the-art.\n\\emph{Besides, we observe one important phenomenon: the MAE of SP-DRFs is even much worse than DRFs.\n\tThis further demonstrates the obvious drawback of the ranking and selecting algorithm in original SPL---incurring seriously biased solutions.}\nIn the first pace of the original SPL, as illustrated in Fig.~\\ref{Uncertainty_efficacy}, the Gaussian means of leaf nodes are concentrated in a small range, leading biased solutions.\nIncorporating underrepresented examples in the early pace of SPUDRFs renders to more reasonable distributions of the leaf nodes.\nFig.~\\ref{biwi_experiment}(b) plots only three CS curves for brevity, \\emph{i.e.}, DRFs, SP-DRFs and SPUDRFs, which is the average of the three angles.\nSPUDRFs also outperform DRFs and SP-DRFs at different error levels.\n\n\n\\section{Conclusion and Future Work}\nThis paper explored how self-paced regime leads deep discriminative models (DDMs) to achieve more robust and less biased solutions on different computer vision tasks (\\emph{e.g.}~facial age estimation and head pose estimation).\nSpecifically, a novel self-paced paradigm, which considers ranking fairness, was proposed.\nThe new ranking scheme jointly considers loss and predictive uncertainty.\nSuch a paradigm was combined with deep regression forests (DRFs), and led to a new model, namely self-paced deep regression forests with consideration on underrepresented examples (SPUDRFs).\nExtensive experiments on two well-known computer vision tasks demonstrated the efficacy of the proposed paradigm.\n\n\n\nWe are currently applying self-paced DDMs for other computer vision tasks, \\emph{e.g.}~viewpoint estimation, indoor scene classification, where the ability to handle ranking unfairness is fundamental to the success.\nThus, investigating the causes of algorithm unfairness in DDMs is a worthy direction.\nObviously, except data imbalance, there exist some other causing factors.\nIn addition to this, exploring how to combine the new self-paced paradigm with other DDMs, including deep regressors and classifiers, will also be our future work.\n\n\n\\noindent \\textbf{Acknowledgement.} The authors gratefully acknowledge the support of China Postdoctoral Science Foundation No.2017M623007.\n\n\\clearpage\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}