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+{"text":"\\section{Field-aligned downward current sheets}\nThe observations concerning auroral electron fluxes and the related currents are the following:\n\n\\subsection{Upward current region properties}\nDownward electrons\/upward currents occupy an extended spatial interval of  low density and barely structured fluxes. The variation of the perpendicular (to the main field) magnetic field is smooth; it changes about linearly from $-\\delta B_\\perp$ to $+\\delta B_\\perp$ signalling that the spacecraft has crossed a homogeneous broad structureless upward sheet current carried by the as well structureless medium energy electron flux which, in the energy-time spectrum occupies a narrow band of constant energy and small energy spread. \n\n\nAbsence of an ionospheric electron background at (FAST) spacecraft altitude either suggests that the ionosphere does not reach up to those altitudes ($\\sim2000- 3000$ km) which sometimes, in a diffusive model of the ionosphere, is interpreted as presence of a field aligned electric potential which holds the ionospheric electrons down while attracting magnetospheric electrons. The validity of such an assumption can be questioned in terms of tail reconnection as the inflow of reconnection accelerated electrons from the tail does not require such an electric potential field, the origin of which is  difficult to justify over a region of upward current extension while being natural when considering tail reconnection where it simply maps the large reconnection affected interval of the cross tail current down into the ionosphere. The small number of downward electrons does not require any presence of electric fields. The flux consists of nearly mono-energetic auroral electrons. These form a field-aligned beam and are accompanied by observed low frequency Langmuir-wave excitation which allows for the determination of the beam density being roughly $N_\\downarrow\\approx 10^6$ m$^{-3}$ (one electron per cubic centimetre). \n\n\\subsection{Upward topside electrons}\nFigure \\ref{fig3} shows simultaneous upward\/downward FAST measurements of electron fluxes when crossing a very active substorm topside auroral ionosphere. The upward-electron downward-current region behaves  differently. Its spatial extension is narrow. In view of the electron flux it consists of a large number of very closely spaced spikes. The flux in each spike (generally) maximizes at the lowest energies $\\epsilon_e\\lesssim 0.1$ keV.  Electron number densities are high estimated to be around $N_\\uparrow \\sim 10^7$ m$^{-3}$ (ten per cubic centimetre) or higher. The total integrated up and down currents must be similar for perfect closure. This is however not guaranteed for the divergence of currents in the ionosphere perpendicular to the magnetic field, current dissipation, and the high spatial structuring of the downward currents such that it cannot be checked whether the indication of the different downward current sheets all belong to closure of the single upward current. Some uncertainty in comparison remains, which however for our purposes does not matter.\n\nThe important observation is the high local structure of the downward currents, their obvious spatial closeness, and their differences in energies and flux level which is reflected in both the flux fluctuations across the narrow downward current region, and in the high spatial fluctuation of the main-field-perpendicular magnetic component $\\mathbf{b}_\\perp$ {(from here on denoting the magnetic variation $\\delta B_\\perp= b_\\perp$)} which indicates the crossing of many downward current sheets or filaments.  All these downward currents flow parallel while being closely spaced in the direction perpendicular to the main field $\\mathbf{B}_0$.  Electrodynamics requires that they should attract each other and merge. Why is this not happening in the auroral downward current region?\n\nOne might argue that the acceleration of electrons in the ionosphere below observation altitude is probably highly localized, depending on processes in the resistive ionospheric plasma.  Therefore there would be no need for upward escaping electrons to merge laterally. This argument is invalid because they transport current. Lorentz attraction forces the currents to approach each other to form a broad unstructured downward current sheet. This is, however, inhibited by the strong main auroral geomagnetic field $B_0$. The argument that this should also happen in the upward current region fails because the current sheet there is broad by its origin from the tail reconnection site.\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.75\\textwidth,clip=]{fig2.jpg}}\n\\caption{Full sequence of FAST measurements across dow-up-auroral current system on 02-01-1997  \\citep[after][]{treumann2011}. {$(a)$ Magnetic field component $b_\\perp$ transverse to main field $B_0$, $(b)$ electric field fluctuation wave form $\\delta E$, $(c)$ low frequency electric fluctuation spectrum, $(d)$ high frequency electric spectrum, showing emission of auroral kilometric radiation bands $(e)$ electron energy spectrum, $(f)$ electron flux versus pitch-angle, $(g)$ ion energy flux, $(h)$ ion flux versus pitch-angle. The most intriguing part here is the smoothness of the magnetic signature of the upward current in its linear course showing that the upward current is a broad homogeneous current sheet. The downward current region (DCR) flanks the upward current region to both its sides, is comparably narrow in its spatial extent, and exhibits strong current and flux variations. This is seen in the electron flux panels $(e-f)$. Downward fluxes around few keV are relatively smooth indicating a relatively stable tail reconnection over observation time, upward fluxes have maximum at low energy and are highly variable in time and space.}The magnetic field being the integrated response to the spatial flux fluctuations exhibits a much smoother course which is inverse with respect to that of the upward current thus indicating the reversed current direction. Note the low energies of the upward electron fluxes in panel five as well as the clear separation of upward and downward fluxes as seen in the left part.} \\label{fig2}\n\\end{figure*}\n\nSince anti-parallel currents reject each other the transition region between upward and downward currents is quiet. {This is seen in panels $(b - f)$ of Fig. \\ref{fig2} and} is in contrast to our previous investigation where we assumed that reconnection would happen there between parallel kinetic Alfv\\'en waves. The Lorentz force between two equally strong sheet currents $\\mathbf{J}_\\|$ is \n\\begin{equation}\n\\mathbf{J}_\\| \\times \\mathbf{b}_\\perp= -\\nabla_\\perp b_\\perp^{2}\/\\mu_0  \n\\end{equation}\nwhere $\\mathbf{b}_\\perp$ is the magnetic field between the two currents {(in the following we suppress the index $\\perp$ on the magnetic field component $\\mathbf{b}$)}, and  $\\nabla_\\perp$ refers to the gradient in the direction from current sheet to current sheet. The current consists, however, of gyrating electrons whose Lorentz force is the cross product of the azimuthal gyration speed times the very strong stationary field with gradient $\\nabla_c$ taken only over the gyro-radius $r_{ce}=v_{e\\perp}\/\\omega_{ce}$ of the electrons. For a separation of the sheet current exceeding the electron gyroradius and low current density the sheet currents will approach each other only on very long diffusive time scales of no interest. For a thin current sheet only a few gyroradii thick the condition for this time to be long is simply that the electron inertial length exceeds the gyroradius or\n\\begin{equation}\nv_{e\\perp}\/c\\ll\\omega_{ce}\/\\omega_e\n\\end{equation}\nwhich holds under very weak conditions in the topside auroral ionosphere. This implies that downward current sheets separated by say an electron inertial length $\\lambda_e=c\/\\omega_e$ will not merge under no circumstance. They remain separated over the observational spacecraft crossing time scales. It is their secondary magnetic field $\\mathbf{b}$ which will undergo reconnection without affecting the ambient magnetic field which just serves as guide field directed along the current flow. This distinguishes topside reconnection from other guide field mediated reconnection. {One may note, however, that sometimes in simulations when plasmoids form \\citep[cf., e.g.,][and others]{malara1991} parallel currents apparently do not attract each other. This happens, when the Lorentz force between the parallel currents does not overcome the mechanical forces exerted by the massive plasmoids, i.e. forces induces by their inertia and impulse. The Lorentz force is then too weak to push the parallel currents toward each other, an effect which can also be observed in highly turbulent plasmas. Such cases, when the currents remain close enough will, by the mechanism proposed below, be subject to reconnection between the opposing fields of the parallel current, leading to a cascade in the current structure towards smaller scales and to local reconnection as a main dissipation process of magnetic and turbulent energy.}\n\n\n\\subsection{Kinetic (shear) Alfv\\'en waves}\nIn the complementary wave picture of field-aligned currents in the auroral region, the current is carried by (kinetic) Alfv\\'en waves in the frequency range well below the local ion-cyclotron frequency. In addition, a large number of low-frequency electromagnetic waves are known to be present there \\citep{labelle2002}. We are in the downward current region with highly sheared upward particle flow along the magnetic field consisting of moderately fast electrons and much slower ions. Such flows are capable of generating Alfv\\'en waves \\citep{hasegawa1982} on perpendicular scales of the ion inertial length $\\lambda_i=c\/\\omega_i=\\lambda_e\\sqrt{m_i\/m_e}$ and below and long wavelength parallel to the ambient field. For the current-carrying electrons such waves are about stationary magnetic structures. \n\nThese Alfv\\'en waves cannot be body waves like in the solar wind \\citep{goldstein2005,narita2020} because they are strictly limited to the narrow field-aligned current sheets. Since they propagate in the strong auroral geomagnetic field, they are rather different from the usual kind of kinetic Alfv\\'en waves which one refers to in solar wind turbulence \\citep{goldstein2005}, where the magnetic field ist very weak and the turbulence is dominated by the mechanics of the flow \\citep{maiorano2020}. There the ion-temperature plays an important role imposing kinetic effects on the wave. \n\nUnder auroral conditions, in particular close to the ionosphere, the magnetic field is so strong that thermal ion effects on the wave are barely important. Their mass effect enters the Alfv\\'en speed. Instead, however, under those conditions electron inertia on scales $\\lambda_i\\sim\\Delta\\gtrsim\\lambda_e= c\/\\omega_e$ below the ion scale comes into play. For sufficiently narrow field-aligned current sheets of width the order of inertial scales, the field does not allow the electrons to leave their flux tube unless they have large perpendicular moment. Field-aligned electrons remain inside their gyration flux tube, and the currents cannot react to merge with neighbouring parallel current sheets. The Lorentz force on the field-aligned current in the magnetic field of its neighbour is not strong enough to move the currents. In this case the kinetic Alfv\\'en waves transporting the currents in pulses become inertially dominated with dispersion relation\n\\begin{equation}\n\\omega^2= \\frac{k_\\|^2V_A^2\\big(1+k_\\perp^2\\rho_i^2\\big)}{1+k_\\perp^2\\lambda_e^2}\n\\end{equation}\nwhere $\\rho_i$ is some modified ion gyro-radius \\citep[cf., e.g.][]{baumjohann1996} containing kinetic temperature contributions. For the cold ions in the topside auroral ionosphere the term containing $\\rho_i$ in the numerator vanishes. The kinetic Alfv\\'en wave under those conditions becomes an inertial or shear wave. It propagates at a reduced though still fast speed along the magnetic field, being of very long parallel wavelength. It also propagates slowly perpendicular to the magnetic field at short wavelength $\\lambda_\\perp\\sim\\lambda_e\/2\\pi$. It is, in principle, this wave which carries the current. Thus the current is not stationary on time scales long compared to the inverse frequency $\\Delta t>\\omega^{-1}$ but can be considered stationary for shorter time scales  $\\omega\\Delta t < 1$. \n\nThe above dispersion relation, neglecting the ion contribution in the numerator, gives the well known relations for the parallel and perpendicular energy transport in the shear wave\n\\begin{equation}\n\\frac{\\partial\\omega}{\\partial k_\\|}=\\frac{V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}},\\qquad \\frac{\\partial\\omega}{\\partial k_\\perp}=-\\frac{\\partial\\omega}{\\partial k_\\|}\\frac{k_\\|}{k_\\perp} \\frac{1}{\\big[1+1\/(k_\\perp\\lambda_e)^{2}\\big]}\n\\end{equation}\nEnergy transport in the perpendicular direction is smaller than parallel by the ratio of wave numbers.\n\nRepeating that we are in the downward current upward electron flux region causality requires that the upward electrons carry information from the ionosphere to the magnetosphere. Hence the kinetic Alfv\\'en waves in this region also propagate upward being produced in the topside by the transverse shear on ion-inertial scales below $\\lambda_i$. \n\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.8\\textwidth,clip=]{fig3.jpg}}\n\\caption{Sequence of downward (top)  and upward (bottom)  auroral electron fluxes observed by FAST on 02-07-1997 in the topside auroral region when crossing a substorm aurora \\citep[after][]{treumann2011a}. The sequence distinguishes nicely between the intense  downward electron fluxes at energies $\\epsilon_e\\sim 10$ keV and downward fluxes at energies $\\epsilon_e<0.1$ keV. Upward fluxes are confined to narrow spatial regions, downward fluxes are distributed over a much wider domain. In the transition regions between both domains one observes flux mixing which indicates that the current systems are not simply two dimension and also that there are many overlapping flux and current sources which the one-domensional path of the spacecraft does not resolve spatially. } \\label{fig3}\n\\end{figure*}\n\n\\section{Reconnection under auroral conditions}\nAssuming stationarity of the field aligned current $\\mathbf{J}=J_\\|\\mathbf{B}_0\/B_0$ in two adjacent but separated parallel sheets and assuming, for simplicity, that the two currents are of equal strength, reconnection will  occur in the central region of separation of the current sheets. (Figure \\ref{fig4} shows a two-cross section schematic of the downward current-field configuration for two closely spaced current sheets.)  Here the two magnetic fields of the field-aligned currents are antiparallel. According to the above discussion, we are in the downward current region \\citep[as in the case of our previous radiation model][]{treumann2011}. In fact an analogue model would apply to the upward current region. The current flows in direction $z$; the direction of $y$ is longitudinal (eastward), $x$ is latitudinal (northward). If the sheet ist extended mostly in $y$ the antiparallel magnetic fields are in $y$ along the sheets. They will touch each other and reconnect between the two sheets thereby forming reconnection X-points with field component $\\pm b_x$ and extended magnetic field free electron exhausts in $x$ and in $y$, which contain the local main-field parallel reconnection electric field, and accelerate electrons along the ambient field. Tjhese exhausts will propagate along the main field together with the wave. Plasmoids might also form in the separation between the sheets perpendicular to the ambient field,  and the presence of the strong ambient field will impose electron gyration and scatter of electrons  causing secondary effects like bursts of field aligned energetic electrons. Moreover, the exhausts will serve as source of radiation and various kinds of electrostatic instabilities (for instance Bernstein modes). \n\n{There are two essential differences between this type of reconnection and ordinary reconnection models. The first is that the ambient field serves as a strong guide field which, as noted, inhibits the adjacent field-aligned current to merge. The second is that initially the set-up lacks the presence of any central current sheet which in conventional models of reconnection is crucial and imposed from the beginning. In topside reconnection such a current flowing along the magnetic field inside the separation region would imply a return current which, however, is absent. Return currents flow through the bottomside ionosphere and close in the upward current region. Nevertheless formally a fictitious return current forms locally and temporarily in the centre of the separator, which can be assumed as distributed over the separating region and belonging to the antiparallel fields $\\pm b$.}\n\n{This current builds up dynamically and locally during the reconnection process itself when the two kinetic Alfv\\'en waves slowly move perpendicular. This is a difficult dynamical problem in that reconnection will set on when the encountering magnetic fields exceed some threshold. Since electrons in this region are magnetized by the strong ambient field, they gyrate but do not take notice of the weak field $b$ of the kinetic Alfv\\'en waves which is transported across the separating region by the perpendicular phase and group speeds of the waves to get into contact and merge.} \n\n{The reconnection process is thus solely between the two waves, primarily not affecting the ambient field and not based on any real central primary current sheet. Observations so far do not resolve the magnetic nor the particle effects of such fictitious return currents though some of the structure seen in the low energy electron fluxes in Fig. \\ref{fig3} could be interpreted as such without proof. In fact, in order to avoid formation of the fictitious return current, which would imply that this current would be equally strong in the gap between the current sheets, reconnection is required over the full length of half a wavelength along $z$. Thus it necessarily generates elongated field-aligned vertical X-lines and electron exhausts in $z$.}\n\n \\subsection{First step}\nAll these effects are of vital interest. However, one particularly interesting question concerns the dissipation produced by this kind of reconnection. It is frequently argued that it leads to sliding of main-field field lines. In order to understand such a mechanism one needs to know the anomalous resistance caused by reconnection. In electrodynamic formulation, reconnection is conventionally dealing solely with the merging and energy transfer of fields. The microscopic mechanism of energy transfer is accounted for in the transport coefficients. Hence the appropriate way of inferring their value is referring to the electromagnetic energy exchange. This leads to the application of Poynting's theorem\n\\begin{equation}\n\\frac{\\partial b^2}{\\partial t}=-\\mu_0\\,\\eta_{an}J_\\|^2-\\nabla_\\perp\\cdot\\big(\\mathbf{E}_\\|\\times\\mathbf{b}\\big)-\\nabla_\\perp\\cdot\\big(\\mathbf{E}_{rec}\\times\\mathbf{B}_0\\big)\n\\end{equation}\nwhere the contribution of the electric field to the left-hand side is neglected as it is relativistically small, and $\\mathbf{b}$ is the magnetic field of the field-aligned current. It allows for a convenient estimate of the anomalous resistivity $\\eta_{an}$ in reconnection without going into any microscopic detail of the mechanism of its generation. The electric field in this expression is along the ambient magnetic field, essentially being the electric field of the kinetic Alfv\\'en wave. Estimates of this parallel field have been provided by \\citet{lysak1996} and were taken as the important agent for accelerating auroral electrons. \n\nThe above expression shows that reconnection in this case is a two-step process. In the first step the parallel field $E_\\|$ along the ambient magnetic guide field sets up reconnection. In the second step the reconnection electric field $E_{rec}$ and exhaust have evolved. The cross-product with the main magnetic field then modifies the dynamics of the exhaust. \n\n\\subsection{Anomalous collision frequency}\nIn this subsection we are not interested in this effect here as it is overwritten once reconnection really sets on but enters in the determination of the perpendicular inflow speed.  It causes it to be different from tailward reconnection. Instead we proceed to an estimate of the anomalous collision frequency.  \n\nThe parallel electric field $E_\\|$ of the kinetic Alfv\\'en wave plays an important role in the first step of the topside reconnection process. Since this field is parallel to the ambient geomagnetic field $\\mathbf{B}_0$, the cross product with the wave magnetic field is responsible for the two current-sheet magnetic field components $\\pm\\mathbf{b}$ to approach each other in the region between the sheets. Hence, referring to this fact, the second term on the right can be expressed through the perpendicular velocity $\\mathbf{V}_\\perp=\\mathbf{E}_\\|\\times\\mathbf{b}\/b^2$, and we have\n\\begin{equation}\n\\nabla_\\perp\\cdot\\big(\\mathbf{E}_\\|\\times\\mathbf{b}\\big)=\\nabla_\\perp\\cdot\\big(\\mathbf{V}_\\perp b^2\\big)\n\\end{equation}\n\nIn order to get some information about the perpendicular velocity $\\mathbf{V}_\\perp$ which according to our coordinate system points to the centre of the region which separates the two current sheets, i.e. along $y$, we refer to the wave picture, noting that these pictures are equivalent: the field-aligned current $J_\\|$ is carried by (upward topside ionospheric) electrons, on the other hand these electrons are transported (or pushed) by the kinetic Alfv\\'en wave. {In fact, of course, $V_\\perp$ is counted from each of the two parallel currents as pointing to the center of the separating sheet. It thus in our water-bag model changes abruptly sign in the center where due to the two antiparallel magnetic fields which collide there a fictitious weak return current of strength $j_\\|\\approx 2b\/\\mu_0\\delta$ arises, with $\\delta$ the fictitious width of this narrow current layer which we do not explicitly consider. The simplest is in our water-bag model to assume that for closely separated parallel current sheets we have essentially \n$\\delta\\to\\alpha\\Delta$, with $\\alpha\\lesssim1$, and a return current distributed over almost the entire separation width. One should also keep in mind that any field-aligned current carried by the Alfv\\'en wave is a current pulse with both $E_\\|$ and $b$ changing direction (oscillating) over half the wavelength. Thus $V_\\perp$ for each current pulse on one ambient field line has same sign over the full wave length while maximizing twice. On using this equivalence the perpendicular velocities $\\pm V_\\perp$ are just the perpendicular phase speeds of the kinetic Alfv\\'en waves on the two adjacent current sheets}\n\\begin{equation}\nV_\\perp\\sim\\frac{\\omega}{k_\\perp}\\approx \\frac{V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|}{k_\\perp}\\ll V_A\n\\end{equation}\n{This velocity apparently diverges for $k_\\perp\\to0$ which, however, is not the case because the kAW is a surface wave being defined only for $k_\\perp\\neq0$ while becoming a body wave for $k_\\perp\\to0$ carrying no current anymore. Its most probable wavenumber is about $k_\\perp\\lambda_e\\sim1$ attributing to the parallel phase and group velocity $\\sim V_A\/\\sqrt{2}$ and a perpendicular group velocity $\\sim-V_A(k_\\|\\lambda_e)\/2^{3\/2}$. However, since $V_\\perp$ indeed transports not only the field but also energy, one may argue that the use of the latter expression would be more appropriate than the phase speed. Since this does not make any big difference for our purposes, we in the following for reasons of simplicity understand $V_\\perp$ as phase speed. For more precise expressions one may replace it in the following with the perpendicular group speed}. \n\nThe velocity $V_\\perp$ is small because $k_\\|\\ll k_\\perp$, i.e. the kinetic Alfv\\'en wave is long-wavelength parallel to the ambient field but of short perpendicular wavelength, a very well-known property. Moreover, $V_\\perp(z)$ may vary along the ambient field but, in the frame of the wave, which corresponds to a water-bag model, is constant in the perpendicular direction. Hence, of the above vector product just remains the variation of the magnetic field $b(x)$ over the distance between the two current sheets. This insight enables us to rewrite Poynting's equation as\n\\begin{equation}\n\\frac{\\partial b^2}{\\partial t}\\approx -\\mu_0\\,\\eta_{an}J_\\|^2-V_\\perp\\frac{\\partial b^2}{\\partial x}\n\\end{equation}\nwhich, assuming a stationary state, enables us to estimate the anomalous resistivity of stationary reconnection (in the wave frame) where the inflow of magnetic energy attributed by the current, i.e. the field-aligned electron flux whose origin is found in reconnection in the magnetotail, is balanced by anomalous energy transfer to the plasma in the region separating the two current sheets. Putting the left-hand side to zero we thus find that in this kind of topside reconnection the anomalous resistivity is bound from above as\n\\begin{equation}\n\\eta_{an}\\lesssim \\frac{4V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|}{k_\\perp}\\frac{b^2}{\\mu_0 \\Delta J_\\|^2}{\\approx \\frac{2V_Ab^2}{\\mu_0\\Delta J_\\|^2} \\frac{k_\\|^2\\lambda_e^2}{(1+k_\\perp^2\\lambda_e^2)^{3\/2}}}\n\\end{equation}\nwhere $\\Delta$ is the spatial separation of the two field-aligned current sheets, and we have taken into account that each of the two identical current layers contributes a field $b$. {The second part of this expression makes use of the perpendicular group speed. This resistivity is small as $k_\\|^2\\lambda_e^2\/\\Delta$ but finite. It gives rise to a finite diffusion coefficient that can be interpreted as an anomalous diffusivity for the ambient magnetic field in the auroral topside ionosphere, caused by topside reconnection between anti-parallel current sheets in the downward current region. We might note at this occasion that the restriction to the downward current region is motivated by the observation of narrow current sheets in the downward current region. Observations do not suggest that similarly narrow current sheets evolve in the upward current region. If this would be the case, the same arguments would apply there, causing reconnection and a similar anomalous resistivity.}  \n\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.75\\textwidth,clip=]{fig4.jpg}}\n\\caption{Schematic of the field configuration between two parallel field aligned flux tubes in the downward current region. \\emph{Left}: Geometry along the ambient field. {Currents are in red. Included would be the (red dashed) fictitious return current which locally would correspond to the antiparallel wave magnetic fields $\\pm b$. This current would be local over the wavelength of the inertial Alfv\\'en wave. In any stationary reconnecting current picture it would be this current whose magnetic field reconnects. However, here this current does not exist in the exhaust. It is completely reconnected and gives rise to the reconnection electric field $E_z$ instead (dashed red) in the exhaust along the main magnetic field.  Electrons are directly accelerated by it along $\\mathbf{B}_0$.} \\emph{Right}: Reconnection geometry with perpendicular velocity $\\mathbf{V}_\\perp$, field free exhaust, reconnection fields $\\pm b_x$ indicated, and $E_z$. {The anomalous collisions caused in the exhaust volume also permit for weak diffusion of the ambient field. This may cause what is believed to be magnetic field diffusion, a  very slow process compared to the wave\/current induced spontaneous reconnection.}} \\label{fig4}\n\\end{figure*}\n\nWhat concerns the spatial separation of the current sheets (see Fig. \\ref{fig3}), the best available observations (FAST) do not resolve any single sheets; it can however be assumed that their scales are the order of or below the ion-inertial length, such that $\\Delta\\lesssim \\lambda_i\\sim$ several to many $\\lambda_e$. This may overestimate the real value but has been accounted for in writing the expression as an upper limit. Determination of the anomalous resistivity thus requires knowledge of the field aligned current density, current sheet separation, and the transverse magnetic field component of the sheet current. We then can estimate the anomalous collision frequency $\\nu_{an}= \\eta_{an}\\epsilon_0\\omega_e^2$ in this kind of reconnection\n\\begin{equation}\n\\nu_{an}\\lesssim \\frac{V_A\/\\alpha^2\\lambda_e}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|\\Delta}{k_\\perp\\lambda_e }\n\\end{equation}\nwhere we used that $J_\\|\\approx 2b\/\\alpha\\Delta$. Note that $V_A=B_0\/\\sqrt{\\mu_0 m_i N}$ is based on the ambient magnetic field and plasma density. This simple estimate shows that reconnection in this case can, under stationary condition be described as being equivalent to a diffusive process based on the anomalous collision frequency which is provided by the merging of the transverse magnetic fields of the two neighbouring field-aligned current sheets. Since the related diffusivity is felt in the entire region it is remarkable that it could effect also the main ambient guide field. In other words, topside reconnection could become responsible for diffusion of the main magnetic field lines in a locally restricted domain possibly causing effects on a larger scale in the auroral region.   \n\nReal reconnection will not occur between field-aligned current sheets of same strength. Thus the above resistivity respectively the collision frequency must be reduced by another factor proportional to the involved current and field fractions. \n\n\\subsection{Second step: Reconnection electric field}\n{So far we just investigated the energy balance in order to obtain an anomalous collision frequency in this kind of reconnection. Reconnection however manifests itself in X points generating transverse magnetic fields and in addition electric fields. Since there is no primary return current flowing, it cannot be used as input into the two-dimensional reconnection equation for the vector potential $A_z$\n\\begin{equation}\n\\nabla^2 A_z=-\\mu_0j_z(x), \\qquad \\nabla= (\\partial_x,\\partial_y,0), \\qquad j_z(x)= -2\\epsilon b(x)\/\\mu_0\\Delta\n\\end{equation}\nwithout prescribing the built-up of the central current profile $j_z(x)$, which is possible only when assuming that the $b$ is independent of $x$, in which case it provides the usual stationary tearing mode solution \\citep[see, e.g.,][]{schindler1974} rewritten for electrons alone. Under these simplifying restrictions the two components of the reconnected magnetic field including the X point are given by $\\mathbf{b}=(\\partial_yA_z,-\\partial_xA_z,0)$, which to refer to suffices for our qualitative considerations. The a priori assumption of a return current is, however, incorrect. On the topside there may weak local return currents exist filling the separations between the narrow downward current sheets, but the main return current flows in the upward current region and is distributed over a wide domain. Hence just a fraction $\\epsilon$ of return current can flow in the gap, as included in the last expression. The electric field in this case primarily has only one component, which is along the main field and is given by $E_z=-\\partial_t A_z-\\nabla U$ where $U$ is the scalar electrostatic gauge potential which may occur if an inhomogeneity exists or the system is not ideally symmetric. This field adds to the field aligned kinetic Alfv\\'en wave electric field and contributes to electron acceleration. It is the wanted reconnection electric field and can be much larger than the small linear wave electric field. Unfortunately its precise knowledge  requires solution of the equation for the vector potential $A_z$ and some interpretation of the time derivative operator. The latter can be transformed into a spatial derivative $\\partial_t= \\pm\\mathbf{V}_\\perp\\cdot\\nabla$, still requiring the solution $A_z(x,y)$. }\n\n{The important conclusion in the case of topside reconnection is rather different from usual reconnection. It tells that the exhaust is, over half the wavelength of the inertial Alfv\\'en wave free of wave magnetic fields $b$, while being bounded by the reconnected wave fields $\\pm b_x$. The exhaust instead contains the reconnection electric field, by being along the main field, does directly contribute to acceleration respectively deceleration of electrons  (and also ions) along the main magnetic field, one of the most important and still unresolved problems in auroral physics. There acceleration is attributed to a variety of waves, reaching from kinetic Alfv\\'en through whistlers and several electrostatic waves to electron and ion holes. Except for the latter nonlinear structures, all wave electric fields are quite weak, and in addition fluctuate. Acceleration thus becomes a second order process.} \n\n{In case of the topside reconnection, a mesoscale first-order electric field $E_z$ is produced which directly accelerates particles, depending on its direction along the main field. Moreover, the source of the accelerated particles is the gap region between the two current sheets, the so-called exhaust, such that the kinetic Alfv\\'en wave electric field and the reconnection electric field do barely interfere. Hence the full strength of the reconnection exhaust field acts accelerating. One may thus conclude that topside reconnection, if it takes place, will substantially contribute to auroral particle acceleration. \n }\n\n{In order to circumvent the above named difficulty of calculating $A_z$ and to obtain an estimate of the reconnection electric field, we may return to the induction equation in its integral form where the electric field is given by the integral over the surface of the reconnection site\n\\begin{equation}\n\\oint \\mathbf{E}\\cdot d\\mathbf{s}=-\\frac{d\\Phi}{dt}= -\\frac{d}{dt}\\int\\mathbf{b}\\cdot d\\mathbf{F}\n\\end{equation}\nand the right-hand side is the exchange of magnetic flux in the reconnection process within the typical time $dt=\\tau_{rec}$. This time is not necessarily the same as the anomalous collision time. The magnetic flux is given by \n$\\Delta\\Phi\\approx 4\\pi b\\Delta\/k_\\|$. The line integral over the boundary of the reconnection site becomes $\\approx4\\pi E_z\/k_\\|+2\\Delta \\delta E_x$. Under ideally symmetric conditions the second term would vanish because the two contributions of the $x$ integration would cancel out. If some asymmetry is retained then a finite component $\\delta E_x$ arises. Taken these together yields dimensionally (not caring for the signs)\n\\begin{equation}\n4\\pi E_z\/k_\\|+2\\Delta \\delta E_x \\approx 4\\pi b\\Delta\/k_\\|\\tau_{rec}\n\\end{equation}\nNeglecting the small second term on the left then gives a simple order of magnitude estimate of the reconnection electric field\n\\begin{equation}\nE_z\\approx \\frac{b\\Delta}{\\tau_{rec}}\n\\end{equation}\nwhich could have been guessed from the beginning. This contains the reconnection time $\\tau_{rec}$ which  so far is undetermined. It can be taken for instance as the above derived anomalous collision time $\\tau_{an}=\\nu_{an}^{-1}$. Below we derive another characteristic time. Which one has to be chosen, cannot decided from these theoretical order of magnitude  estimates. It is either due to observation or numerical simulations.} \n\nThe small additional term $2\\Delta\\delta E_x =-U$ is a potential field produced by a possibly present asymmetry between the original current sheets or some gradient in the particle density. Such a gradient can be produced, if a substantial part of the electron component in the gap is accelerated away along the main field, causing a dilution of plasma in the exhaust. Being perpendicular to the magnetic fields $\\mathbf{B}_0$ and $\\mathbf{b}$ it leads to weak shear motions and circulation of the electrons inside the gap-exhaust region, which should observationally be detectable. \n\n\n\\subsection{Reconnection time}\nIn the above we have made use of the notion of reconnection time $\\tau_{rec}$. Here we attempt a clarification of this time. Topside reconnection will not be stationary. It should vary on the time scale of the kinetic Alfv\\'en frequency respectively moving together with the latter along the magnetic field. This motion should mainly be upward since causality requires that the wave transports information back upward with the upward moving electrons in the downward current region. It will thus be modulated and lead to quasi-periodic acceleration and generate medium energy electron bursts ejected from the local electron exhaust reconnection region along the sheet current magnetic field. These bursts flow perpendicular to the ambient field, start gyrating and immediately become scattered along the ambient field spiralling mainly upward into the weak ambient field region. Their pitch-angle distribution should obey a well defined downward loss-cone. \n\nWith the above estimate of the anomalous resistivity in this kind of reconnection, we can proceed asking for the typical reconnection time scale. For this purpose we return to Poynting's full theorem and take its variation with respect to the stationary state, indexing the latter with 0 while keeping the slow perpendicular velocity $V_\\perp$ fixed but varying the resistivity. We need to express the parallel current through the resistivity. This can be done via the electric field $E_\\|$ to obtain\n\\begin{equation}\nJ_\\|^2=\\eta^{-2}E_\\|^2=\\eta^{-2}b^2V_\\perp^2\n\\end{equation}\nThis procedure, after some straightforward and simple algebra and rearranging, leads to the following expressions\n\\begin{eqnarray}\n\\frac{d(\\delta b)^2}{dt}\\equiv\\Big(\\frac{\\partial}{\\partial t}+V_\\perp\\nabla_\\perp\\Big)(\\delta b)^2&=&- 2\\mu_0J_{\\|0}^2\\delta\\eta\\\\\n\\delta\\eta&=&-\\frac{V_\\perp}{\\mu_0J^2_{\\|0}\\Delta} (\\delta b)^2\n\\end{eqnarray}\nand we obtain dimensionally for the typical time of reconnection\n\\begin{equation}\n\\tau_{rec}\\sim\\frac{2\\Delta}{V_\\perp}\n\\end{equation}\nThis seems a trivial result, but it tells that reconnection is a process which annihilates the excess magnetic field which is provided by the perpendicular inflow under the condition that we are close to a stationary state. This time can be compared with the times of energy flow in the shear Alfv\\'en wave. Since clearly $V_\\perp\\approx\\partial\\omega\/\\partial k_\\perp$, one obtains\n\\begin{equation}\n\\tau_{rec}\\approx\\frac{2\\Delta}{V_A}\\frac{k_\\perp}{k_\\|}\\frac{\\big(1+k_\\perp^2\\lambda_e^2\\big)^\\frac{3}{2}}{k_\\perp^2\\lambda_e^2}\n\\end{equation}\na time the length of which depends essentially on the spacing of the current sheets. Since $V_A$ is large, there will be a balance between the spacing and the domain of the kinetic Alfv\\'en wave spectrum which allows reconnection to occur in the topside. Let the vertical topside width be $L_z$ and the Alfv\\'en time $\\tau_A=L_z\/V_A$ then we have the condition \n\\begin{equation}\n\\frac{\\tau_{rec}}{\\tau_A}\\approx \\frac{2\\Delta}{L_z}\\frac{k_\\perp}{k_\\|}\\frac{\\big(1+k_\\perp^2\\lambda_e^2\\big)^\\frac{3}{2}}{k_\\perp^2\\lambda_e^2}< 1\n\\end{equation}\nfor reconnection to occur in topside parallel field-aligned current sheets. This essentially is a condition on the spacing $\\Delta$ of the sheets, meaning that \n\\begin{equation}\n\\frac{\\sqrt{32}\\,\\Delta}{L_z}< \\frac{k_\\|}{k_\\perp}=\\frac{\\lambda_\\perp}{\\lambda_\\|}\\ll 1\n\\end{equation}\nAny current sheet separation is strictly limited. Since it must be larger than the upward electron gyro-radius we have $\\Delta>r_{ce}$. Both conditions are easily satisfied.\n\n\n\n\\subsection{Conclusions}\n{In the present letter we propose that reconnection might occur not only in given current sheets but also in the topside ionosphere-magnetosphere auroral transition region  where the main magnetic field is very strong, almost vertical, and directly connects to the tail reconnection region. It serves  as a guide for any particle flow exchange between the topside ionosphere and the tail plasma sheet, exchange between low frequency electromagnetic waves (in our case kinetic Alfv\\'en waves) trapped in flux tubes and the accompanying field-aligned current sheets, and ultimately as an inhibitor for the field-aligned parallel current sheets to merge. This  enables reconnection in the gap between the current sheets between the oppositely directed magnetic field of the sheets respectively the kinetic Alfv\\'en wave magnetic fields.} \n\n{Dealing with reconnection, one is not primarily interested in the change of magnetic topology but in energy transformation from magnetic into kinetic, diffusion of plasma and magnetic field across the reconnection region, generation of electric fields, and ultimately selective particle acceleration as these are the observed effects. }\nThe generality of reconnection is not the best argument. The decades old claim that reconnection converts magnetic energy into mechanical energy is no fundamental insight; in all processes involving reconnection, the main energy is stored in the basic mechanical motion and by no means in the magnetic field. This motion, convection in inhomogeneous media with boundaries, like the magnetotail or the magnetopause, or turbulence necessarily produces currents and transports magnetic fields to let them get into contact. The amount of energy released by reconnection is in all cases just the minor electromagnetic part, a fraction of the mechanical energy. \n\n{Topside reconnection is expected predominantly in the downward current region, which observationally seems to be highly structured, consisting of several adjacent parallel current sheets. Similar  conditions may also occur in the upward current region though no such structuring is obvious from observations. If it exist, then the physics will be similar. We have shown that topside reconnection is possible, generates a elongated field-aligned regions (exhausts) where the fields of parallel current sheets merge, anomalous collisions are generated, energy is exchanged and dissipated, and most important a first order reconnection electric field $E_{rec}$ is produced in the exhaust along the ambient magnetic field but  restricted to the gap region between the current sheets. This field is capable of accelerating electrons along the main field, as is most desired by all auroral physics. Here it comes out as a natural result of topside reconnection.} Topside reconnection generates parallel electron beams, it lifts the escaping electrons in the exhaust into an elevated parallel energy level. These beams then cause a wealth of auroral effects in the environment and when impinging onto the upper ionosphere. Acceleration of electrons by the reconnection electric field leaves behind an electron depleted exhaust mainly containing only an anisotropic electron component whose pitch angle distribution peaks at perpendicular energies.  \n\n{It is instructive to briefly inspect Fig. \\ref{fig3}. It shows the downward (upper panel) and upward (lower panel) electron fluxes. In addition to the temporally\/spatially highly structured fluxes, still obeying the spatial differences between the downward and upward current regions imposed by the tail-source of the downward fluxes, resulting  from  variations in tail-reconnection, or several tail-reconnection sites, one occasionally observes the simultaneous presence of upward and downward fluxes in the downward current region. One particular case it at $t\\approx 60$ s. The upward electron fluxes maximize below $\\sim0.1$ keV. Simultaneously a banded flux of downward electrons with central energy $\\sim 0.3$ keV appears in the upper panel. This event  is indicated as flux mixing. It could also be understood as acceleration of electrons resulting from the local reconnection in the gap between current sheets. }\n\nAside of acceleration, radiation generation may be taken as signature of topside reconnection. Radiation is preferrably generated by the electron cyclotron maser mechanism. It requires low electron densities, strong magnetic fields, and a rather particular particle distribution with excess energy in its component perpendicular to the ambient magnetic field \\citep{sprangle1977,melrose1985}. Such a state in dilute plasmas lacks sufficiently many electrons for re-absorbing the spontaneously emitted radiation while the excited state causes inversion of the absorption coefficient. These conditions allow for the plasma to become an emitter \\citep{twiss1958,schneider1959,gaponov1959} by the electron cyclotron maser mechanism \\citep{wu1979} based on a loss-cone distribution \\citep{louarn1996}. It requires weakly relativistic electrons \\citep[see][for reviews]{melrose1985,treumann2006} and a low density electron background embedded into a strong field. It nicely comes up for the weak auroral kilometric background radiation but fail explaining the intense narrow band observed and drifting emission seen in panel $d$ of Fig. \\ref{fig2}. \n\nTo explain the latter, in earlier work we referred to electron hole formation \\citep{pottelette2005,treumann2011}. Hole models favourably apply to electron depleted exhausts in topside reconnection where densities become low \\citep[see, e.g.][]{treumann2013} and the remaining trapped electron component  maximizes at perpendicularly speeds having large anisotropy. Intense narrow band drifting emissions in the frequency range 300-600 kHz may be a signature of topside reconnection in the strong main auroral field. They were originally attributed to Debye scale electrostatic electron holes \\citep{ergun1998b,pottelette1999} observed by Viking \\citep{deferaudy1987} and  FAST \\citep{carlson1998,ergun1998a,pottelette2005} but are to small-scale for radiation sources. Topside reconnection exhausts instead have dimensions along the magnetic field of half a kinetic Alfv\\'en length and transverse scales of few ion inertial lengths $\\lambda_i$ or $\\sim 100\\lambda_e$. Such scales can host and amplify one or more radiation wave lengths.\n\nOf course, details of this process should be developed both analytically as far as possible, and by numerical simulations. If confirmed, this mechanism would also map to any astrophysical moderately or strongly magnetized object with appropriate modification. \n\nThe present qualitative considerations which we spiced with a few simple estimates based on energy conservation arguments just propose that reconnection in the topside auroral ionosphere is a process which has so far been missed and probably is that mechanism which releases the largest amount of so-called magnetically stored energy available and from the smallest spatial regions. Reconnection in much weaker fields like in turbulence and broad current sheets will be substantially less efficient because of the weakness of the reconnecting magnetic fields. Nevertheless in very large extended systems with reconnection proceeding on the microscales  \\citep{treumann2015} with the total number of reconnection regions very large, the emission measure is large as well, and radiation from reconnection may become a non-negligible signature even in weak fields. However, in very strong fields like those in magnetized planets and magnetized stars (predominantly neutron stars, white dwarfs but also including outer atmospheres of magnetized stars like the sun) reconnection following our argumentation may be more important than so far assumed. \n\n\n\\section*{Acknowledgments}\n This work was part of a brief Visiting Scientist Programme at the International Space Science Institute Bern. RT acknowledges the interest of the ISSI directorate as well as the generous hospitality of the ISSI staff, in particular the assistance of the librarians Andrea Fischer and Irmela Schweitzer, and the Systems Administrator Saliba F. Saliba. We acknowledge discussions with R. Nakamura, and Y. Narita. RT acknowledges the cooperation with R. Pottelette two decades ago  on the data reduction and the radiation and electron hole problems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCollections of interacting self-motile objects\nfall into the class of systems known as active\nmatter \\cite{1,2},\nwhich can be biological in nature\nsuch as swimming bacteria \\cite{3} or animal herds \\cite{4},\na social system such as pedestrian or traffic flow \\cite{5},\nor a robotic swarm \\cite{6,7}.\nThere are also a wide range of artificial active matter\nsystems such as self-propelled colloidal particles \\cite{8,9,10}.\nStudies of these systems have  generally focused on the case\nwhere the motile objects interact with either a  smooth or\na static substrate; however, the field is now advancing to a point where\nit is possible to ask\nhow such systems behave\nin more complex static or dynamic environments.\n\nOne subclass of active systems\nis a collection of interacting disks that undergo either run-and-tumble \\cite{11,12} or\ndriven diffusive \\cite{13,14,15} motion.\nSuch systems\nhave been shown to exhibit a transition\nfrom a uniform density liquid state\nto a motility-induced phase separated state\nin which the disks form dense clusters surrounded\nby a low density gas phase \\cite{9,10,11,12,13,14,15,16,17,18}.\nRecently it was shown that when phase-separated run-and-tumble disks are\ncoupled to a random pinning substrate, a transition to a uniform density liquid state\noccurs as a function of the maximum force exerted by the substrate \\cite{19}.\nIn other studies of run-and-tumble disks driven over an obstacle array by a dc driving force,\nthe onset of clustering coincides with a drop in the net disk transport since a large cluster\nacts like a rigid object that can only move through the obstacle array with difficulty; in\naddition, it was shown that the disk transport was maximized at an optimal\nactivity level or disk running time \\cite{20}.\nStudies of flocking or swarming disks that obey modified Vicsek models of\nself-propulsion  \\cite{21}\ninteracting with obstacle arrays\nindicate \nthat there is an optimal\nintrinsic noise level at which collective swarming occurs \\cite{22,23},\nand that transitions between swarming and non-swarming states can occur as\na function of increasing substrate disorder \\cite{24}.\nThe dynamics in such swarming models differ from those of\nthe active disk systems, so it is not clear whether the  same behaviors will occur across\nthe two different systems.\n\nA number of studies have already considered\nactive matter such as bacteria or run-and-tumble disks\ninteracting with periodic obstacle arrays \\cite{25} or\nasymmetric arrays  \\cite{26,27,28,29}.\nSelf-ratcheting behavior occurs for the asymmetric arrays\nwhen the combination of broken detailed balance and the substrate asymmetry\nproduces\ndirected or ratcheting motion of the active matter particles\n\\cite{30,31},\nand it is even possible to couple passive particles to the active matter particles\nin such arrays in order to shuttle cargo across the sample \\cite{29}.\nIn the studies described above, the substrate is static, and external driving\nis introduced via fluid flow or chemotactic effects; however, it is also possible\nfor the substrate itself to be dynamic, such as in the case of\ntime dependent optical traps  \\cite{32,33} or a traveling wave substrate.\nTheoretical and experimental studies of colloids in traveling wave potentials\nreveal a rich variety of dynamical phases,\nself-assembly behaviors, and directed transport \\cite{34,35,36,37,38,39,40}.\n\nHere we examine a two-dimensional system of run and tumble active\nmatter disks that can exhibit motility induced phase separation\ninteracting with a periodic quasi-one dimensional (q1D) traveling wave substrate.\nIn the low activity limit, the substrate-free system forms a\nuniform liquid state, while in the presence of a substrate,\nthe disks are readily trapped by the substrate minima\nand swept through the system by the traveling wave.\nAs the activity increases, a partial decoupling transition of the disks and the substrate\noccurs, producing a drop in the net effective transport.  This transition is correlated\nwith the onset of the phase separated state,\nin which the clusters act as large scale composite objects that cannot be transported\nas easily as individual disks by the traveling wave.\nWe also find that the net disk transport is optimized at particular\ntraveling wave speeds, disk run length, and substrate strength.\nIn the phase separated state we observe an interesting effect where\nthe center of mass of each cluster moves in the direction opposite to that in which\nthe traveling wave is moving, and we also find reversals\nto states in which the clusters and the traveling wave move in the same direction.\nThe reversed motion of the clusters arises due to asymmetric growth and shrinking rates\non different sides of the cluster.\nThe appearance of backward motion of the cluster center of mass\nsuggests that certain biological or social active systems can move against biasing\ndrifts by forming large collective objects or swarms.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig1.png}\n\\caption{Schematic of the system.\n  Red spheres represent the active run and tumble disks in a two-dimensional system\n  interacting with a periodic q1D traveling wave potential\n  which is moving in the positive $x$-direction (arrow) with a wave speed of $v_{w}$.\n}\n\\label{fig:1}\n\\end{figure}\n\n\n\\section{Simulation}\nWe model a two-dimensional system of $N$ run and tumble disks\ninteracting with a q1D traveling wave periodic substrate,\nas shown in the schematic in Fig.~\\ref{fig:1} where the substrate\nmoves to the right at a constant velocity $v_{w}$.\nThe dynamics of each disk is governed by the following overdamped equation of motion:\n\\begin{equation}\n\\eta \\frac{d{\\bf r}_i}{dt} = {\\bf F}^{\\rm inter}_i + {\\bf F}^m_i + {\\bf F}^{s}_i ,\n\\end{equation}\nwhere the damping constant is $\\eta = 1.0$.\nThe disk-disk repulsive interaction force ${\\bf F}^{\\rm inter}_i$\nis modeled as a harmonic spring,\n${\\bf F}^{\\rm inter}_i=\\sum_{j\\neq i}^N\\Theta(d_{ij}-2R)k(d_{ij}-2R){\\bf \\hat d}_{ij}$,\nwhere $R=1.0$ is the disk radius, $d_{ij}=|{\\bf r}_i-{\\bf r}_j|$ is the distance between\ndisk centers, ${\\bf \\hat d}_{ij}=({\\bf r}_i-{\\bf r}_j)\/d_{ij}$, and the spring constant\n$k=20.0$\nis large enough to prevent significant disk-disk overlap under the conditions we\nstudy\nyet small enough to permit a computationally efficient time step\nof $\\delta t=0.001$ to be used.\nWe consider a sample of size $L \\times L$ with $L=300$, and describe the disk density in\nterms of the area coverage\n$\\phi = N\\pi R^2\/L^2$.\nThe run and tumble self-propulsion is modeled with a motor force  ${\\bf F}^{m}_i$\nof fixed magnitude $F^m=1.0$ that acts in a randomly chosen direction during\na run time of $\\tilde{t}_{r}$.\nAfter this run time, the motor force instantly reorients into a new\nrandomly chosen direction for the next run time.\nWe take $\\tilde{t}_r$ to be uniformly distributed over the range\n[$t_r,2t_r$], using run times ranging from $t_r=1 \\times 10^3$ to $t_r=3 \\times 10^5$.\nFor convenience we describe \nthe activity in terms of the run length $r_l=F^mt_r\\delta t$,\nwhich is the distance a disk would move during a single run time\nin the absence of a substrate or other disks.\nThe substrate is modeled as a time-dependent\nsinusoidal force $F^s_i(t)=A_{s}\\sin(2\\pi x_i-v_w t)$\nwhere $A_s$ is the substrate strength and $x_i$ is the $x$ position of disk $i$.  We take\na substrate periodicity of $a = 15$ so that the system contains\n$20$ minima.\nThe substrate travels at a constant velocity of $v_{w}$ in the positive $x$-direction.\nWe measure the average drift velocity of the disks in the direction of the traveling wave,\n$\\langle V\\rangle = N^{-1}\\sum^{N}_{i}{\\bf v}_{i}\\cdot {\\hat {\\bf x}}$.\nWe  vary the run length, substrate strength, disk density, and wave speed.\nIn each case   \nwe wait for a fixed time\nof $5 \\times 10^6$ simulation time steps\nbefore taking measurements to avoid any transient effects.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig2.png}\n\\caption{(a) The average velocity per disk $\\langle V\\rangle$\n  vs wave speed $v_{w}$ for a system with\n  $N=13000$ disks, $\\phi  = 0.45376$, and\n  $r_l = 300$ for varied substrate strengths of $A_s=0.5$ to 3.0.\n  The dashed line indicates the limit in which all the disks move at the wave\n  speed, $\\langle V\\rangle = v_{w}$.\n  (b) The corresponding number $\\tilde{C}_L$ of disks that are in a cluster\nvs wave speed. \nThe inset shows the regions of cluster and non-cluster (NC) states\nas a function of $v_{w}$ vs $A_{s}$. \n(c) The number $\\tilde{P}_6$  of sixfold-coordinated\ndisks vs wave speed $v_w$.\n}\n\\label{fig:2}\n\\end{figure}\n\n\\section{Results}\nIn Fig.~\\ref{fig:2}(a) we plot the average velocity per disk\n$\\langle V\\rangle$ versus wave speed $v_{w}$ at different substrate strengths\n$A_{s}$ for a system containing $N=13000$ active disks,\ncorresponding to $\\phi = 0.45376$,\nat $r_l = 300$, a running length at which the substrate-free system\nforms a phase separated state.\nThe number of disks that are in\nthe largest cluster, $\\tilde{C}_L$, serves as an effective measure\nof whether the system is in a phase separated state or not.  We measure $\\tilde{C}_L$\nusing the cluster identification algorithm described in Ref.~\\cite{Hermann}, and call\nthe system phase separated when $\\tilde{C}_L\/N>0.55$.  In Fig.~\\ref{fig:2}(b) we plot\n$\\tilde{C}_L$ versus $v_w$ at varied $A_s$,\nand in Fig.~\\ref{fig:2}(c) we show the corresponding number of sixfold-coordinated\ndisks, $\\tilde{P}_6=\\sum_i^N\\delta(z_i-6)$, where $z_i$ is the coordination number of\ndisk $i$ determined from a Voronoi construction \\cite{cgal}.  In phase separated states,\nmost of the disks within a cluster have $z_i=6$ due to the triangular ordering of the\ndensely packed state.\nIn Fig.~\\ref{fig:2}(a), the linearly increasing dashed line denotes\nthe limit in which all the disks\nmove with the substrate so that $\\langle V\\rangle = v_{w}$ .\nAt $A_{s} = 3.0$, $\\langle V\\rangle$ initially increases linearly, following the dashed\nline, up to $v_{w} = 1.25$, \nindicating that there is a complete locking of the\ndisks to the substrate.\nFor $v_w > 1.25$, there is a slipping process in which the disks\ncannot keep up with the traveling wave and jump to the next well.\nA maximum in $\\langle V\\rangle$ appears near $v_{w} = 2.0$,\nand there is a sharp drop in $\\langle V\\rangle$ near $v_{w} = 5.0$,\nwhich also coincides with a sharp increase in  $\\tilde{C}_L$ and $\\tilde{P}_{6}$.\nThe  $\\langle V\\rangle$ versus $v_{w}$ curves for $A_{s} > 1.0$\nall show similar trends, with a sharp drop\nin $\\langle V\\rangle$ accompanied by an increase in\n$\\tilde{C}_{L}$ and $\\tilde{P}_{6}$, showing that the onset of clustering\nresults in a sharp decrease in $\\langle V\\rangle$.\nFor $A_{s} \\leq 1.0$, the substrate is weak enough that the system\nremains in a cluster state even at $v_{w} = 0$,\nindicating that a transition from a cluster to a non-cluster state\ncan also occur as a function of substrate strength.\nIn the inset of Fig.~\\ref{fig:2}(b) we show the regions in which\nclustering and non-clustering states appear as a function of $v_w$ versus $A_s$.\nAt $v_{w} = 0$, there is a substrate-induced transition from a\ncluster to a non-cluster state near $A_{s} = 1.0$,\nwhile for higher $A_s$, the location of the transition shifts linearly to higher $v_w$\nwith increasing $A_s$.\nSince the motor force is $F^m=1.0$,\nwhen $A_{s} < 1.0$ individual disks\ncan escape from the substrate minima,\nso provided that $r_{l}$ is large enough, the disks can freely move\nthroughout the entire system and form a cluster state.\nFor $A_{s} > 1.0$, the disks are confined by the substrate minima,\nbut when $v_w$ becomes large enough, the disks can readily escape\nthe minima and again form a cluster state.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig3.png}\n\\caption{(a) $\\langle V\\rangle\/v_{w}$\n  vs $A_{s}$ for the system in Fig.~\\ref{fig:2}\n  at $v_{w} = 0.6$.\n  (b) The corresponding normalized $C_{L}$ showing\n  that the  transition from a cluster to a non-cluster state\n  coincides with an increase in $\\langle V\\rangle\/v_w$.\n  (c) $\\langle V\\rangle\/v_{w}$ vs $A_s$ for the same system with $v_w=2.0$\n  where the cluster to non-cluster transition occurs at a higher value of $A_{s}$.\n  (d) The corresponding normalized $C_{L}$ vs $A_s$.\n}\n\\label{fig:3}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig4.png}\n\\caption{ The real space positions of the active disks for the system in Fig.~\\ref{fig:3}(a,b)\n  with $v_{w} = 0.6$.\n  (a) At $A_{s} = 0.75$, a phase separated state appears.\n(b) At $A_{s} = 2.5$, the disks are strongly localized in the substrate minima and move\nwith the substrate.\n}\n\\label{fig:4}\n\\end{figure}\n\nTo highlight the correlation between the changes in\nthe transport and the onset of clustering, in\nFig.~\\ref{fig:3}(a,b)\nwe plot $\\langle V\\rangle\/v_{w}$ and the normalized\n$C_{L}=\\tilde{C}_L\/N$ versus $A_{s}$ at a fixed value of  $v_{w} = 0.6$\nfrom the system in Fig.~\\ref{fig:2}.\nHere\nthe cluster to non-cluster transition occurs at $A_{s} = 1.25$,\nas indicated by the drop in $C_{L}$\nwhich also coincides with a jump in $\\langle V\\rangle\/v_{w}$.\nFor this value of $v_{w}$, a complete locking between the disks and the traveling\nwave occurs for $A_{s} \\geq 3.0$, where\n$\\langle V\\rangle\/v_{w} = 1.0$.\nIn Fig.~\\ref{fig:3}(c,d) we plot $\\langle V\\rangle\/v_w$ and $C_L$ versus $A_s$ for the\nsame system at\n$v_{w} = 2.0$, where the cluster to non-cluster transition\noccurs at a higher value of $A_{s} = 2.0$.\nThis transition again coincides with a sharp\nincrease in $\\langle V\\rangle\/v_{w}$.\nIn Fig.~\\ref{fig:4}(a) we show images of the disk configurations for\nthe system in Fig.~\\ref{fig:3}(a,b) with $v_w=0.6$ at\n$A_{s} = 0.75$,\nwhere the disks form a cluster state,\nwhile in Fig.~\\ref{fig:4}(b), at $A_s=2.5$ in the same system,\nthe clustering is lost and the disks are\nstrongly trapped in the substrate minima, forming\nchain like states that move with the substrate.\nThese results indicate that the clusters act as composite objects that\nonly weakly couple to the substrate.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig5.png}\n\\caption{(a) $\\langle V\\rangle$ vs $r_{l}$\n  in samples with $A_{s} = 2.0$ and $\\phi = 0.453$ for varied $v_{w}$ from\n  $v_w=0.25$ to $v_w=6.0$.\n(b) The corresponding $\\tilde{C}_{L}$ vs $r_{l}$.}\n\\label{fig:5}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig6.png}\n\\caption{A sample with $A_s=2.0$ and $\\phi=0.453$\n  for $r_{l} = 5$ (red squares) and $r_l=200$ (blue circles).\n  (a) $\\langle V\\rangle$ vs $v_{w}$.\n(b) $C_{L}$ vs $v_{w}$.\n}\n\\label{fig:6}\n\\end{figure}\n\nWe next examine the case with a fixed\nsubstrate strength of $A_{s} = 2.0$ and varied $r_{l}$.\nFigure~\\ref{fig:5}(a) shows\n$\\langle V\\rangle$ versus $r_{l}$\nfor $v_{w}$ values ranging from $v_w=0.25$\nto $v_w=6.0$, and Fig.~\\ref{fig:5}(b) shows the corresponding $\\tilde{C}_{L}$\nversus $r_{l}$.\nFor $v_{w} < 3.0$ the system remains in a non-cluster state for all values of $r_{l}$, while\nfor $v_{w} \\geq 3.0$ there is a transition\nfrom a non-cluster to a cluster state\nwith increasing $r_l$ as indicated by the simultaneous drop in\n$\\langle V\\rangle$ and increase in $\\tilde{C}_{L}$.\nIn Fig.~\\ref{fig:6}(a,b) we plot $\\langle V\\rangle$ and\n$C_{L}$ versus $v_{w}$ at $A_{s} = 2.0$\nfor $r_{l} = 200$ and $r_l=5.0$.\nThe system is in a non-cluster state for all $v_w$ when\n$r_{l} = 5.0$,\nand there is a peak in $\\langle V\\rangle$ near $v_{w} = 1.0$,\nwhile for $r_{l} = 200$ there is a transition to a cluster state\nclose to $v_{l} = 3.0$\nwhich coincides with a drop in $\\langle V\\rangle$ that is much sharper than the decrease\nin $\\langle V\\rangle$ with increasing $v_w$ for the $r_{l} = 5$ system.\nIn general, when $r_{l}$ is small, the net transport of disks through the sample is\ngreater than in samples with larger $r_l$.\nThe fact that the net disk transport varies with varying $r_l$ suggests\nthat traveling wave substrates could be used as a method for separating\ndifferent types of active matter, such as clustering and non-clustering species.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig7.png}\n\\caption{\n  (a) $\\langle V\\rangle$ vs $v_{w}$ at $\\phi = 0.56$ and\n  $r_{s} = 300$ for varied $A_{s} = 0.5$ to $A_s=4.0$.\n(b) The corresponding $\\tilde{C}_{L}$ vs $v_{w}$.\n(c) $\\langle V\\rangle$ vs $\\phi$ for $A_{s} = 2.5$, $r_{l} = 300$ and $v_{w} = 2.0$.\n  (d) The corresponding\n  $\\tilde{C}_{l}$ (blue circles) and $\\tilde{P}_{6}$ (red squares) vs $\\phi$\n  where the onset of clustering occurs\n  near $\\phi = 0.6$ at the same point for which there is a drop in\n $\\langle V\\rangle$ in panel (c).\n}\n\\label{fig:7}\n\\end{figure}\n\nWhen we vary the disk density $\\phi$ while holding $r_l$ fixed, we find\nresults similar to those described above.\nIn Fig.~\\ref{fig:7}(a) we plot\n$\\langle V\\rangle$ versus $v_{w}$\nat $\\phi = 0.56$ for varied $A_{s}$ from $A_s=0.5$ to $A_s=4.0$, where we find\na similar trend in which\n$\\langle V\\rangle$ increases with increasing wave speed\nwhen the disks are strongly coupled to the substrate.\nA transition to a cluster state occurs at higher $v_w$ as shown in\nFig.~\\ref{fig:7}(b) where we plot $\\tilde{C}_{L}$ versus $v_{w}$ for the same\nsamples.  The increase in $\\tilde{C}_{L}$ at the cluster state onset\ncoincides\nwith a drop in $\\langle V\\rangle$.\nIn Fig.~\\ref{fig:7}(c) we plot $\\langle V\\rangle$ versus $\\phi$\nfor a system with fixed $v_{w} = 2.0$, $A_{s} = 2.5$, and $r_{l} = 300$, while\nin Fig.~\\ref{fig:7}(d) we show the corresponding\n$\\tilde{C}_{L}$ and $\\tilde{P}_{6}$ versus $\\phi$.\nA transition from the\nnon-cluster to the cluster state occurs\nnear $\\phi=0.6$, which correlates with a sharp drop in $\\langle V\\rangle$ and\na corresponding increase\nin $\\tilde{C}_{L}$ and $\\tilde{P}_{6}$.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig8.png}\n\\caption{\n  The disk positions on the traveling wave substrate for the system in\n  Fig.~\\ref{fig:7}(a) at $\\phi = 0.56$.  Colors indicate disks belonging to\n  the five largest clusters.\n  (a) Complete locking\n  at $A_{s} = 4.0$ and $v_{w} = 1.0$,\n  where the transport efficiency is $\\langle V\\rangle\/v_{w} = 0.998$.\n  (b) Partial locking\n  at $A_{s} = 2.0$ and $v_{w} = 1.5$, \n  where $\\langle V\\rangle\/v_{w} =  0.41$.\n  (c) Weak locking\n  at $A_{s} = 1.0$ and $v_{w} = 0.6$  \n  with $\\langle V\\rangle\/v_{w} = 0.078$.\n}\n\\label{fig:8}\n\\end{figure}\n\nIn Fig.~\\ref{fig:8}(a) we show the disk configurations from the system in\nFig.~\\ref{fig:7}(a) at $A_{s} = 4.0$ and $v_{w} = 1.0$.\nHere $\\langle V\\rangle\/v_{w} = 0.998$,\nindicating that the disks are almost completely locked with\nthe traveling wave motion\nand there is little to no slipping of the disks out of the substrate minima.  \nIn Fig.~\\ref{fig:8}(b), the same system at\n$A_{s} = 2.0$ and $v_{w} = 1.5$\nhas a transport efficiency of $\\langle V\\rangle\/v_{w} = 0.41$.\nNo clustering occurs but there are numerous disks that \nslip as the traveling wave moves.\nAt $A_{s} = 1.0$ and $v_{w} = 0.6$ in\nFig.~\\ref{fig:8}(c)\nthere is a low transport efficiency of\n$\\langle V\\rangle\/v_w=0.078$.\nThe system forms a cluster state and smaller numbers of individual disks\noutside of the cluster are transported by the traveling wave.\n\n\\section{Forward and Backward Cluster Motion}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig9.png}\n\\caption{\n  The center of mass $X_{\\rm COM}$ location of a cluster vs\n  time in simulation time steps for a system with $\\phi = 0.454$ and $r_{l} = 300$.\n(a) At $A_{s} = 1.25$ and $v_{w} = 0.6$, the cluster moves in the negative $x$-direction,\n  against the direction of the traveling wave.\n  (b) At $A_{s} = 0.5$ and $v_{w} = 4.0$, the cluster\n  is stationary.\n  (c) At $A_{s} = 3.0$ and $v_{w} = 7.0$, the cluster moves in the positive\n  $x$-direction, with the traveling wave.\n  The dip indicates the point at which the center of mass\n  passes through the periodic boundary conditions.\n}\n\\label{fig:9}\n\\end{figure}\n\nIn general, we find that when the traveling wave is moving in the positive $x$-direction,\n$\\langle V\\rangle > 0$; however, within the\ncluster phase,\nthe center of mass motion of a cluster can be\nin the positive or negative $x$ direction or the cluster can be almost stationary.\nBy using the cluster algorithm we can track the $x$-direction motion of the\ncluster center of mass $X_{\\rm COM}$ over fixed time periods,\nas shown in Fig.~\\ref{fig:9}(a) for a system with\n$\\phi = 0.454$, $r_{l} = 300$, $A_{s} = 1.25$, and $v_{w} = 0.6$.\nDuring the course of $6\\times 10^6$ simulation time steps\nthe cluster moves in the negative $x$-direction a distance of\n$235$ units, corresponding to a space containing 16 potential minima.\nEven though the net disk flow is in the positive\n$x$ direction, the cluster itself drifts in the negative $x$ direction.\nIn Fig.~\\ref{fig:9}(b) at $A_{s} = 0.5$ and $v_{w} = 4.0$,\nthe disks are weakly coupled to the substrate\nand the cluster is almost completely stationary.\nFigure~\\ref{fig:9}(c) shows that at $A_{s} = 3.0$ and $v_{w} = 7.0$,\nthe cluster center of mass motion is now\nin the positive $x$ direction, and the cluster\ntranslates a distance equal to almost $20$ substrate minima\nduring the time period shown.\nThe apparent dip in the center of mass motion\nis due to the periodic boundary conditions.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig10.png}\n\\caption{\n  Height field of the direction and magnitude of the center of mass motion $V_{\\rm COM}$\n  as a function of $A_s$ vs $v_w$ for the cluster\n  obtained after $4\\times 10^6$ simulation steps.\n  The gray area indicates a regime in which there is no cluster formation.\n}\n\\label{fig:10}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig11.png}\n\\caption{ The disk positions for the system in Figs.~\\ref{fig:9} and\n  ~\\ref{fig:10}.\n  (a)\n  At  $A_{s} = 1.0$ and $v_{w} = 0.4$,  the cluster drifts in the negative $x$-direction.\n  (b)\n  At $A_{s} = 3.0$ and $v_{w} = 5.0$, \n  the cluster drifts in the positive $x$-direction.\n}\n\\label{fig:11}\n\\end{figure}\n\n\nWe have conduced a series of simulations and measured\nthe direction and amplitude $V_{\\rm COM}$ of the center\nof mass motion, as plotted\nin Fig.~\\ref{fig:10} as a function of $A_{s}$  versus wave speed\nfor the system in Fig.~\\ref{fig:7}.\nThe gray area indicates a region in which clusters do not occur,\nand in general we find that the negative cluster motion occurs\nat lower wave speeds while the positive motion occurs\nfor stronger substrates and higher wave speeds.\nThere are two mechanisms that control the cluster center of mass motion.\nThe first is the motion of the substrate itself,\nwhich drags the cluster in the positive $x$ direction, and\nthe second is the manner in which the cluster grows or shrinks\non its positive $x$ and negative $x$ sides.\nAt lower substrate strengths and low wave speeds,\nthe disks in the cluster are weakly coupled to the substrate\nso the cluster does not move with the substrate.\nIn this case the disks can leave or\njoin the cluster anywhere around its edge; \nhowever, disks tend to join the cluster at a higher rate on its\nnegative $x$ side since individual disks, driven by the moving\nsubstrate, collide with the negative $x$ side of the cluster and can become\ntrapped in this higher density area.  The positive $x$ side of the cluster\ntends to shed disks at a higher rate since the disks can be carried\naway by the moving substrate into the low density gas region.\nThe resulting asymmetric growth rate causes the cluster to drift in\nthe negative $x$ direction.\nThere is a net overall transport of disks in the positive $x$ direction due to the\nlarge number of gas phase disks outside of the cluster region which follow\nthe motion of the substrate.\nFigure~\\ref{fig:11}(a) shows the\ndisk positions at $A_{s} = 1.0$ and $v_{w} = 0.4$, where the cluster\nis drifting in the negative $x$ direction.     \nFor strong substrate  strengths, all the disks that are outside of the cluster become\nstrongly confined in the q1D substrate minima,\nand the disk density inside the cluster itself starts to become modulated by the substrate.\nUnder these conditions, the cluster is dragged along with the traveling substrate\nin the positive $x$ direction, as illustrated\nin Fig.~\\ref{fig:11}(b) for $A_{s} = 3.0$ and $v_{w} = 5.0$. \nThese results\nsuggest that it may be possible for\ncertain active matter systems to \ncollectively form a cluster state in order to \nmove against an external bias\neven when isolated individual particles on average move with the\nbias.\n\n\n\\section{Summary}\n\nWe have examined run and tumble active matter disks interacting with\ntraveling wave periodic substrates.\nWe find that\nin the non-phase separated state,\nthe disks couple to the traveling waves,\nand that at the transition\nto the cluster state,\nthere is a partial decoupling from the substrate and the net transport of disks by\nthe traveling wave  is strongly reduced.\nWe also find a transition from a cluster\nstate to a periodic quasi-1D liquid\nstate for increasing substrate strength,\nas well as a transition back to a cluster state\nfor increasing traveling wave speed.\nWe show that there is a transition\nfrom a non-cluster to a cluster state as a function of increasing\ndisk density which is correlated with a drop in the net disk transport.\nSince disks with different run times drift with different velocities,\nour results indicate that traveling wave substrates\ncould be an effective method for separating active matter particles with different mobilities.\nWithin the regime in which the system forms a cluster state,\nwe find that as a function of wave speed and substrate strength,\nthere are weak substrate regimes where the center of mass of the\ncluster moves in the opposite direction from that of the traveling wave,\nwhile for stronger substrates,\nthe cluster center of mass moves in the same direction as the traveling wave.\nThe reversed cluster motion occurs due to the\nspatial asymmetry of the rate at which disks leave or join the cluster.\nThis suggests that collective clustering could be an effective method\nfor forming an emergent object that\ncan move  against gradients or drifts even\nwhen individual disks on average move with the drift.\n\n\\acknowledgments\nThis work was carried out under the auspices of the \nNNSA of the \nU.S. DoE\nat \nLANL\nunder Contract No.\nDE-AC52-06NA25396.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the past few years, the field of text generation witnesses many significant advances, including but not limited to neural machine translation \\cite{Transformer:17,gehring2017convolutional}, dialogue systems \\cite{liu-etal-2018-knowledge,zhang2019memory} and text generation \\cite{clark-etal-2018-neural,guo2018long}.\nBy utilizing the power of the sequence-to-sequence (S2S) framework \\cite{Sutskever:14}, generation models can predict the next token based on the previous generated outputs and contexts.\nHowever, S2S models are not perfect.\nOne of the obvious drawbacks is that S2S models tend to be short-sighted on long context and are unaware of global knowledge.\nTherefore, how to incorporate global or local knowledge into S2S models has been a long-standing research problem.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/figure0.png}\n  \\caption{\\label{example-table} An example from a novel called ``Fights Break Sphere''. The relations between \\textcolor{y}{Yanxiao}, \\textcolor{x}{Xuner} and \\textcolor{n}{Nalanyanran} are evolutionary. And the characteristic of \\textcolor{y}{Yanxiao} changes over time.}\n\\end{figure}\n\nThere lie two different directions to include knowledge into S2S models.\nOn the one hand, many efforts \\cite{zhang-etal-2018-improving,guan2019story,li-etal-2019-incremental} have been taken to address the short-sighted problem in text generation by explicitly modeling unstructured context. Nevertheless, these approaches rely heavily on the quality and scale of unstructured context, and become intractable when applying to scenarios where the context length increases drastically (e.g., commenting a full-length novel).\nOn the other hand, researchers~\\citep{beck-etal-2018-graph,marcheggiani-perez-beltrachini-2018-deep,li-etal-2019-coherent} try to combine knowledge with S2S by employing pre-processed structured data~(e.g., knowledge graph) which naturally avoid the difficulty for context length. However, those models are oriented to static knowledge, and hence hardly model events where temporal knowledge evolution occurs.\n\n\n\n\nThe dynamic knowledge evolution is very common in full-length novels. In a novel, a knowledge graph can be constructed by using entities (characters, organizations, locations etc.) as vertices together with the entity relations as edges. Obviously, a single static knowledge graph is hardly to represent the dynamic story line full of dramatic changes. For example, a naughty boy can grow up into a hero, friends may become lovers, etc. In this paper, we proposes \\textbf{E}volutionary \\textbf{K}nowledge \\textbf{G}raph~(\\textbf{EKG}) which contains a series of sub-graph for each time step. Figure 1 illustrates EKG for the novel ``Fights Break Sphere''. At three different scenes, ``Yanxiao'', the leading role of the novel, is characterized as ``proud boy'', ``weak fighter'', and ``magic master'' separately. At the same time, the relation between ``Yanxiao'' and ``Xuner'' is evolved from friend into lovers, and finally get married, and the relation between ``Yanxiao'' and ``Nalanyanran'' is changed over time as ``engagement$\\rightarrow$divorce$\\rightarrow$friend''.\n\nEKG is important for commenting passage of novels since it is the dramatic evolution and comparison in the storyline but not the static facts resonate the readers most. As illustrated in Figure~\\ref{example-table}. When commenting passage sampled from the $T$-th chapter of a novel, the user-A refers to a historical fact that \"Nalanyanran\" has abandoned ``Yanxiao'', while the user-B refers to the future relation between ``Yanxian'' and a related entity ``Xuner'' that they will go through difficulties together. In this paper, EKG is trained within a multi-task framework to represent the latent dynamic context, and then a novel graph-to-sequence model is designed to select the relevant context from EKG for comment generation.\n\n\\subsection{Related Work}\ngraph-to-sequence model has been proposed for text generation.~\\citet{song-etal-2018-graph}, ~\\citet{beck-etal-2018-graph}, and~\\citet{guo-etal-2019-densely} used the graph neural networks to solve the AMR-to-text problem.~\\citet{bastings-etal-2017-graph} and ~\\citet{GraphSeq2Seq:18} utilized graph convolutional networks to incorporate syntactic structure into neural attention-based encoder-decoder models for machine translation. In comment generation, Graph2Seq~\\citep{li-etal-2019-coherent} is proposed to generate comments by modeling the input news as a topic graph. These methods are using static graph, and did not involve the dynamic knowledge evolution.\n\nRecently, more research attention has been focused on dynamic knowledge modeling.~\\citet{taheri2019www} used gated graph neural networks to learn the temporal dynamics of an evolving graph for dynamic graph classification.~\\citet{KnowEvolve:17, trivedi2018dyrep, kumar2019kdd} learned evolving entity representations over time for dynamic link prediction. Unlike the EKG in this paper, they did not model the embeddings of the relations between dynamic entities.\n~\\citet{iyyer-etal-2016-feuding} proposed an unsupervised deep learning algorithm to model the dynamic relationship between characters in a novel without considering the entity embedding. Unlike these methods, our EKG-based model represents the temporal evolution of entities and relations simultaneously by learning their temporal embeddings, and hence has an advantage in supporting text generation tasks.\n\nTo our knowledge, few studies make use of evolutionary knowledge graph for text generation. This may due to the lack of datasets involving dynamic temporal evolution. We observed that novel commenting need to understand long context full of dramatic changes, and hence build such a dataset by collecting full-length novels and real user comments. The dataset with its EKG will be made publicly available, and more details can be found in Section~\\ref{sect:Dataset}.\n\nThe main contributions of our work are three-fold:\n\n\\begin{itemize}\n  \\item We build a new dataset to facilitate the research of evolutionary knowledge based text generation.\n  \\item We propose a multi-task framework for the learning of evolutionary knowledge graph to model the long and dynamic context.\n  \\item We propose a novel graph-to-sequence model to incorporate evolutionary knowledge graph for text generation.\n\n\\end{itemize}\n\n\\section{Dataset Development and Evolutionary Knowledge Graph Building}\\label{sect:Dataset}\nTo facilitate the research of modeling knowledge evolution for text generation, we build a dataset called \\textit{GraphNovel} by collecting full-length novels and real user comments. Together with the corresponding EKG embeddings, we will make the dataset public available soon.\nWe detail the collection of the dataset below.\n\n\\subsection{Data collection}\n\\label{sub:collect}\nThe data is collected from well-known Chinese novel websites. To increase the diversity of data, top-1000 hottest novels are crawled with different types including science fiction, fantasy, action, romance, historical, and so on. Then we filter out novels due to the following 3 considerations: 1) the number of chapters is less than 10, 2) few entities are mentioned and 3) lack of user comments. Each remained novel includes chapters in chronological order, a set of user-underlined passages, and user comments for the passages.\n\nThen, we use the lexical analysis tool~\\citep{jiao2018LAC} to recognize three types of entities (persons, organizations, locations) from each novel. Due to many of the nickname in novels, the identified entities from the tool contains much noise. To improve the knowledge quality, human annotators are asked to verify the entities, and add missing ones. Then all the paragraphs containing mentions of two entities are identified, and will later serve as a representation of the entity relations at that specific time step.\n\nAs for the highlighted novel passage and user comments, three criteria are used to select high quality and informative data: 1) The selected passage must contain at least one entity; 2) the selected passage must be commented by at least three users; 3) comments related to the same passage are ranked according to the upvotes, and the bottom 20\\% are dropped. Notably, those highlighted passages have a degree of redundancy because users tend to highlight and comment passages at very similar positions. Thus we merge passages which have more than 50\\% overlapping rate. This operation can effectively reduce the quantity of passages by 30\\%.\n\n\n\n\\subsection{Core Statistics}\nThe dataset contains 203 novels, 349,695 highlighted passages and 3,136,210 comments totally. Due to diverse genre of novels in our dataset, the number of entities and relations per novel varies widely with range [$10^2$, $10^5$]. And the number of comments per passage changes a lot with a range of [$3$, $2\\times{10^3}$], because it depends on how interesting the corresponding passage is.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{images\/figure2.png}\n  \\caption{Architecture of our model. First, the EKG is trained under a multi-task learning framework. Then a graph-to-sequence model is traind to utilize the EKG for text generation.}\\label{fig:model}\n\\end{figure*}\n\nWe partitioned the dataset into non-overlapping training, validation, and test portions, along novels (See Table~\\ref{dataset-table} for detailed statistics). Five most relevant comments for each passage in validation and test set are selected by human annotators. While the comments in train set are all preserved in order to ensure its flexible use.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{l|r|r|r}\n\\hline\n & \\textbf{train} & \\textbf{valid} & \\textbf{test} \\\\\n\\hline\n{\\small{\\# novels}} & 173 &  10 & 20 \\\\\n\\hline\n{\\small{\\# passages}} & 324,803 & 7,976 & 16,916 \\\\\n\\hline\n{\\small{\\# comments}} & 3,011,750 & 39,880 & 84,580 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. length}\\\\\\small{~~of context}} & 520,571 & 305,492 & 277,847 \\\\\n\\hline\n{\\small{Avg. \\# entities}} & 383.4 & 720.6 & 281.7 \\\\\n\\hline\n{\\small{Avg. \\# relations}} & 9,013 & 1,9919 & 7,084 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# comments}\\\\\\small{~~per passage}} & 9.3 & 5.0 & 5.0 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# entities}\\\\\\small{~~per passage}} & 2.6 & 3.1 & 3.4 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# relations}\\\\\\small{~~per passage}} & 4.4 & 9.2 & 11.0 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{dataset-table} Statistics of Dataset. }\n\\end{table}\n\n\\subsection{Build up Evolutionary Knowledge Graph}\n\\label{sub:collect}\n\n\nThen for each novel, we build up a knowledge graph which consists of a sequence of sub-graphs. Obviously, it is sensible to build up a sub-graph for each import scene of the novel, and build up more sub-graphs around the critical transitions in the stroryline. In this paper, we assume each chapter usually represents an integral scene, and hence build a sub-graph for each chapter of the novel.\n\nThen for each chapter, the entities being mentioned in the chapter are the vertices of the corresponding sub-graph. And if a paragraph consists two of the entities, an edge is created between the two entities. In such a way, a sequence of sub-graphs are constructed, and form our EKG. In the next section, we will formulate the embedding computation of the EKG, and its application for comment generation.\n\n\\section{Model Formalism}\\label{sec:formalism}\nIn this section, we formulate our approach in details. First, the training of EKG embedding is presented. Then a graph-to-sequence model is shown to utilize the EKG for comment generation. The architecture of the model is shown in Figure~\\ref{fig:model}.\n\n\\subsection{Definition}\nFor a novel with $n_e$ entities and $n_r$ relations, define a global evolutionary knowledge graph: $G_{ekg}^{global} = \\{G(t)\\}|_{t:1\\rightarrow{T}}$,\nwhere $T$ is the number of chapters\\footnote{We will cluster successive chapters into a longer one if they are too short.} (or time periods); $G(t)=\\langle{V(t), E(t)}\\rangle$, is a temporal knowledge graph of the chapter $t$; $V(t)=\\{v_1(t), v_2(t),...,v_{n_e}(t)\\}$ is the set of vertices and $E(t)=\\{e_1(t), e_2(t),...,e_{n_r}(t)\\}$ is the set of edges between two vertices.\n\nGiven a passage $C$ from the chapter $t$ with $c_e$ entities and $c_r$ relations, a local EKG related to it can is a sub-graph of the global EKG: $G_{ekg}^{local}=\\{G_c(t)\\}|_{t:1\\rightarrow{T}}$, where $G_c(t)=\\langle{V_c(t), E_c(t)}\\rangle$; $V_c(t)$ is a subset of $V(t)$ with size of $c_e$ and $E_c(t)$ is a subset of $E(t)$ with size of $c_r$. Then the local EKG of the passage is a sequence of local temporal knowledge sub-graphs with $T \\times c_e$ vertex embeddings and $T \\times c_r$ edge embeddings.\n\\subsection{EKG Embedding Training}\nInspired by the consistent state-of-the-art performance in language understanding tasks, we use off-the-shelf Chinese BERT model~\\citep{devlin-etal-2019-bert} to calculate the initial semantic representation of sentences. Considering the fact that entities are either out-of-vocabulary or associated with special semantics within the novel context, we propose the following algorithm to jointly learn entity and relation embeddings in EKG:\n\n\\paragraph{Vertex embedding learning.} The passages containing mentions of entity $v$ will contribute to learn the embedding of $v$. Specifically, the $i$-th passage is tokenized while the entity mention $v_i$ is masked with token ``[MASK]''. Then resulted tokens are fed into the pre-trained Chinese BERT model, and $f_{v_i}$ is obtained as the output feature corresponding to the mask token.\nWithin the chapter $t$, there exists $N_{v}^{t}$ sentences containing vertices $v$. The embedding of $v$ is learned by optimizing the following softmax loss summation which models the probabilities to predict the masked entities as $v$.\n\\begin{equation}\\label{node_learn}\n  L_{v}^{t} = -\\sum_{i=1}^{N_{v}^{t}}{\\log{p_{v_i}^{t}}}\n\\end{equation}\n\\begin{equation}\\label{node_softmax}\n  p_{v_i}^{t} = softmax(W_{v}^{t}\\cdot{f_{v_i}})\n\\end{equation}\nwhere $W_v^t$ is learnable parameter and denotes the embedding of $v$ from the chapter $t$. Usually the semantic representations of entities change smoothly over time, so we propose a temporally smoothed softmax loss to retain the similarity of entity embeddings from successive time periods:\n\\begin{equation}\\label{node_improve}\n   \\tilde{L}_{v}^{t} = -\\sum_{i=1}^{N_{v}^{t}}{(\\lambda{_0}\\log{p_{v_i}^{t-1}}+\\lambda{_1}\\log{p_{v_i}^{t}}+\\lambda{_2}\\log{p_{v_i}^{t+1}})}\n \n\\end{equation}\nwhere $\\lambda_0$, $\\lambda_1$ and $\\lambda_2$ are smooth factors; and only valid probabilities are included when $t=1$ or $T$. Then the overall loss for all time periods and all vertices is:\n\\begin{equation}\\label{node_overall}\n  L_{vertex} = \\sum_{t=1}^{T}\\sum_{v=1}^{n_e}{\\tilde{L}_{v}^{t}}\n\\end{equation}\n\n\n\\paragraph{Edge embedding learning.} Since the number of relations equal the number of co-occurrence for any two entities, it is infeasible to employ an embedding matrix to model the relation. Therefore, a \\textit{Relation Network} (RN) is proposed to learn the edge embeddings in the TKGs as shown in Figure~\\ref{fig:rn}. Specifically, the RN takes two vertex embeddings as input, and feed them into the first hidden layer to obtain the embedding of the edge $r$. Then the embeddings of two vertices and one edge are concatenated and fed into second hidden layer to reconstruct the sentence. We also use the pre-trained BERT~\\citep{devlin-etal-2019-bert} to obtain the representation $f_c$ for the whole sentence. The $f_c$ is taken from the final hidden state corresponding to ``[CLS]'' token because it aggregates sequence representation.\n\nA reconstruction loss applied to the network is optimized to jointly learn RN and edge embeddings:\n\\begin{equation}\\label{edge_l0}\n  L_r = \\max(d(f_{p^+}, f_c)- d(f_{p^-}, f_c) + \\alpha, 0)\n\\end{equation}\nwhere $p$ stands for a pair of vertices; $p^+$ represents a positive pair with two vertex related to the edge, and $p^-$ represents a negative pair with one vertex related to the edge and the other unrelated. The overall loss for learning edge embedding is:\n\\begin{equation}\\label{edge_loss}\n  L_{edge} = \\sum_{r=1}^{N_r}L_r\n\\end{equation}\n\nCombining the vertex and edge loss above, our final multi-task loss is:\n\\begin{equation}\\label{tkg_loss}\n  L_{EKG} = L_{vertex} + \\lambda_{r}L_{edge}\n\\end{equation}\nwhere $\\lambda_{r}$ is a hyperparameter to be tuned.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/rn.png}\n  \\caption{Relation Network with a reconstruction loss.The edge embedding is shown in \\textcolor{red}{red color}.}\\label{fig:rn}\n\\end{figure}\n\n\\subsection{Graph-to-sequence modeling}\nAfter EKG embedding learning, we propose a graph encoder to utilize the embeddings of the EKG for comment generation. From the learned $G_{ekg}^{local}$, we can obtain vector sequences $V_i=\\{v_i(1),..,,v_i(2), v_i(T)\\}$ for each vertex, and $E_i=\\{e_i(1),e_i(2),...e_i(T)\\}$ for each edge. All these sequences are fed into Bi-LSTM to integrate information from all time periods. Then final representation of the vertices and edges are taken from the final hidden state of Bi-LSTM corresponding to the time step $t$.\n\nFurther, our graph encoder employs graph convolutional networks~\\citep{GNN:18} to aggregate the structured knowledge from the EKG and then is combined into a widely used encoder-decoder framework~\\citep{Transformer:17} for generation.\n\nour graph encoder is based on the implementation of GAT~\\citep{v2018graph}. The input to a single GAT layer is a set of vertices and edge features, denoted as $\\mathbf{F}_v=\\{\\vec{v}_1, \\vec{v}_2,..., \\vec{v}_{c_e}\\}$, and $\\mathbf{F}_r=\\{\\vec{r}_1, \\vec{r}_2,..., \\vec{r}_{c_r}\\}$, where ${c_e}$ is the number of vertices, and ${c_r}$ is the number of edges from the passage. The layer produces a new set of vertex features, $\\mathbf{F'}_v=\\{\\vec{v'}_1, \\vec{v'}_2,..., \\vec{v'}_{c_e}\\}$ as its output.\n\nIn order to aggregate the structured knowledge from input features and transform them into higher-level features, we then perform self-attention on both the vertices and the edges to compute attention coefficients\n\\begin{equation}\\label{gat_atten1}\n  \\alpha_{ij}^e = g(\\mathbf{W}\\vec{v}_i, \\mathbf{W}\\vec{v}_j)\n\\end{equation}\n\n\\begin{equation}\\label{gat_atten2}\n  \\alpha_{ij}^r = h(\\mathbf{W}\\vec{v}_i, \\mathbf{W}\\vec{r}_{ij})\n\\end{equation}\nwhere $\\mathbf{W}$ is learnable parameter; $g$ and $h$ are mapping functions; $\\alpha_{ij}^e$ and $\\alpha_{ij}^r$ indicate the importance of neighbor features to vertex $i$; they are normalized using softmax.\n\nOnce obtained, the normalized attention coefficients are used to compute a linear combination of the features corresponding to them, to serve as the final output features for every vertex:\n\\begin{equation}\\label{gat_output}\n  \\vec{v'}_i=\\sum_{j\\in{\\mathcal{N}_i}}\\alpha_{ij}^e\\mathbf{W}\\vec{v}_j + \\alpha_{ij}^r\\mathbf{W}\\vec{r}_{ij}\n\\end{equation}\n\nThen the graph encoder is combined into the encoder-decoder framework~\\citep{Transformer:17}, in which a self-attention based encoder is used to encode the passage. To aggregate structured knowledge, the encoding of all vertices from graph encoder are concatenated with the output of the passage encoder, and fed into a Transformer decoder for text generation. The whole graph-to-sequence model is trained to minimize the negative log-likelihood of the related comment.\n\\section{Experiment}\nIn this section, we first introduce the experimental details, and then present results from automatic and human evaluations.\n\\subsection{Details}\nWe train all the models using training set and tune hyperparamters based on validation set. The automatic and human evaluations are carried out based on test set.\nDuring the training of the EKG, we first learn the vertex embeddings and fix them during the subsequent training of edge embeddings. Our GAT-based graph encoder is based on the entities from the passage. we keep $K$ entities for each passage: if the number of entities\\footnote{Note that the number will not be zero because the passage contains as least one entity in our dataset.} is smaller than $K$, we use breadth-first searching on the global graph to fill the gap, otherwise we filter the low-order entities according to entity frequency. We set $K$ to 5 which is selected by validation. Label smoothing is used in the smoothed softmax loss. We denote our full model as ``\\textbf{EKG+GAT(V+E)}'', its variant which only use the first term in Equ.~\\ref{gat_output} as ``\\textbf{EKG+GAT(V)}'' and name the other variant ``\\textbf{EKG}'', which does not use GAT-based graph encoder and feeds the encoding of vertices into the Transformer decoder directly.\n\\subsection{Hyperparameters}\nIn our model, we set $\\lambda_1=0.5$, $\\lambda_2=1.0$, $\\lambda_3=0.3$ for the smoothed softmax loss; set $\\alpha$ to 0.0 for the reconstruction loss and $\\lambda_r$ to 1.0 for the multi-task loss; the number of self-attention layers in our passage encoder is 6; the number of Bi-LSTMs is 2 and the length of its hidden state is 768; and the GAT-based graph encoder has two layers. To stabilize the training, we use Adam optimizer~\\citep{adam:14} and follow the learning rate strategy in~\\citet{klein-etal-2017-opennmt} by increasing the learning rate linearly during the first $5000$ steps for warming-up and then decaying it exponentially. For inference, the maximum length of decoding is 50; beam searching is used with beam size 4 for all the models.\n\\subsection{Evaluation metrics}\nwe use both automatic metrics and human evaluations to evaluate the quality of generated novel comments.\n\n\\noindent{\\textbf{Automatic metrics}}:~1) \\textbf{BLEU} is commonly employed in evaluating translation systems. It is also introduced into comment generation task~\\citep{qin-etal-2018-automatic, yang-etal-2019-read}. we use $multi{-}bleu.perl$\\footnote{https:\/\/github.com\/moses-smt\/mosesdecoder\/blob\/master\/scripts\/generic\/multi-bleu.perl} to calculate the BLEU score. 2) \\textbf{ROUGE-L}(\\citep{lin-2004-rouge}) uses longest common subsequence to calculate the similar score between candidates and references. For calculation, we use a python package called $pyrouge$\\footnote{https:\/\/pypi.org\/project\/pyrouge\/}. These metrics also support the multi-reference evaluations on our dataset.\n\n\\noindent{\\textbf{Human evaluations}}:~1) \\textbf{Relevance}: This metric evaluates how relevant is the comment to the passage. It measures the degree that the comment is about the main storyline of the novel.2) \\textbf{Fluency}: This metric evaluates whether the sentence is fluent and judges whether the sentence follows the grammar and whether the sentence has clear logic. 3) \\textbf{Informativeness}: This metric evaluates how much structured knowledge the comment contains. It measures whether the comment reflects the evolution of entities and relations, or is just a general description that can be used for many passages. All these metrics have three gears, the final scores are projected to 0$\\sim$3.\n\n\\subsection{Baseline Models}\nWe describe three kinds of models used as baselines. All the baselines are implemented according to the related works and tuned on the validation set.\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{lccccc}\n\\hline \\textbf{Model} & \\textbf{BLEU} &\\textbf{ROUGE-L}\n& \\textbf{Relevance} & \\textbf{Fluency} & \\textbf{Informativeness}\\\\ \\hline\n\\textbf{Seq2Seq}\\citep{qin-etal-2018-automatic} & 2.59 & 14.71 & 0.12 & 1.71 & 0.09\\\\\n\\textbf{Attn}\\citep{qin-etal-2018-automatic} & 3.71 & 16.44 & 0.34 & 1.70 & 0.33 \\\\\n\\textbf{Trans}\\citep{Transformer:17} & 6.11 & 19.21 & 0.57 & 1.62 & 0.58 \\\\\n\\textbf{Trans.+CTX}\\citep{zhang-etal-2018-improving} & 6.52 & 19.11 & 0.68 & 1.68 & 0.67\\\\\n\\textbf{Graph2Seq}\\citep{li-etal-2019-coherent} & 4.93 & 16.91 & 0.35 & 1.69 & 0.31 \\\\\n\\textbf{Graph2Seq++}\\citep{li-etal-2019-coherent} & 5.56 & 17.51 & 0.85 & 1.67 & 0.60\\\\\n\\hline\n\\textbf{EKG} & 6.59 & 20.00 & 0.81 & \\textbf{1.83} & 0.64 \\\\\n\\textbf{EKG+GAT(V)} & 6.72 & 20.09 & 0.88 & 1.77 & 0.70 \\\\\n\\textbf{EKG+GAT(V+E)} & \\textbf{7.01} & \\textbf{20.10} & \\textbf{0.89}  & 1.74 & \\textbf{0.75}  \\\\\n\\hline\n\\textbf{Human Performance} & 100 & 100 & 1.09 & 1.85 & 1.04\\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{metric-table} Automatic metrics and human evaluations. }\n\\end{table*}\n\\begin{itemize}\n  \\item \\textbf{Seq2Seq models}~\\citep{qin-etal-2018-automatic}:~those models generate comments for news either from the title or the entire article. Considering there are no titles in our dataset, We compare two kinds of models from their work. 1) \\textbf{Seq2Seq:} it is a basic sequence-to-sequence model~\\citep{Sutskever:14} that generate comments from the passage; 2) \\textbf{Attn:} sequence-to-sequence model with an attention mechanism\\citep{Bahdanau:14}. For the input of the attention model, we append the related entities to the back of the passage.\n\n  \\item \\textbf{Self-attention models}:~our model includes a graph encoder to encode knowledge from graph, and a passage encoder use multiple self-attention layers. To show the power of graph encoder, we use the encoder-decoder framework (\\textbf{Trans.})~\\citep{Transformer:17} for passage-based comparison. Also we introduce an improved Transformer~\\citep{zhang-etal-2018-improving} with a context encoder to represent document-level context and denote it \\textbf{Trans.+CTX}. For the context input, we use up to 512 tokens before the passage as context.\n\n  \\item \\textbf{Graph2Seq}~\\citep{li-etal-2019-coherent}:~this is a graph-to-sequence model that builds a static topic graph for the input and generates comments based on representations of entities only. A two-layer transformer encoder is used in their work. For fair comparison, we use 6-layer transformer encoder to replace the original and denote the new model as \\textbf{Graph2Seq++}.\n\\end{itemize}\n\n\n\\subsection{Evaluation results}\nTable~\\ref{metric-table} shows the results of both automatic metrics and human evaluations.\n\nIn automatic metrics, our proposed model has best BLEU and ROUGE-L scores.\nFor BLEU, our full model EKG+GAT(V+E) achieves 7.01 score, which is 0.59 higher than that of the best baseline Trans.+CTX.\nThe Graph2Seq++ has a BLEU score 5.56 which is obviously lower than the EKG+GAT(V+E). The main reason is that the Graph2Seq++ is based on static graph and cannot make use of the dynamic knowledge.\nFor Rouge-L, our models all have ROUGE-L scores higher than 20\\%, and is 0.79\\% slightly better than that of Trans., which is the best among all baselines; the ROUGE-L score of Trans. is higher than of the Trans.+CTX, which is opposite of that in BLEU.\n\nIn human evaluations, we randomly select 100 passages from the test set and run all the models in Table~\\ref{metric-table} to generate respective comments. We also provide one user comment for each passage to get evaluations of human performance. All these passage-comment pairs are labeled by human annotators.\nIn relevance metric, our full model EKG+GAT(V+E) has better relevance score than all the baselines. It means our model can generate more relevant comments and better reflect the main storyline of the novel. For all this, there still exists significant gaps when compared to the human performance.\nIn fluency and informativeness metrics, our EKG+GAT(V+E) model has achieves higher score compared to all baselines. It illustrates that the generated comments by our proposed model are more fluent and contains more attractive information.\n\n\\begin{table*}[!t]\\small\n\\centering\n\\begin{tabular}{|l|}\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P1:}~\\textbf{\u8fd9\u4eba\u5bb6\u59d3\u66fe,\u4f4f\u5728\u53bf\u57ce\u4ee5\u5357\u4e00\u767e\u4e09\u5341\u91cc\u5916\u7684\u8377\u53f6\u5858\u90fd\u3002}\\\\\n(This family, surnamed Zeng, lives in Heyetangdu, 130 miles \\textcolor{orange}{south of the county}.)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T1:}~\u539f\u6765\u662f\u8fd9\u4e48\u6765\u7684\u3002~(That's how it turned out.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G1:}~\u8fd9\u4e2a\u5730\u65b9\u5440!~(This is the place !)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E1:}~\u4ed6\u4e00\u76f4\u601d\u5ff5\u7740\u5bb6\u91cc\u4eba\u3002~(He has been \\textcolor{cyan}{missing his family}.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P2]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P2:}~\\textbf{\u56fd\u85e9\u4eca\u65e5\u4e43\u6234\u5b5d\u4e4b\u8eab,\u8001\u6bcd\u5e76\u672a\u5b89\u846c\u59a5\u5e16,\u600e\u5fcd\u79bb\u5bb6\u51fa\u5c71?}\\\\\n(Today Guofan is wearing mourning. \\textcolor{cyan}{My mother hasn't been buried yet}. How can I leave home ?)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T2:}~\u771f\u662f\u4e00\u4e2a\u806a\u660e\u4eba~(He is so clever.)\\\\}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G2:}~\u8001\u592a\u592a\u4e5f\u662f\u4e2a\u597d\u4eba~(The old lady is also a good person.)\\\\}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E2:}~\u8fd9\u4e2a\u65f6\u5019\u7684\u56fd\u5bb6\u5df2\u7ecf\u6709\u4e86\u53d8\u5316~(\\textcolor{green}{The country is changing} at this time.)}\\end{CJK*} ~\\textcolor{blue}{\\textbf{[P3]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P3:}~\\textbf{\u9762\u4e34\u5927\u654c,\u66fe\u6697\u81ea\u4e0b\u5b9a\u51b3\u5fc3,\u4e00\u65e6\u57ce\u7834,\u7acb\u5373\u81ea\u520e,\u8ffd\u968f\u5854\u9f50\u5e03\u3001\u7f57\u6cfd\u5357\u4e8e\u5730\u4e0b\u3002}}\\end{CJK*} \\\\\n(\\textcolor{green}{Facing the enemy}, Zeng made up his mind to \\textcolor{red}{commit suicide as soon as the city broke}, following Ta Qibu and Luo Zenan.)\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T3:}~\u8fd9\u5c31\u662f\u539f\u6765\u6218\u4e89\u7684\u6837\u5b50}\\end{CJK*} (This is what the war looks like.)\\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G3:}~\u4e00\u4e2a\u4eba\u7684\u547d\u8fd0\u603b\u662f\u5982\u6b64}\\end{CJK*}(One's destiny is always like this.)\\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E3:}~\u81ea\u7acb\u4e8e\u5357\u57ce,\u81ea\u7834\u800c\u7acb~(He established himself in \\textcolor{orange}{the south of the country} throught constant breakthroughs.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P1]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P4:}~\\textbf{\u66fe\u56fd\u85e9\u7684\u8138\u4e0a\u9732\u51fa\u4e00\u4e1d\u6d45\u6d45\u7684\u7b11\u610f,\u5934\u4e00\u6b6a,\u5012\u5728\u592a\u5e08\u6905\u4e0a.}\\\\\n(Zeng Guofan smiled slightly. His head tilted and fell on the chair.)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T4:}~\u8fd9\u4e00\u6bb5\u63cf\u5199\u771f\u7684\u5f88\u6709\u753b\u9762\u611f~(This description is really picturesque.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G4:}~\u8fd9\u4e2a\u4eba\u7684\u5fc3\u601d\u7f1c\u5bc6~(This man is very thoughtful.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E4:}~\u4ed6\u4e00\u751f\u5fe0\u541b\u4e3a\u56fd,\u5c31\u8fd9\u6837\u8d70\u4e86~(He was \\textcolor{red}{loyal to his country} all his life; he is gone.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P3]}}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{case-table}Comments generated by Trans.+CTX~(\\textbf{T}), Graph2Seq++~(\\textbf{G}) and our EKG+GAT(V+E)~(\\textbf{E}). The passages (i.e., P1, P2, P3, P4) are extracted from the same novel called \\emph{Zeng Guofan}. We highlight the passage corresponding to the generated comment from our model~\\textbf{E} with \\textcolor{blue}{blue color}. Moreover, the relevant fragments are marked with a same color.}\n\\end{table*}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/ablation.png}\n  \\caption{Ablation results about number of entities (a) and number of time periods (b).}\\label{fig:ablation}\n\\end{figure}\n\\subsection{Analysis and Discussion}\n\\paragraph{Ablation study:} we compare the results of EKG, EKG+GAT(V), and EKG+GAT(V+E). The EKG, which does not use graph encoder, can achieve 6.59 BLEU score, which is 1.03 higher than that of Graph2Seq++. Then the BLEU score can be further improved to 6.72 by introducing a vertex-only variant EKG+GAT(V). Comparing EKG+GAT(V) and EKG+GAT(V+E) to the EKG, the BLEU scores increase 0.13 and 0.42 respectively; it indicates the usefulness of the graph encoder and that the evolutionary knowledge from edges can be treated as a good supplement to that of vertices. In human evaluations, EKG+GAT(V+E) and EKG+GAT(V) have higher relevance and informativeness scores than that of the EKG. It also indicates that the graph encoder can effectively utilize the evolutionary knowledge from vertices and edges, and make the generated comments more relevant and informative.\n\\paragraph{Analysis of the number of entities:} the corresponding local EKG is constructed based on the entities from the passage. To explore the influence of the number ($N$) of entities, we report BLEU scores of our full model based on different number\\footnote{We do not report the BLEU score of the full model when $N=1$ because there are no edges included.} in Figure~\\ref{fig:ablation}(a). The best BLEU score is achieved at $N=5$. The BLEU score at $N=0$ belongs to the Transformer(Trans.). And our full model is robust to the number of entities because the BLEU scores are stable when N is in the range $[2, 7]$.\n\\paragraph{Analysis of the number of time periods:} we also report the BLEU scores under the different number of time periods in Figure~\\ref{fig:ablation}(b). Our full model achieves the best BLEU score of 7.01 at $N=4$, which is 0.41 higher than that of the static graph at $N=1$. It illustrates that the dynamic knowledge is useful for improving the performance.\n\n\\paragraph{Case study:} we provide a case study here. Four passages that need to be commented are extracted from a novel chronologically and shown in Table~\\ref{case-table}. For comparison, we use \\textbf{T}rans.+CTX and \\textbf{G}raph2Seq++, which have the best relevance and informativeness scores among baselines respectively. To start with, within each case, we find that the generated comments from our model are more informative, while the generated outputs from other models tend to be general or common replies, which proves the effectiveness of our knowledge usage.\n\nFrom another perspective, we observe that our model can make use of the dynamics of knowledge. Let us take a look at \\textbf{P3}, our generated comment describes that \\textit{Zeng Guofan} was born in the south of the country, which is in accordance with the passage described in \\textbf{P1}. Similar interactions can be found in all four cases, which support our claims above.\n\\section{Conclusion}\nIn this paper, we propose to encode evolutionary knowledge for automatic commenting long novels. We learn an \\textit{Evolutionary Knowledge Graph} under a multi-task framework and then design a graph-to-sequence model to utilize the EKG for generating comments. In addition, we collect a new generation dataset called \\textit{GraphNovel} to advance the corresponding research. Experimental results show that our EKG-based model is superior to several strong baselines on both automatic metrics and human evaluations. In the future, we plan to develop new graph-based encoders to generate personalized comments with this dataset.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nAnisotropy of molecular interactions plays an important role in many\nphysical, chemical and biological processes. Attractive forces are\nresponsible for the tendency toward particle association, while the\ndirectionality of the resulting bonds determines the geometry of the\nresulting clusters. Aggregation may thus lead to very different structures:\nin particular, chains, globular forms, and bi- or three-dimensional\nnetworks. Understanding the microscopic mechanisms underlying such phenomena\nis clearly very important both from a theoretical and a technological point\nof view. Polymerization of inorganic molecules, phase behaviour of\nnon-spherical colloidal particles, building up of micelles, gelation,\nformation of $\\alpha $-helices from biomolecules, DNA-strands, and other\nordered structures in living organisms, protein folding and crystallization,\nself-assembly of nanoparticles into composite objects designed for new\nmaterials, are all subjects of considerable interest, belonging to the same\nclass of systems with anisotropic interactions.\n\nModern studies on these complex systems strongly rely upon computer\nsimulations, which have provided a number of useful information about many\nproperties of molecular fluids.\n\nNevertheless, analytic models with explicit expressions for structural and\nthermodynamic properties still represent an irreplaceable tool, in view of\ntheir ability of capturing the essential features of the investigated\nphysical systems.\n\nAt the lowest level in this hierarchy of minimal models on assembling\nparticles lies the problem of the formation of linear aggregates, from\ndimers \\cite{Spinozzi02,Giacometti05} up to polymer chains. This topic has\nbeen extensively investigated, through both computer simulations and\nanalytical methods. In the latter case a remarkable example is Wertheim's\nanalytic solution of the \\textit{mean spherical approximation} (MSA)\nintegral equation for dipolar hard spheres (DHS), i.e. hard spheres (HS)\nwith a point dipole at their centre \\cite{Wertheim71} (hereafter referred to\nas I). For the DHS model, several studies predict chain formation, whereas\nlittle can be said about the existence of a fluid-fluid coexistence line,\nsince computer simulations and mean field theories provide contradictory\nresults \\cite{Weis93,Leeuwen93,Sear96,Camp00,Tlusty00}. On the other hand,\nfor mesoscopic fluids the importance of combining \\textit{short-ranged}\nanisotropic attractions and repulsions has been well established \\cite%\n{Gazzillo06,Fantoni07}, and hence the long-range of the dipolar interaction\nis less suited for the mesoscopic systems considered here, at variance with\ntheir atomistic counterpart.\n\nThe aim of the present paper is to address both the above points, by\nstudying a model with anisotropic surface adhesion that is amenable to an\nanalytical solution, within an approximation which is expected to be valid\nat significant experimental regimes.\n\nIn the isotropic case, the first model with `surface adhesion' was\nintroduced long time ago by Baxter \\cite{Baxter68,Baxter71}. The interaction\npotential of these `sticky hard spheres' (SHS) includes a HS repulsion plus\na spherically symmetric attraction, described by a square-well (SW) which\nbecomes infinitely deep and narrow, according to a limiting procedure\n(Baxter's sticky limit) that keeps the second virial coefficient finite.\n\nPossible anisotropic variations include `sticky points'\\ \\cite%\n{Sciortino05,Bianchi06,Michele06,Lomakin99,Starr03,Zhang04,Glotzer04a,Glotzer04b,Sciortino07}%\n, `sticky patches'\\ \\cite%\n{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07} and, more\nrecently, `Gaussian patches'\\ \\cite{Wilber06,Doye07}. The most common\nversion of patchy sticky models refers to HS with one or more `uniform\ncircular patches', all of the same species. This kind of patch has a\nwell-defined circular boundary on the particle surface, and is always\nattractive, with an `uniform' strength of adhesion, which does not depend on\nthe contact point within the patch \\cite{Jackson88}.\n\nIn the present paper we consider a `dipolar-like' SHS model, where the sum\nof a uniform surface adhesion (isotropic background) plus an appropriate\ndipolar sticky correction -- which can be both positive or negative,\ndepending on the orientations of the particles -- yields a nonuniform\nadhesion. Although the adhesion varies continuously and no discontinuous\nboundary exists, the surface of each molecule may be regarded as formed by\ntwo hemispherical `patches' (colored red and blue, respectively, in the\nonline Figure 1). One of these hemispheres is `stickier' than the other, and\nthe entire molecular surface is adhesive, but its stickiness is nonuniform\nand varies in a dipolar fashion. By varying the dipolar contribution, the\ndegree of anisotropy can be changed, in such a way that the total sticky\npotential can be continuously tuned from very strong attractive strength\n(twice the isotropic one) to vanishing adhesion (HS limit). The physical\norigin of this model may be manifold (non-uniform distribution of surface\ncharges, or hydrophobic attraction, or other physical mechanisms), one\nsimple realization being as due to an `extremely screened' attraction. The\npresence of a solvent together with a dense ionic atmosphere could induce\nany electrostatic interaction to vanish close to the molecular surface, and\n-- in the idealized sticky limit -- to become \\textit{truncated }exactly at\ncontact.\n\nFor this model, we solve analytically the molecular Ornstein-Zernike (OZ)\nintegral equation, by using a truncated \\textit{Percus-Yevick} (PY)\napproximation, \\textit{with orientational linearization }(PY-OL), since it\nretains only the lowest order terms in the expansions of the correlation\nfunctions in angular basis functions. This already provides a clear\nindication of the effects of anisotropy on the adhesive adhesion.\n\nThe idea of an anisotropic surface adhesion is not new. In a series of\npapers on hydrogen-bonded fluids such a water, Blum and co-workers \\cite%\n{Cummings86,Wei88,Blum90} already studied models of spherical molecules with\nanisotropic pair potentials, including both electrostatic multipolar\ninteractions and sticky adhesive terms of multipolar symmetry. Within\nappropriate closures, these authors outlined the general features of the\nanalytic solutions of the OZ equation by employing a very powerful formalism\nbased upon expansions in rotational invariants. In particular, Blum,\nCummings and Bratko \\cite{Blum90} obtained an analytic solution within a\nmixed MSA\/PY closure (extended to mixtures by Protsykevich \\cite%\n{Protsykevich03}) for molecules which have surface adhesion of dipolar\nsymmetry and at most dipole-dipole interactions. From the physical point of\nview, our model -- with `dipolar-like' adhesion resulting from the sum of an\nisotropic plus a dipolar term -- is different and more specifically\ncharacterized with respect to the one of Ref. [32], whose adhesion has a\nsimpler, strictly `dipolar', symmetry. From the mathematical point of view,\nhowever, the same formalism employed by Blum \\textit{et al.} \\cite{Blum90}\ncould also be applied to our model. Unfortunately, the solution given in\nRef. [32] is not immediately usable for the actual computation of\ncorrelation functions, since the explicit determination of the parameters\ninvolved in their analytical expressions is lacking.\n\nIn the present paper we adopt a simpler solution method, by extending the\nelegant approach devised by Wertheim for DHS within the MSA closure \\cite%\n{Wertheim71}, and, most importantly, we aim at providing a \\textit{complete}\nanalytic solution -- including the determination of all required parameters\n-- within our PY-OL approximation.\n\nThe paper is organized as follows. Section II defines the model. In Section\nIII we recall the molecular OZ integral equation and the basic formalism. In\nSection IV we present the analytic solution. Numerical exact results for\nsome necessary parameters, as well as very accurate analytic approximations\nfor them, will be shown in Section V. Some preliminary plots illustrating\nthe effects of the anysotropic adhesion on the local structure are reported\nin Section VI. Phase stability is breafly discussed in Section VII, while\nfinal remarks and conclusions are offered in Section VIII.\\bigskip\n\n\\bigskip\n\n\\section{HARD SPHERES WITH\\ ADHESION OF\\ DIPOLAR-LIKE\\ SYMMETRY}\n\nLet the symbol $i\\equiv \\left( \\mathbf{r}_{i},\\Omega _{i}\\right) $ (with $%\ni=1,2,3,\\ldots $) denote both the position $\\mathbf{r}_{i}$ of the molecular\ncentre and the orientation $\\Omega _{i}$ of molecule $i$; for linear\nmolecules, $\\Omega _{i}\\equiv \\left( \\theta _{i},\\varphi _{i}\\right) $\nincludes the usual polar and azimuthal angles. Translational invariance for\nuniform fluids allows to write the dependence of the pair correlation\nfunction $g\\left(1,2\\right)$ as \n\\begin{equation*}\n(1,2)=(\\mathbf{r}_{12},\\Omega _{1},\\Omega _{2})=(r,\\Omega _{1},\\Omega _{2},%\n\\widehat{\\mathbf{r}}_{12})=(r,\\Omega _{1},\\Omega _{2},\\Omega _{r}),\n\\end{equation*}%\nwith $\\mathbf{r}_{12}=\\mathbf{r}_{2}-\\mathbf{r}_{1}$, $r=|\\mathbf{r}_{12}|$,\nand $\\Omega _{r}$ being the solid angle associated with $\\widehat{\\mathbf{r}}%\n_{12}=\\mathbf{r}_{12}\/r.$\n\nIn the spirit of Baxter's isotropic counterpart \\cite{Baxter68,Gazzillo04},\nour model is defined by the Mayer function given by \n\\begin{equation}\nf^{\\mathrm{SHS}}(1,2)=f^{\\mathrm{HS}}(r)+t\\ \\epsilon (1,2)\\ \\sigma \\delta\n\\left( r-\\sigma \\right) ,  \\label{eq2}\n\\end{equation}%\nwhere $f^{\\mathrm{HS}}(r)=\\Theta \\left( r-\\sigma \\right) -1$ is its HS\ncounterpart, $\\Theta $ is the Heaviside step function ($\\Theta (x<0)=0$, $%\n\\Theta (x>0)=1$) and $\\delta \\left( r-\\sigma \\right) $ the Dirac delta\nfunction, which ensures that the adhesive interaction occurs only at contact\n($\\sigma $ being the hard sphere diameter). An appropriate limit of the\nfollowing particular square well potential of width $R-\\sigma $ \n\\begin{equation*}\n\\Phi ^{\\mathrm{SW}}\\left( 1,2\\right) =\\left\\{ \n\\begin{array}{ccc}\n+\\infty \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ } &  & 0<r<\\sigma \\\\ \n-k_{B}T\\ \\ln \\left[ 1+t\\ \\epsilon (1,2)\\ \\frac{\\sigma }{R-\\sigma }\\ \\ \\right]\n&  & \\sigma <r<R \\\\ \n0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ } &  & \\text{ \\ \\ \\ \\ \\ \\ \\ }r>R\\text{ ,}%\n\\end{array}%\n\\right.\n\\end{equation*}%\ncan be shown to lead to Eq. (\\ref{eq2}).\n\nThe angular dependence is buried in the angular factor%\n\\begin{equation}\n\\epsilon (1,2)=1+\\alpha D(1,2),\n\\end{equation}%\nincluding the dipolar function%\n\\begin{equation*}\nD(1,2)=D(\\Omega _{1},\\Omega _{2},\\Omega _{r})=3(\\mathbf{u}_{1}\\cdot \\hat{%\n\\mathbf{r}})(\\mathbf{u}_{2}\\cdot \\hat{\\mathbf{r}})-\\mathbf{u}_{1}\\cdot \n\\mathbf{u}_{2}\n\\end{equation*}%\nwhich stems from the dipole-dipole potential $\\phi ^{\\mathrm{dip-dip}%\n}(1,2)=-\\mu ^{2}D(1,2)\/r^{3}$ ($\\mu $ is the magnitude of the dipole moment)\nand is multiplied by the tunable \\textit{anisotropy parameter} $\\alpha $. In\nthe isotropic case, $\\alpha =0$, one has $\\epsilon (1,2)=1$. Here and in the\nfollowing, $\\hat{\\mathbf{r}}$ coincides with $\\hat{\\mathbf{r}}_{12}=-\\hat{%\n\\mathbf{r}}_{21}$ , while $\\mathbf{u}_{i\\text{ }}$is the versor attached to\nmolecule $i$ (drawn as yellow arrow in Figure 1) which completely determines\nits orientation $\\Omega _{i}$. Note the symmetry $D(2,1)=D(1,2)$.\n\nThe condition $\\epsilon (1,2)\\geq 0$ must be enforced in order to preserve a\ncorrect definition of the sticky limit, ensuring that the total sticky\ninteraction remains attractive for all orientations, and the range of\nvariability $-2\\leq D(1,2)\\leq 2$ yields the limitation $0\\leq \\alpha \\leq \n\\frac{1}{2}$ on the anisotropy degree. The stickiness parameter $t$ -- equal\nto $\\left( 12\\tau \\right) ^{-1}$ in Baxter's original notation \\cite%\n{Baxter68} -- measures the strength of surface adhesion relatively to the\nthermal energy $k_{B}T$ ($k_{B}$ being the Boltzmann constant, $T$ the\nabsolute temperature) and increases with decreasing temperature.\n\nIf we adopt an `inter-molecular reference frame' (with both polar axis and\ncartesian $z$-axis taken along $\\mathbf{r}_{12}$), then the cartesian\ncomponents of $\\hat{\\mathbf{r}}$ and $\\mathbf{u}_{i}$ are $(0,0,1)$ and $%\n(\\sin \\theta _{i}\\cos \\varphi _{i}$, $\\sin \\theta _{i}\\sin \\varphi _{i}$, $%\n\\cos \\theta _{i})$, respectively, and thus%\n\\begin{equation}\nD(1,2)=2\\cos \\theta _{1}\\cos \\theta _{2}-\\sin \\theta _{1}\\sin \\theta\n_{2}\\cos \\left( \\varphi _{1}-\\varphi _{2}\\right) .  \\label{eq4b}\n\\end{equation}\n\nThe strength of adhesion between two particles $1$ and $2$ at contact\ndepends -- in a continuous way -- on the relative orientation of $\\mathbf{u}%\n_{1}$ and $\\mathbf{u}_{2}$ as well as on the versor $\\widehat{\\mathbf{r}}%\n_{12}$ of the intermolecular distance. We shall call \\textit{parallel} any\nconfiguration with $\\mathbf{u}_{1}\\cdot \\mathbf{u}_{2}=1$, while \\textit{%\nantiparallel} configurations are those with $\\mathbf{u}_{1}\\cdot \\mathbf{u}%\n_{2}=-1$ (see Figure 1). For all configurations with $D(1,2)>0$, the\nanisotropic part of adhesion is attractive and adds to the isotropic one.\nThus, the surface adhesion is maximum, and larger than in the isotropic\ncase, when $\\mathbf{u}_{1}=\\mathbf{u}_{2}=$ $\\widehat{\\mathbf{r}}_{12}$ and\nthus $\\epsilon (1,2)=1+2\\alpha $ (head-to-tail parallel configuration, shown\nin Figure 1b). On the contrary, when $D(1,2)<0$ the anisotropic contribution\nis repulsive and subtracts from the isotropic one, so that the total sticky\ninteraction still remains attractive. Then, the stickiness is minimum, and\nmay even vanish for $\\alpha =1\/2$, when $\\mathbf{u}_{1\\text{ }}=-$ $\\mathbf{u%\n}_{2}=\\widehat{\\mathbf{r}}_{12}$ and thus $\\epsilon (1,2)=1-2\\alpha $\n(head-to-head or tail-to-tail antiparallel configurations, reported in\nFigure 1c). The intermediate case of \\textit{orthogonal }configuration ($%\n\\mathbf{u}_{2}$ perpendicolar to $\\mathbf{u}_{1}$) corresponds to $D(1,2)=0,$\nwhich is equivalent to the isotropic SHS interaction.\n\nIt proves convenient to `split' $f^{\\mathrm{SHS}}(1,2)$ as \n\\begin{equation}\n\\text{\\ }f^{\\mathrm{SHS}}(1,2)=f_{0}(r)+f_{\\mathrm{ex}}(1,2),  \\label{eq6a}\n\\end{equation}%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nf_{0}(r)=f^{\\mathrm{HS}}(r)+t\\ \\sigma \\delta \\left( r-\\sigma \\right) \\equiv\nf^{\\mathrm{isoSHS}}(r) \\\\ \nf_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ \\sigma \\delta \\left( r-\\sigma\n\\right) \\ D(1,2),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{eq6b}\n\\end{equation}%\nwhere the spherically symmetric $f_{0}(r)$ corresponds to the `reference'\nsystem with isotropic background adhesion, while $f_{\\mathrm{ex}}(1,2)$ is\nthe orientation-dependent `excess' term.\n\nWe remark that, as shown in Ref. I (see also Table I in Appendix A of the\npresent paper), convolutions of $f^{\\mathrm{SHS}}$-functions generate\ncorrelation functions with a more complex angular dependence. Therefore, in\naddition to $D(1,2)$, it is necessary to consider also \n\\begin{equation}\n\\Delta (1,2)=\\mathbf{u}_{1}\\cdot \\mathbf{u}_{2}\\text{ }=\\cos \\theta _{1}\\cos\n\\theta _{2}+\\sin \\theta _{1}\\sin \\theta _{2}\\cos \\left( \\varphi _{1}-\\varphi\n_{2}\\right) ,\\ \n\\end{equation}%\nwhere the last equality holds true in the inter-molecular frame. The limits\nof variation for $\\Delta (1,2)$ are clearly $-1\\leq \\Delta (1,2)\\leq 1$.\n\n\\section{BASIC FORMALISM}\n\nThis section, complemented by Appendix A, presents the main steps of \\\nWertheim's formalism, as well as its extension to our model.\n\n\\subsection{Molecular Ornstein-Zernike equation}\n\nThe \\textit{molecular OZ integral equation} for a pure and homogeneous fluid\nof molecules interacting via non-spherical pair potentials is\n\n\\begin{equation}\nh(1,2)=c(1,2)+\\rho \\int d\\mathbf{r}_{3}\\ \\left\\langle \\ c(1,3)\\ h(3,2)\\\n\\right\\rangle _{\\Omega _{3}}\\ ,  \\label{oz4}\n\\end{equation}%\nwhere $h(1,2)$ and $c(1,2)$ are the total and direct correlation functions,\nrespectively, $\\rho $ is the number density, and $g(1,2)=1+h(1,2)$ is the\npair distribution function \\cite{Friedman85,Lee88,Hansen06}. Moreover, the\nangular brackets with subscript $\\Omega $ denote an average over the\norientations, i.e. $\\left\\langle \\cdots \\right\\rangle _{\\Omega }=\\left( 4\\pi\n\\right) ^{-1}\\int d\\Omega \\ \\cdots .$\n\nThe presence of convolution makes convenient to Fourier transform (FT) this\nequation, by integrating with respect to the space variable $\\mathbf{r}$\nalone according to\n\n\\begin{equation}\n\\widehat{F}\\left( \\mathbf{k},\\Omega _{1},\\Omega _{2}\\right) =\\int d\\mathbf{r}%\n\\ F(\\mathbf{r},\\Omega _{1},\\Omega _{2})\\ \\exp (i\\mathbf{k\\cdot r}).\n\\label{oz5}\n\\end{equation}%\nThe $\\mathbf{r}$-space convolution becomes a product in $\\mathbf{k}$-space,\nthus leading to%\n\\begin{equation}\n\\widehat{h}(\\mathbf{k},\\Omega _{1},\\Omega _{2})=\\widehat{c}(\\mathbf{k}%\n,\\Omega _{1},\\Omega _{2})+\\rho \\ \\left\\langle \\widehat{c}(\\mathbf{k},\\Omega\n_{1},\\Omega _{3})\\ \\widehat{h}(\\mathbf{k},\\Omega _{3},\\Omega\n_{2})\\right\\rangle _{\\Omega _{3}}\\ .  \\label{oz6}\n\\end{equation}\n\nAs usual the OZ equation involves two unknown functions, $h$ and $c$, and\ncan be solved only after adding a closure, that is a second (approximate)\nrelationship among $c$, $h$ and the potential.\n\n\\subsection{Splitting of the OZ equation: reference and excess part}\n\nThe particular form of our potential, as defined by the Mayer function of\nEq. (\\ref{eq2}), gives rise to a remarkable exact splitting of the original\nOZ equation. Using diagrammatic methods \\cite{Friedman85,Lee88,Hansen06} it\nis easy to see that both $c$ and $h$ can be expressed as graphical series\ncontaining the Mayer function $f$ as \\textit{bond function}. If $\\ f^{%\n\\mathrm{SHS}}=f_{0}+f_{\\mathrm{ex}}$ is substituted into all graphs of the\nabove series, each diagram with $n$ $f$-bonds will generate $2^{n}$ new\ngraphs. In the cluster expansion of $c$, the sum of all graphs having only $%\nf_{0}$-bonds will yield $c_{0}(r)=c^{\\mathrm{isoSHS}}(r)$, i.e. the the\ndirect correlation function (DCF) of the reference fluid with isotropic\nadhesion. On the other hand, all remaining diagrams have \\textit{at least one%\n} $f_{\\mathrm{ex}}$-bond, whose expression is given by Eq. (\\ref{eq6b}).\nThus, in the sum of this second subset of graphs it is possible to factorize \n$\\alpha t$, and we can write \n\\begin{equation}\n\\text{\\ }c^{\\mathrm{SHS}}(1,2)=c_{0}(r)+c_{\\mathrm{ex}}(1,2),  \\label{oz7a}\n\\end{equation}%\n\\begin{equation}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\left\\{ \n\\begin{array}{c}\nc_{0}(r)=c^{\\mathrm{isoSHS}}(r),\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\n\\\\ \nc_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ c^{\\dagger }(1,2).\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{oz7b}\n\\end{equation}%\nSimilarly, for $h$ we get\n\n\\begin{equation}\n\\text{\\ }h^{\\mathrm{SHS}}(1,2)=h_{0}(r)+h_{\\mathrm{ex}}(1,2),  \\label{oz8a}\n\\end{equation}%\n\\begin{equation}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\left\\{ \n\\begin{array}{c}\nh_{0}(r)=h^{\\mathrm{isoSHS}}(r),\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\n\\\\ \nh_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ h^{\\dagger }(1,2).\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{oz8b}\n\\end{equation}%\n\\ \\ \\ \\ \\ \n\nNote that this useful separation into reference and excess part may also be\nextended to other correlation functions, such as $\\gamma (1,2)\\equiv\nh(1,2)-c(1,2)$, $g(1,2)=1+h(1,2)$, and the `cavity' function $%\ny(1,2)=g(1,2)\/e(1,2)$. The function $\\gamma $ coincides with the OZ\nconvolution integral, without singular $\\delta $-terms. Similarly $y$ is\nalso `regular', and its exact expression reads $y\\left( 1,2\\right) =\\exp %\n\\left[ \\ \\gamma \\left( 1,2\\right) +B(1,2)\\ \\right] $, where the `bridge'\nfunction $B$ is defined by a complicated cluster expansion \\cite%\n{Friedman85,Lee88,Hansen06}.\n\nFrom Eqs. (\\ref{oz7a})-(\\ref{oz8b}), which are merely a consequence of the\nparticular form of $f_{\\mathrm{ex}}$ in the splitting of $f^{\\mathrm{SHS}}$,\none immediately sees that, if the anisotropy degree\\textit{\\ }$\\alpha $\ntends to zero, then%\n\\begin{equation}\n\\lim_{\\alpha \\rightarrow 0}c_{\\mathrm{ex}}(1,2)=\\lim_{\\alpha \\rightarrow\n0}h_{\\mathrm{ex}}(1,2)=\\lim_{\\alpha \\rightarrow 0}y_{\\mathrm{ex}}(1,2)=0.\n\\label{oz9}\n\\end{equation}\n\nNote that the spherically symmetric parts $c_{0}$ and $h_{0}$ must be\nrelated through the OZ equation for the reference fluid with isotropic\nadhesion (\\textit{reference OZ equation})\n\n\\begin{equation}\nh_{0}(r)=c_{0}(r)+\\rho \\int d\\mathbf{r}_{3}\\ c_{0}(r_{13})\\ h_{0}(r_{32}).\\ \n\\label{ozeq1}\n\\end{equation}%\nThus, substituting $c$ and $h$ of Eq. (\\ref{oz4}) with $c_{0}+c_{\\mathrm{ex}%\n} $ and $h_{0}+h_{\\mathrm{ex}}$, respectively, and subtracting Eq. (\\ref%\n{ozeq1}), we find that $c_{\\mathrm{ex}}$ and $h_{\\mathrm{ex}}$ must obey the\nfollowing relation\n\n\\begin{eqnarray*}\nh_{\\mathrm{ex}}(1,2) &=&c_{\\mathrm{ex}}(1,2)+\\rho \\int d\\mathbf{r}_{3}\\ %\n\\left[ \\ c_{0}(r_{13})\\ \\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle\n_{\\Omega _{3}}\\right. \\\\\n&&\\left. +\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ \\right\\rangle _{\\Omega\n_{3}}h_{0}(r_{32})+\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ h_{\\mathrm{ex}%\n}(3,2)\\ \\right\\rangle _{\\Omega _{3}}\\ \\right] \\ .\n\\end{eqnarray*}%\nand when \n\\begin{equation}\n\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ \\right\\rangle _{\\Omega\n_{3}}=\\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle _{\\Omega _{3}}=0\n\\label{eq_cond}\n\\end{equation}%\nthe orientation-dependent excess parts $c_{\\mathrm{ex}}$ and $h_{\\mathrm{ex}%\n} $ satisfy the equality \n\\begin{equation}\nh_{\\mathrm{ex}}(1,2)=c_{\\mathrm{ex}}(1,2)+\\rho \\int d\\mathbf{r}_{3}\\\n\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle\n_{\\Omega _{3}}\\ ,  \\label{ozeq2}\n\\end{equation}%\nwhich is decoupled from that of the reference fluid and may be regarded as\nan OZ equation for the excess part (\\textit{excess OZ equation}). As we\nshall see, condition (\\ref{eq_cond}) is satisfied in our scheme.\n\nWe stress that, in principle, the closures for Eq. (\\ref{ozeq1}) and Eq. (%\n\\ref{ozeq2}), respectively, might be \\textit{different}. In addition,\nalthough the two OZ equations\\ are decoupled, a suitably selected closure\nmight establish a relationship between $F_{0}$ and $F$ $(F=c,h).$\n\n\\subsection{Percus-Yevick closure with orientational linearization}\n\nFor hard-core fluids, $h$ and $c$ inside the core are given by \n\\begin{equation}\n\\left\\{ \n\\begin{array}{ccc}\nh(1,2)=-1\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } &  & \\text{for \\ }0<r<\\sigma ,\n\\\\ \nc(1,2)=-\\left[ 1+\\gamma (1,2)\\right] &  & \\text{ for \\ }0<r<\\sigma .%\n\\end{array}%\n\\right.  \\label{oz6a}\n\\end{equation}%\nAt the same time, we have the following \\textit{exact} relations \n\\begin{eqnarray*}\nh(1,2) &=&g(1,2)-1=e(1,2)y\\left( 1,2\\right) -1, \\\\\nc(1,2) &=&f(1,2)\\left[ 1+\\gamma (1,2)\\right] +e(1,2)\\left[ y\\left(\n1,2\\right) -1-\\gamma (1,2)\\right] .\n\\end{eqnarray*}\n\nSince $c$, $h$ and $g$ are discontinuous for hard-core fluids and involve $%\n\\delta $-terms for sticky particles, it is more convenient to define\nclosures in terms of $y$ and $\\gamma $, which are still continuous and\nwithout $\\delta $-singularities. The \\textit{Percus-Yevick} approximation\nfor molecular fluids with orientation-dependent interactions corresponds to\nassuming%\n\\begin{equation}\ny^{\\mathrm{PY}}\\left( 1,2\\right) =1+\\gamma (1,2)\\text{ \\ \\ \\ everywhere,}\n\\label{w3a}\n\\end{equation}%\nand thus, for the DCF,%\n\\begin{equation}\nc^{\\mathrm{PY}}\\left( 1,2\\right) =f(1,2)\\ \\left[ \\ 1+\\gamma (1,2)\\ \\right] \n\\text{,}  \\label{w3b}\n\\end{equation}%\nwhich implies that $c$ vanishes beyond the range of the potential.\n\nHowever, the dependence of $\\gamma (1,2)$ on angles may still be very\ncomplex. A possible procedure is to perform a series expansion of all\ncorrelation functions in terms of an \\textit{infinite} set of \\textit{%\nrotational invariants}, which are angular basis functions -- related to the\nspherical harmonics -- having the property of rotational invariance valid\nfor homogeneous fluids \\cite{Blum71}. Unfortunately, the full PY\napproximation requires an infinite number of expansion coefficients for both \n$c(1,2)$ and $h(1,2)$. This approach is usually impracticable, but sometimes\neven unnecessary, as it is possible that the most significant angular basis\nfunctions are included in a small closed subset of that infinite set. Indeed\nthis happens, for instance, in the DHS model within the MSA \\cite{Wertheim71}%\n, where the set $\\{1,\\Delta ,D\\}$ is the required subset. Although this does \n\\textit{not} happen in our model, we shall argue that the same truncation is\nsufficient due to the dipolar symmetry of the anisotropic adhesion.\n\nIndeed, a natural assumption is that the only nonzero harmonics in $c(1,2)$\nand $h(1,2)$ are those contained in $f(1,2)$ and those which can be obtained\nfrom that set by convolution \\cite{Cummings86}. Now, the angular basis\nfunctions included in our $f$-bond are only $1$ and $D$, but the convolution\nof two $f$-bonds involves the angular average of two $D^{\\prime }$s, which\nyields \\cite{Wertheim71}%\n\\begin{equation*}\n\\ \\left\\langle \\ D(\\mathbf{k},\\Omega _{1},\\Omega _{3})\\ D(\\mathbf{k},\\Omega\n_{3},\\Omega _{2})\\ \\right\\rangle _{\\Omega _{3}}=\\frac{1}{3}\\left[ D(\\mathbf{k%\n},\\Omega _{1},\\Omega _{2})+2\\Delta (\\mathbf{k},\\Omega _{1},\\Omega _{2})%\n\\right]\n\\end{equation*}%\nin $\\mathbf{k}$-space, and thus generates also $\\Delta $. Consequently, we\nwill expand any angle-dependent correlation function $F$ as%\n\\begin{equation}\nF(1,2)=F_{0}(r)+F_{\\Delta }(r)\\Delta (1,2)+F_{D}(r)D(1,2)+\\cdots \\ ,\n\\label{oz7}\n\\end{equation}%\nneglecting all higher order terms. In other words, we assume that all\nangular series expansions can be \\textit{truncated} after these first three\nterms, \\textit{linear} with respect to the angular basis functions. Using\nthis spirit in the PY approximation, given by Eq. (\\ref{w3b}), we obtain the\nfollowing PY correlation functions with \\textit{orientational linearization}\n(OL):%\n\\begin{eqnarray}\nc^{\\mathrm{PY-OL}}(1,2) &=&c_{0}(r)+c_{\\Delta }(r)\\Delta (1,2)+c_{D}(r)D(1,2)\n\\notag \\\\\n&=&c_{0}(r)+\\left( \\alpha t\\right) \\ \\left[ \\ c_{\\Delta }^{\\dagger\n}(r)\\Delta (1,2)+c_{D}^{\\dagger }(r)D(1,2)\\ \\right] ,  \\label{oz10aa}\n\\end{eqnarray}%\nand\n\n\\begin{eqnarray}\nh^{\\mathrm{PY-OL}}(1,2) &=&h_{0}(r)+h_{\\Delta }(r)\\Delta (1,2)+h_{D}(r)D(1,2)\n\\notag \\\\\n&=&h_{0}(r)+\\left( \\alpha t\\right) \\ \\left[ \\ h_{\\Delta }^{\\dagger\n}(r)\\Delta (1,2)+h_{D}^{\\dagger }(r)D(1,2)\\ \\right] .  \\label{oz10bb}\n\\end{eqnarray}%\nwhere%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nc_{0}(r)=\\Lambda _{0}\\text{\\ }\\sigma \\delta \\left( r-\\sigma \\right) \\\\ \nc_{\\Delta }(r)=\\Lambda _{\\Delta }\\text{\\ }\\sigma \\delta \\left( r-\\sigma\n\\right) \\\\ \nc_{D}(r)=\\Lambda _{D}\\ \\sigma \\delta \\left( r-\\sigma \\right)%\n\\end{array}%\n\\right\\} \\qquad \\text{for \\ }r\\geq \\sigma ,  \\label{closure}\n\\end{equation}%\nwith%\n\\begin{equation}\n\\text{\\ \\ }\\left\\{ \n\\begin{array}{l}\n\\Lambda _{0}=y_{0}^{\\mathrm{PY}}(\\sigma )\\ t\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{\\Delta }=y_{\\Delta }^{\\mathrm{PY}}(\\sigma )\\ t\\text{\\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{D}=\\left[ \\ y_{D}^{\\mathrm{PY}}(\\sigma )+\\alpha \\ y_{0}^{\\mathrm{PY%\n}}(\\sigma )\\right] \\ t = y_{D}^{\\mathrm{PY}}(\\sigma )\\ t+\\alpha \\ \\Lambda\n_{0} ,%\n\\end{array}%\n\\right.  \\label{oz21}\n\\end{equation}%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\ny_{0}^{\\mathrm{PY}}\\left( r\\right) =1+\\gamma _{0}(r)\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \ny_{\\Delta }^{\\mathrm{PY}}(r)=\\gamma _{\\Delta }(r)=\\left( \\alpha t\\right) \\\ny_{\\Delta }^{\\dagger }(r)\\text{ \\ \\ } \\\\ \ny_{D}^{\\mathrm{PY}}(r)=\\gamma _{D}(r)=\\left( \\alpha t\\right) \\\ny_{D}^{\\dagger }(r).\\text{ \\ }%\n\\end{array}%\n\\right.  \\label{ysigma}\n\\end{equation}\n\nClearly for $f(1,2)$ no truncation is required, as the expansion \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nf_{0}(r)=f^{\\mathrm{isoSHS}}(r)=f^{\\mathrm{HS}}(r)+t\\text{\\ }\\sigma \\delta\n\\left( r-\\sigma \\right) \\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nf_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nf_{D}(r)=\\left( \\alpha t\\right) \\ \\sigma \\delta \\left( r-\\sigma \\right) \n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.\n\\end{equation}%\nis exact.\n\nIt can be shown that $c(1,2)$ and $h(1,2)$ must have the same approximate\nform in view of the OZ equation, Eq. (\\ref{oz4}).\n\nThe solution of the original OZ equation (\\ref{oz4}) is then equivalent to\nthe calculation of the radial coefficients $c_{0}(r),c_{\\Delta }(r),c_{D}(r)$\nand $h_{0}(r),h_{\\Delta }(r),h_{D}(r)$, which are the projections of $c(1,2)$\nand $h(1,2)$ onto the angular basis $\\left\\{ 1,\\Delta ,D\\right\\} $. The \n\\textit{core condition} on $h$, Eq. (\\ref{oz6a}), becomes%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nh_{0}(r)=-1 \\\\ \nh_{\\Delta }(r)=0 \\\\ \nh_{D}(r)=0%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma .  \\label{oz10a}\n\\end{equation}\n\nNote that in the zero density limit $\\gamma (1,2)=\\rho \\int d\\mathbf{r}_{3}\\\n\\left\\langle \\ c(1,3)\\ h(3,2)\\ \\right\\rangle _{\\Omega _{3}}$ must vanish,\nand thus $y^{\\mathrm{PY}}(1,2)\\rightarrow 1$, i.e.\n\n\\begin{equation*}\n\\lim_{\\rho \\rightarrow 0}y_{0}^{\\mathrm{PY}}(r)=1\\text{, \\ \\ \\ \\ \\ \\ \\ }%\n\\lim_{\\rho \\rightarrow 0}y_{\\Delta }^{\\mathrm{PY}}(r)=\\lim_{\\rho \\rightarrow\n0}y_{D}^{\\mathrm{PY}}(r)=0,\n\\end{equation*}%\nwhile both $c\\left( 1,2\\right) $ and $h\\left( 1,2\\right) $ must reduce to $%\nf(1,2)$: \n\\begin{equation*}\n\\begin{array}{c}\n\\lim_{\\rho \\rightarrow 0}F_{0}(r)=f_{0}(r)\\text{ } \\\\ \n\\text{\\ }\\lim_{\\rho \\rightarrow 0}F_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\lim_{\\rho \\rightarrow 0}F_{D}(r)=f_{D}(r)\\text{ }%\n\\end{array}%\n\\text{\\ \\ \\ \\ \\ \\ }(F=c,h),\n\\end{equation*}%\nand%\n\\begin{equation}\n\\lim_{\\rho \\rightarrow 0}\\Lambda _{0}=t\\text{, \\ \\ \\ \\ \\ \\ \\ }\\lim_{\\rho\n\\rightarrow 0}\\Lambda _{\\Delta }=0\\text{, \\ \\ \\ \\ \\ \\ }\\lim_{\\rho\n\\rightarrow 0}\\Lambda _{D}=\\alpha t\\text{.}\n\\end{equation}\n\nMoreover as $\\alpha \\rightarrow 0$ all $\\Delta $- and $D$-coefficients of $%\nc,h$ and $y$ vanish, so that the isotropic adhesion case is recovered.\nFinally, it is also worth stressing that the \\textit{same} $\\delta $-term\narises in $c,$ $h$ and $g$, that is\n\n\\begin{equation*}\nF(1,2)=F_{\\mathrm{reg}}(1,2)+F_{\\text{sing}}(1,2)\\text{ \\ \\ \\ \\ \\ \\ (}%\nF=c,h,g),\n\\end{equation*}%\nwhere $F_{\\mathrm{reg}}$ is the `regular' part (i.e., the part with no $%\n\\delta $-singularity, and -- at most -- some step discontinuities), while $%\nF_{\\text{sing}}(1,2)=\\ \\sigma \\delta \\left( r-\\sigma \\right) \\Lambda (1,2)$\nis the singular term representing the anisotropic surface adhesion ( with$\\\n\\Lambda (1,2)=\\Lambda _{0}+\\Lambda _{\\Delta }\\Delta (1,2)+\\Lambda _{D}D(1,2)$\n\\ ).\n\n\\subsection{Integral equations for the projections of $c$ and $h$}\n\nIn the following, we extend Wertheim theory \\cite{Wertheim71} to our model,\nin order to obtain the radial projections of $c$ and $h$.\n\n\\ \\ The PY-OL approximation to the excess anisotropic part of the\ncorrelation functions is \n\\begin{equation}\n\\begin{array}{c}\nc_{\\mathrm{ex}}^{\\mathrm{PY-OL}}(1,2)=c_{\\Delta }(r)\\Delta\n(1,2)+c_{D}(r)D(1,2)\\text{ } \\\\ \nh_{\\mathrm{ex}}^{\\mathrm{PY-OL}}(1,2)=h_{\\Delta }(r)\\Delta\n(1,2)+h_{D}(r)D(1,2),%\n\\end{array}%\n\\end{equation}%\nthus verifying the required property $\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\\n\\right\\rangle _{\\Omega _{3}}=\\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\\n\\right\\rangle _{\\Omega _{3}}=0$ described in Section III, and allowing the\nsplitting of the molecular OZ equation into a reference and an excess part.\n\nThe first part is the \\textit{reference} PY equation, and coincides with\nthat solved by Baxter for the fluid with isotropic adhesion \\cite%\n{Baxter68,Baxter71}:\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{cc}\nh_{0}(r)=c_{0}(r)+\\rho \\ (h_{0}\\star c_{0})\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ } &  \\\\ \nh_{0}(r)=-1\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ } & 0<r<\\sigma \\\\ \nc_{0}(r)=\\Lambda _{0}\\ \\sigma \\delta \\left( r-\\sigma \\right) \\text{ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } & r\\geq \\sigma ,%\n\\end{array}%\n\\right.  \\label{w1}\n\\end{equation}%\nwhere the symbol $\\star $ denotes spatial convolution, i.e. $(A\\star\nB)(r_{12})=\\int A(r_{13})B(r_{32})\\;d\\mathbf{r}_{3}$.\n\nThe second part is the \\textit{excess PY-OL equation}, given by Eq. (\\ref%\n{ozeq2}) coupled with the PY-OL closure. Following an extension of\nWertheim's approach, as described in detail in Appendix A, Eq. (\\ref{ozeq2})\ncan be splitted into the following system for the $\\Delta -$ and $D-$%\nprojections of $c$ and $h:$ \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nh_{\\Delta }(r)=c_{\\Delta }(r)+\\frac{1}{3}\\rho ~\\left( \\ c_{\\Delta }\\star\nh_{\\Delta }+2\\ c_{D}^{0}\\star h_{D}^{0}\\ \\right) \\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n} \\\\ \nh_{D}^{0}(r)=c_{D}^{0}(r)+\\frac{1}{3}\\rho ~\\left( \\ c_{\\Delta }\\star\nh_{D}^{0}+c_{D}^{0}\\star h_{\\Delta }+c_{D}^{0}\\star h_{D}^{0}\\ \\right) ,%\n\\end{array}%\n\\right.\n\\end{equation}%\nwhere $c_{D}^{0}(r)$ and $h_{D}^{0}(r)$ are defined by the relationship%\n\\begin{equation}\nF_{D}^{0}(r)=F_{D}(r)-3\\int_{r}^{\\infty }\\frac{F_{D}(x)}{x}\\ dx\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ }(F=c,h),  \\label{oz12}\n\\end{equation}%\nwhose inverse is \\cite{Wertheim71} \n\\begin{equation}\nF_{D}(r)=F_{D}^{0}(r)-\\frac{3}{r^{3}}\\int_{0}^{r}F_{D}^{0}(x)\\ x^{2}\\ dx.\n\\label{oz12a}\n\\end{equation}\n\nThe core conditions become%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\n\\text{ \\ }h_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ } \\\\ \nh_{D}^{0}(r)=-3K%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma ,  \\label{oz14a}\n\\end{equation}%\nwith \n\\begin{equation}\nK=\\int_{\\sigma ^{-}}^{\\infty }\\frac{h_{D}(x)}{x}\\ dx=K_{\\mathrm{reg}%\n}+\\Lambda _{D}\\ ,  \\label{oz14b}\n\\end{equation}%\n\\begin{equation}\nK_{\\mathrm{reg}}=\\int_{\\sigma }^{\\infty }\\frac{h_{D,\\text{\\textrm{reg}}}(r)}{%\nr}\\ dr.  \\label{oz14bb}\n\\end{equation}%\nNote that the presence of the $\\delta $-singularity in $h_{D}(x)$ requires\nthe specification of $\\sigma ^{-}$ as lower integration limit, unlike the\ncase of Ref. I where only the regular part $K_{\\mathrm{reg}}$ is present.\nMoreover, since $h_{D}(r)=\\alpha t\\ \\ h_{D}^{\\dagger }(r)$, from Eq. (\\ref%\n{oz14b}) one could also write%\n\\begin{equation}\nK=\\alpha t\\ \\mathcal{K}\\text{ ,}\n\\end{equation}%\nwhich shows that $K$ is related to the anisotropy degree, and vanishes both\nin the symmetric adhesion case ($\\alpha =0$) and in the HS limit ($t=0$).\nSince $h_{D}(r)\\rightarrow f_{D}(r)=\\left( \\alpha t\\right) \\sigma \\delta\n(r-\\sigma )$ in the zero density limit, one then finds that\n\n\\begin{equation}\n\\lim_{\\eta \\rightarrow 0}K=\\alpha t.\n\\end{equation}\n\nFinally, the PY-OL closure for the new DCFs reads%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nc_{\\Delta }(r)=\\Lambda _{\\Delta }\\ \\sigma \\delta \\left( r-\\sigma \\right) \\\\ \nc_{D}^{0}(r)=\\Lambda _{D}\\ \\sigma \\delta \\left( r-\\sigma \\right)%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ }r\\geq \\sigma  \\label{oz14c}\n\\end{equation}%\n(for simplicity, here and in the following we omit the superscript PY-OL).\n\n\\subsection{Decoupling of the integral equations}\n\nIt is possible to \\textit{decouple} the two equations for $\\Delta $- and $D$%\n-coefficients, by introducing two new unknown functions, which are linear\ncombinations of the previous ones. As shown in Appendix A, if we define $%\nF_{1}(r)$ and $F_{2}(r)$ $\\left( F=c,h\\right) $ through the relations\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\nF_{1}\\left( r\\right) =\\left( 3\\mathcal{L}_{1}\\right) ^{-1}\\left[ F_{\\Delta\n}(r)-F_{D}^{0}(r)\\right] \\text{ } \\\\ \nF_{2}\\left( r\\right) =\\left( 3\\mathcal{L}_{2}\\right) ^{-1}\\left[ F_{\\Delta\n}(r)+2F_{D}^{0}(r)\\right]%\n\\end{array}%\n\\right. \\text{\\ \\ \\ \\ }\\left( F=c,h\\right) ,\n\\end{equation*}%\nthen we get the OZ equations\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\nh_{1}(r)=c_{1}(r)+\\rho _{1}\\ (h_{1}\\star c_{1})\\text{ } \\\\ \nh_{2}(r)=c_{2}(r)+\\rho _{2}\\ (h_{2}\\star c_{2}),%\n\\end{array}%\n\\right.\n\\end{equation*}%\nwith the following densities and core conditions\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{1}=\\mathcal{L}_{1}\\ \\rho \\text{ } \\\\ \n\\rho _{2}=\\mathcal{L}_{2}\\ \\rho ,%\n\\end{array}%\n\\right. \\qquad \\qquad \\left\\{ \n\\begin{array}{c}\nh_{1}(r)=K\/\\mathcal{L}_{1}\\text{ \\ \\ \\ } \\\\ \nh_{2}(r)=-2K\/\\mathcal{L}_{2}%\n\\end{array}%\n\\right. \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma .\n\\end{equation*}%\n\\ \\ \n\nThe decoupling of the three different projections of $c$ and $h$ is\nremarkable: the molecular anisotropic OZ equation reduces to a set of three\nradial integral relations, which may be regarded as OZ equations for three\n`hypothetical' fluids (labelled as $0,1,2$) with \\textit{spherically\nsymmetric} interactions. We stress that there is not a unique solution to\nthe decoupling problem, since - in principle - there exist infinite possible\nchoices for $\\left( \\mathcal{L}_{1},\\mathcal{L}_{2}\\right) $. The final\nresults are clearly independent of the values of $\\left( \\mathcal{L}_{1},%\n\\mathcal{L}_{2}\\right) $.\n\nIn the present paper, we adopt Wertheim's choice, i.e. $\\mathcal{L}_{1}=-K$\nand $\\mathcal{L}_{2}=2K$, which leads to%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{1}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D}^{0}(r)-F_{\\Delta }(r)\\right] \n\\text{ } \\\\ \nF_{2}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D}^{0}(r)+\\frac{1}{2}F_{\\Delta\n}(r)\\right]%\n\\end{array}%\n\\right. \\text{\\ \\ \\ \\ }\\left( F=c,h\\right) ,\n\\end{equation}\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{1}=-K\\rho \\text{ } \\\\ \n\\rho _{2}=2K\\rho ,%\n\\end{array}%\n\\right. \\qquad \\qquad \\left\\{ \n\\begin{array}{c}\nh_{1}(r)=-1 \\\\ \nh_{2}(r)=-1%\n\\end{array}%\n\\right. \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma\n\\end{equation}%\n(in Ref. I, $F_{1}$ and $F_{2}$ were denoted as $F_{-}$ and $F_{+}$,\nrespectively).\n\nNote that the auxiliary fluids have densities different from that of the\nreference fluid (the negative sign of $\\rho _{1}$ poses no special\ndifficulty).\n\nWe can also write\n\n\\begin{equation}\nF_{m}\\left( r\\right) =F_{m,\\mathrm{reg}}(r)+\\Lambda _{m}\\ \\sigma \\delta\n\\left( r-\\sigma \\right) ,  \\label{oz15a}\n\\end{equation}%\nwith\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{1,\\mathrm{reg}}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D,\\mathrm{reg}%\n}^{0}(r)-F_{\\Delta ,\\mathrm{reg}}(r)\\right] \\text{ } \\\\ \nF_{2,\\mathrm{reg}}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D,\\mathrm{reg}%\n}^{0}(r)+\\frac{1}{2}F_{\\Delta ,\\mathrm{reg}}(r)\\right]%\n\\end{array}%\n\\right.  \\label{oz15b}\n\\end{equation}%\nand%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\Lambda _{1}=\\frac{1}{3K}\\left( \\Lambda _{D}-\\Lambda _{\\Delta }\\text{\\ }%\n\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n=\\frac{1}{3K}\\left[ h_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})-h_{\\Delta ,\\text{%\n\\textrm{reg}}}(\\sigma ^{+})\\right] \\ t\\text{ }+\\alpha \\ \\frac{1}{3K}\\\n\\Lambda _{0}\\text{ \\ } \\\\ \n\\text{\\ \\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{2}=\\frac{1}{3K}\\left( \\Lambda _{D}+\\frac{1}{2}\\Lambda _{\\Delta\n}\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\text{ \\ \\ }=\\frac{1}{3K}\\left[ h_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})+\\frac{%\n1}{2}h_{\\Delta ,\\text{\\textrm{reg}}}(\\sigma ^{+})\\right] \\ t\\text{ }+\\alpha\n\\ \\frac{1}{3K}\\ \\Lambda _{0}\\text{.\\ \\ }%\n\\end{array}%\n\\right.  \\label{oz16}\n\\end{equation}%\n(since $\\gamma _{\\ldots }(\\sigma )=h_{\\ldots ,\\text{\\textrm{reg}}}(\\sigma\n^{+})-c_{\\ldots ,\\text{\\textrm{reg}}}(\\sigma ^{+})$, and $c_{\\ldots ,\\text{%\n\\textrm{reg}}}(\\sigma ^{+})=0$ within the PY-OL closure).\n\nKnowing the correlation functions $F_{1}(r)$ and $F_{2}(r)$ $\\left( \\text{%\nwith }F=c,h\\right) $, one can derive $F_{\\Delta }\\left( r\\right)\n,F_{D}^{0}\\left( r\\right) $, i.e. \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{\\Delta }\\left( r\\right) =2K\\left[ F_{2}\\left( r\\right) -F_{1}\\left(\nr\\right) \\right] \\ \\text{\\ \\ \\ \\ } \\\\ \nF_{D}^{0}\\left( r\\right) =2K\\left[ F_{2}\\left( r\\right) +\\frac{1}{2}%\nF_{1}\\left( r\\right) \\right] \\ ,%\n\\end{array}%\n\\right.  \\label{so11}\n\\end{equation}%\nand \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\Lambda _{\\Delta }=2K\\left( \\Lambda _{2}-\\Lambda _{1}\\right) \\text{\\ \\ \\ \\ }\n\\\\ \n\\Lambda _{D}=K\\left( 2\\Lambda _{2}+\\Lambda _{1}\\right) \\text{ .\\ \\ }%\n\\end{array}%\n\\right.  \\label{c10}\n\\end{equation}\n\nFinally, from $F_{\\Delta }\\left( r\\right) ,F_{D}^{0}\\left( r\\right) $ one\nhas to evaluate $F_{\\Delta }\\left( r\\right) ,F_{D}\\left( r\\right) $, by\nemploying Eq. (\\ref{oz12a}). We note the following points:\n\ni) Insertion of $h_{D}^{0}(r)=h_{D,\\text{\\textrm{reg}}}^{0}(r)+\\Lambda _{D}\\\n\\sigma \\delta (r-\\sigma )$ into Eq. (\\ref{oz12a}) yields $h_{D}(r)=h_{D,%\n\\text{\\textrm{reg}}}(r)+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma ),$ with%\n\\begin{equation}\nh_{D,\\text{\\textrm{reg}}}(r)=\\left\\{ \n\\begin{array}{cc}\n0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }0<r<\\sigma ,\\text{\\ \\ \\ \\ \\ \\ } &  \\\\ \nh_{D,\\text{\\textrm{reg}}}^{0}(r)+3r^{-3}\\left[ K_{\\mathrm{reg}}\\sigma\n^{3}-\\int_{\\sigma }^{r}h_{D,\\text{\\textrm{reg}}}^{0}(x)\\ x^{2}\\ dx\\right] \n\\text{ \\ \\ \\ \\ \\ \\ \\ } & r>\\sigma .%\n\\end{array}%\n\\right.  \\label{c11}\n\\end{equation}%\nAt $r=2\\sigma $ $h_{D,\\text{\\textrm{reg}}}$ and $h_{D,\\text{\\textrm{reg}}%\n}^{0}$ have the same discontinuity. We also get\n\n\\begin{equation}\nh_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})=h_{D,\\text{\\textrm{reg}}}^{0}(\\sigma\n^{+})+3K_{\\mathrm{reg}}.  \\label{f7}\n\\end{equation}%\nClearly, these results must agree with those obtained from Eq. (\\ref{oz12}),\ni.e. \n\\begin{equation*}\n\\begin{array}{c}\nh_{D}^{0}(r)=h_{D}(r)-3\\psi (r),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\psi (r)\\equiv \\int_{r}^{\\infty }h_{D}(x)\\ x^{-1}\\ dx=\\Lambda _{D}\\ \\theta\n\\left( \\sigma -r\\right) +\\int_{r}^{\\infty }h_{D,\\text{\\textrm{reg}}}(x)\\\nx^{-1}\\ dx.%\n\\end{array}%\n\\text{\\ }\n\\end{equation*}%\nIn order to recover Eq. (\\ref{f7}) along this second route, note that $\\psi\n(r)$ is not continuous at $r=\\sigma $. In fact, from Eqs. (\\ref{oz14b}) and (%\n\\ref{oz14bb}) follows $\\psi (\\sigma ^{-})=K$ whereas $\\psi (\\sigma ^{+})=K_{%\n\\mathrm{reg}}.$\n\nii) Similarly, for $c_{D}(r)$ we obtain $c_{D}(r)=c_{D,\\text{\\textrm{reg}}%\n}(r)+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma )$, with\n\n\\begin{equation}\nc_{D,\\text{\\textrm{reg}}}(r)=c_{D,\\text{\\textrm{reg}}}^{0}(r)-3r^{-3}\\left[\n\\int_{0}^{r}\\ c_{D,\\text{\\textrm{reg}}}^{0}(x)x^{2}\\ dx+\\Lambda _{D}\\sigma\n^{3}\\ \\theta (r-\\sigma )\\right] ,  \\label{c12}\n\\end{equation}%\nsince $\\int_{0}^{r}\\ \\delta (x-\\sigma )x^{2}dx=\\sigma ^{2}\\theta (r-\\sigma )$%\n. On the other hand, from Eq. (\\ref{oz12}) one easily finds that%\n\\begin{equation}  \\label{cDcD0}\nc_{D}(r)=c_{D}^{0}(r)\\text{ \\ \\ \\ for \\ }r\\geq \\sigma .\n\\end{equation}%\n\\ \\ \\ \n\niii) By applying the relationship (\\ref{oz12a}) to $c_{D}(r)$, using Eq. (%\n\\ref{cDcD0}) and noticing that $c_{D}(x)=0$ for $r>\\sigma $ within the PY-OL\napproximation, leads to a \\textit{sum rule}: \n\\begin{equation}\n\\int_{0}^{\\infty }c_{D}^{0}(x)\\ x^{2}\\ dx=\\int_{0}^{\\sigma }\\ c_{D,\\text{%\n\\textrm{reg}}}^{0}(x)x^{2}\\ dx+\\Lambda _{D}\\sigma ^{3}\\ =0,  \\label{f4b}\n\\end{equation}%\nthat we will exploit later.\n\n\\section{ANALYTIC\\ SOLUTION}\n\nWe have seen that the molecular PY-OL integral equation (IE) for our \\textit{%\nanisotropic}-SHS model splits into three IE's \n\\begin{equation}\n\\left\\{ \n\\begin{array}{cc}\nh_{m}(r)=c_{m}(r)+\\rho _{m}\\ (h_{m}\\star c_{m})\\text{\\ \\ } &  \\\\ \nh_{m}(r)=-1\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } & \n0<r<\\sigma \\\\ \nc_{m}(r)=\\Lambda _{m}\\ \\sigma \\delta \\left( r-\\sigma \\right) \\text{ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ } & r\\geq \\sigma%\n\\end{array}%\n\\right. \\qquad \\left( m=0,1,2\\right) ,  \\label{gpy1}\n\\end{equation}%\nwhere\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{0}=\\rho \\text{ \\ \\ \\ \\ \\ } \\\\ \n\\rho _{1}=-K\\rho \\text{ } \\\\ \n\\rho _{2}=2K\\rho ,%\n\\end{array}%\n\\right.  \\label{gpy2}\n\\end{equation}%\nand the `amplitudes' of the adhesive $\\delta $-terms are \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\text{ }\\Lambda _{0}=\\left[ 1+h_{0,\\text{\\textrm{reg}}}(\\sigma ^{+})\\right]\n\\ t=y_{0}^{\\mathrm{PY}}(\\sigma )t\\text{ }\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\text{ }\\Lambda _{m}=h_{m,\\text{\\textrm{reg}}}(\\sigma ^{+})\\ t\\text{ }+%\n\\mathcal{P} = y_{m}^{\\mathrm{PY}}(\\sigma )t+\\mathcal{P}\\text{ \\ \\ \\ \\ \\ }%\n\\left( m=1,2\\right) ,\\text{ }%\n\\end{array}%\n\\right.  \\label{gpy3}\n\\end{equation}%\nwith%\n\\begin{equation}\n\\mathcal{P}=\\frac{1}{3}\\ \\frac{\\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\ }{K}+%\n\\frac{K_{\\mathrm{reg}}}{K}\\ t=\\frac{1}{3}\\ \\frac{\\alpha \\ \\Lambda _{0}\\ }{K}+%\n\\frac{K_{\\mathrm{reg}}}{K}\\ t.  \\label{gpy4}\n\\end{equation}%\nHere, the new expressions of $\\Lambda _{1}$ and $\\Lambda _{2}$ have been\nobtained from Eqs. (\\ref{oz16}) with the help of Eqs. (\\ref{f7}) and (\\ref%\n{oz15b}).\n\nThe essential difference with respect to Ref. I lies in the closure, which\nis -- of course -- related to the model potential. While Wertheim's paper on\nDHS \\cite{Wertheim71} employed the MSA closure, which performs properly for\nlong-ranged electrostatic potentials at low strength of interaction, our\nPY-OL closure is more appropriate for the short-ranged potential of the\npresent model.\n\nThe first integral equation IE0 is fully independent, whereas IE1 and IE2\ndepend on the solution of IE0 (unlike the case of Ref. I), because of the\npresence of $\\Lambda _{0}$ inside $\\Lambda _{1\\text{ }}$and $\\Lambda _{2}$.\nWhile IE0 is exactly the PY equation for the \\textit{reference} SHS with \n\\textit{isotropic} adhesion solved by Baxter \\cite{Baxter68,Baxter71}, IE1\nand IE2 are different from both Wertheim's MSA solution for DHS and Baxter's\nPY solution for SHS. We remark that the closures for IE1 and IE2 are not PY\nas $\\Lambda _{1\\text{ }}$and $\\Lambda _{2}$ -- given by Eq. (\\ref{gpy3}) --\ndiffer, by the term $\\mathcal{P}$, from those appropriate for the PY choice,\ncorresponding to $\\Lambda _{m}^{\\mathrm{PY}}=y_{m}^{\\mathrm{PY}}(\\sigma )t$.\n\nConsequently, IE1 and IE2 can be rekoned as belonging to a class of \\textit{%\ngeneralized PY }\\ (GPY) \\textit{approximations}, introduced in Ref. [39],\nwhich admit an analytic solution. Thus, the PY-OL closure for $c(1,2)$ leads\nto a PY integral equation for $c_{0}(r)$, coupled to a two GPY integral\nequations for $c_{1}(r)$ and $c_{2}(r)$ (which are linear combinations of $%\nc_{\\Delta }(r)$ and $c_{D}^{0}(r)$).\n\nOn comparing the three IE's and their closures given by Eq. (\\ref{gpy1}), it\nis apparent that they have exactly the same form, but differ by the density $%\n\\rho _{m}$ and the expression for $\\Lambda _{m}$. The first integral\nequation IE0 corresponds to an isotropic SHS fluid with density $\\rho $. On\nthe other hand, IE1 and IE2 refer to `auxiliary' isotropic SHS fluids with\ndensities $\\rho _{1}$ and $\\rho _{2}$, and adhesion parameters $\\Lambda _{1}$\nand $\\Lambda _{2}$, respectively. Note that, according to Eqs. (\\ref{oz16})\n, $\\Lambda _{m}$ is not evaluated at the actual density $\\rho _{m}$ of the\nauxiliary fluid, but at the real density $\\rho $ . These remarks strongly\nsuggests that the solutions of IE0, IE1 and IE2 can be expressed in terms of\nan \\textit{unique} solution -- the PY one for isotropic SHS -- by changing\nonly $\\rho _{m}$ and $\\Lambda _{m}$. This can be achieved by the formal\nmapping \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{0}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{0},\\Lambda _{0})\\text{ } \\\\ \nF_{1}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{1},\\Lambda _{1})\\text{ } \\\\ \nF_{2}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{2},\\Lambda _{2}),%\n\\end{array}%\n\\right. \\qquad \\quad (F=q,c,h)  \\label{gpy5}\n\\end{equation}%\nwhere $\\eta _{0}=\\eta $ is the \\textit{real} volume fraction, while $\\eta\n_{1}$ and $\\eta _{2}$ are `modified volume fractions' of the `auxiliary'\nfluids $1$ and $2$, i.e.,\\ \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\eta _{0}=\\eta \\equiv \\left( \\pi \/6\\right) \\rho \\sigma ^{3}\\text{ \\ \\ } \\\\ \n\\eta _{1}=-K\\eta \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\eta _{2}=2K\\eta \\text{\\ .\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{gpy6}\n\\end{equation}%\nIn Eqs. (\\ref{gpy5}) $q(r)$ denotes the Baxter factor correlation function,\nintroduced in the next Subsection.\n\nIt is worth noting that this result for SHS mirrors the analog of the MSA\nsolution for DHS \\cite{Wertheim71} where all the three harmonic coefficients\nof can be expressed similarly, in terms of a single PY solution for the\nreference HS fluid.\n\n\\bigskip\n\n\\subsection{Baxter factorization}\n\nWe shall now solve Eqs. ($\\ref{gpy1}$) by using the Wiener-Hopf\nfactorization due to Baxter \\cite{Baxter71}. Let us recall its basic steps.\nAfter Fourier transforming the OZ equation for a one-component fluid with\nspherically symmetric interactions, one assumes the following factorization: \n\\begin{equation}\n\\begin{array}{c}\n1-\\rho \\widetilde{c}\\left( k\\right) =Q(k)Q(-k), \\\\ \nQ(k)=1-2\\pi \\rho \\int_{0}^{\\infty }q(r)\\ e^{ikr}dr.%\n\\end{array}\n\\label{bf1}\n\\end{equation}%\nThen it can be shown that the introduction of the `factor correlation\nfunction' $q(r)$ allows the OZ equation to be cast into the form \\cite%\n{Baxter71}\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nrc\\left( r\\right) =-q^{\\prime }(r)+2\\pi \\rho \\int_{0}^{\\infty }du\\ q\\left(\nu\\right) q^{\\prime }\\left( r+u\\right) ,\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nrh\\left( r\\right) =-q^{\\prime }(r)+2\\pi \\rho \\int_{0}^{\\infty }du\\ q\\left(\nu\\right) \\left( r-u\\right) h\\left( |r-u|\\right) ,%\n\\end{array}%\n\\right.  \\label{ie2b}\n\\end{equation}%\nwhere the prime denotes differentiation with respect to $r$. Solving these\nBaxter equations is tantamount to determining -- within a chosen closure --\nthe function $q(r)$, from which $c\\left( r\\right) $ and $h\\left( r\\right) $\ncan be easily calculated. It is also necessary to remember that, for \\textit{%\nall} closures leading to $c\\left( r\\right) =0$ for $r>\\sigma $, one finds $%\nq\\left( r\\right) =0$ for $r>\\sigma $ \\cite{Gazzillo04}.\n\nOn applying Baxter's factorization to Eqs. ($\\ref{gpy1}$), we get \n\\begin{equation}\nrh_{m}\\left( r\\right) =-q_{m}^{\\prime }(r)+2\\pi \\rho _{m}\\int_{0}^{\\sigma\n}du\\ q_{m}\\left( u\\right) \\left( r-u\\right) h_{m}\\left( |r-u|\\right) .\n\\label{ie3}\n\\end{equation}%\nwith $m=0,1,2$. Now the closure $c_{m}(r)=\\Lambda _{m}\\ \\sigma \\delta \\left(\nr-\\sigma \\right) $ for \\ $r\\geq \\sigma $ implies that the same $\\delta $%\n-term must appear in $h_{m}\\left( r\\right) $. Thus, for $0\\leq r\\leq \\sigma $%\n, using $h_{m}(r)=-1+\\Lambda _{m}\\ \\sigma \\delta \\left( r-\\sigma \\right) $,\nwe find\n\n\\begin{equation*}\nq_{m}^{\\prime }(r)=a_{m}r+b_{m}\\sigma -\\Lambda _{m}\\ \\sigma ^{2}\\delta\n\\left( r-\\sigma \\right) ,\n\\end{equation*}%\nwith%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\na_{m}=\\ 1-2\\pi \\rho _{m}\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ , \\\\ \nb_{m}\\sigma =\\ 2\\pi \\rho _{m}\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ u\\\n.\\text{ \\ \\ }%\n\\end{array}%\n\\right.  \\label{ie4}\n\\end{equation}\n\nThe $\\delta $-term of $q_{m}^{\\prime }(r)$ means that $q_{m}(r)$ has a\ndiscontinuity $q_{m}(\\sigma ^{+})-q_{m}(\\sigma ^{-})=-\\Lambda _{m}\\sigma\n^{2} $, with $q_{m}(\\sigma ^{+})=0.$ Integrating $q_{m}^{\\prime }(r)$,\nsubstituting this result into Eqs. ($\\ref{ie4}$), and solving the\ncorresponding algebraic system, we find the following solution\n\n\\begin{equation}\nq_{m}(r)=\\left\\{ \n\\begin{array}{cc}\n\\ \\frac{1}{2}a_{m}(r-\\sigma )^{2}+\\left( a_{m}+b_{m}\\right) \\sigma (r-\\sigma\n)\\ +\\Lambda _{m}\\ \\sigma ^{2} & \\text{ \\ \\ \\ \\ }0\\leq r\\leq \\sigma , \\\\ \n0 & \\text{ \\ \\ \\ otherwise,}%\n\\end{array}%\n\\right.  \\label{so1}\n\\end{equation}%\n\\begin{eqnarray}\na_{m} &=&\\ a^{\\mathrm{HS}}(\\eta _{m})-\\frac{12\\eta _{m}\\ \\Lambda _{m}\\ }{%\n1-\\eta _{m}}\\   \\label{so2} \\\\\n&&  \\notag \\\\\nb_{m} &=&\\ b^{\\mathrm{HS}}(\\eta _{m})+\\frac{6\\eta _{m}\\ \\Lambda _{m}\\ }{%\n1-\\eta _{m}}\\ \\   \\label{so3} \\\\\n&&  \\notag \\\\\n\\text{\\ }\\eta _{m} &=&\\left( \\pi \/6\\right) \\rho _{m}\\sigma ^{3}\\text{ \\ \\ \\ }\n\\\\\n&&  \\notag \\\\\na^{\\mathrm{HS}}(x) &=&\\frac{1+2x}{\\left( 1-x\\right) ^{2}},\\text{ \\ \\ \\ \\ \\ \\\n\\ }b^{\\mathrm{HS}}(x)=-\\frac{3x}{2\\left( 1-x\\right) ^{2}},\n\\end{eqnarray}\n\nFrom the first of Eqs. ($\\ref{ie2b}$) we get the DCFs $c_{m}(r)=c_{m,\\text{%\n\\textrm{reg}}}(r)+\\ \\Lambda _{m}\\ \\sigma \\delta (r-\\sigma )$, where $c_{m,%\n\\text{\\textrm{reg}}}(r)=0$ for $r\\geq \\sigma ,$ and for $0<r<\\sigma $\n\n\\begin{eqnarray}\nc_{m,\\text{\\textrm{reg}}}(r) &=&-\\frac{1}{2}\\eta _{m}\\ a_{m}^{2}\\left( \\frac{%\nr}{\\sigma }\\right) ^{3}+6\\eta _{m}\\ \\left[ \\ (a_{m}+b_{m})^{2}-2a_{m}\\\n\\Lambda _{m}\\ \\right] \\left( \\frac{r}{\\sigma }\\right)  \\notag \\\\\n&&-a_{m}^{2}-12\\eta _{m}\\ \\Lambda _{m}^{2}\\left( \\frac{r}{\\sigma }\\right)\n^{-1}.\n\\end{eqnarray}\n\nThe second of Eqs. ($\\ref{ie2b}$) yields the total correlation functions $%\nh_{m}(r)=h_{m,\\text{\\textrm{reg}}}(r)+\\ \\Lambda _{m}\\ \\sigma \\delta\n(r-\\sigma ).$ For $r>\\sigma $, Eqs. ($\\ref{ie3}$) becomes%\n\\begin{equation}\nH_{m,\\text{\\textrm{reg}}}\\left( r\\right) =12\\eta _{m}\\ \\sigma ^{-3}\\left\\{ \n\\begin{array}{cc}\n\\begin{array}{c}\n\\int_{0}^{r-\\sigma }du\\ q_{m}\\left( u\\right) \\ H_{m,\\text{\\textrm{reg}}%\n}\\left( r-u\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ } \\\\ \n+\\ \\int_{r-\\sigma }^{\\sigma }du\\ q_{m}\\left( u\\right) \\left( u-r\\right)\n+\\Lambda _{m}\\sigma ^{2}\\ q_{m}\\left( r-\\sigma \\right) \\text{ \\ \\ \\ }%\n\\end{array}\n& \\sigma <r<2\\sigma , \\\\ \n&  \\\\ \n\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ H_{m,\\text{\\textrm{reg}}}\\left(\nr-u\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }r>2\\sigma ,\\text{\\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ } & \n\\end{array}%\n\\right.  \\label{so7}\n\\end{equation}%\nwhere $H_{m}\\left( r\\right) \\equiv rh_{m}\\left( r\\right) $. Due to the last\nterm of Eq. ($\\ref{so7}$) and the discontinuity of $q_{m}\\left( r\\right) $\nat $r=\\sigma $, $h_{m,\\text{\\textrm{reg}}}(r)$ has a jump of at $r=2\\sigma $ \n\\cite{Kranendonk88,Miller04}: $h_{m,\\text{\\textrm{reg}}}(2\\sigma ^{+})-h_{m,%\n\\text{\\textrm{reg}}}(2\\sigma ^{-})=-6\\eta _{m}\\ \\Lambda _{m}^{2}.$\n\n\\bigskip\n\n\\subsection{An important relationship}\n\nIn Appendix B it is shown that a remarkable consequence of the sum rule (\\ref%\n{f4b}) is the condition%\n\\begin{equation}\na_{2}=a_{1}\\text{ ,}  \\label{so10}\n\\end{equation}%\nthat will play a significant role in the determination of the unknown\nparameters $\\Lambda _{1},$ $\\Lambda _{2}$ and $K$ \\ (see Appendix B).\n\n\\bigskip\n\n\\subsection{Reference fluid coefficients}\n\nThe $m=0$ case corresponds to Baxter's PY results for the reference fluid of\nisotropic SHS particles \\cite{Baxter68,Baxter71}. We have: $q_{0}(r)=q^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0}),$ and%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{0}(r)=c^{\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})=c_{\\text{\\textrm{reg}}}^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})+\\Lambda _{0}\\ \\sigma \\delta (r-\\sigma\n)\\text{ } \\\\ \nh_{0}(r)=h^{\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})=h_{\\text{\\textrm{reg}}}^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})+\\Lambda _{0}\\ \\sigma \\delta (r-\\sigma\n)%\n\\end{array}%\n\\right.  \\label{f1}\n\\end{equation}%\n(for simplicity, we omit -- here and in the following -- the superscript PY).\n\n\\subsection{$\\Delta -$ and $D-$coefficients}\n\nWe write $q_{m}(r)=q^{\\mathrm{isoSHS}}(r;\\eta _{m},\\Lambda _{m})$ with $%\nm=1,2 $. Then,\n\ni) For the $\\Delta $\\textit{-coefficients}, after recalling Eq. (\\ref{oz15a}%\n) and exploiting Eqs. (\\ref{c10}), we end up with:%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{\\Delta }(r)=2K\\left[ c_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda\n_{2})-c_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] +\\Lambda\n_{\\Delta }\\ \\sigma \\delta (r-\\sigma )\\text{\\ \\ \\ } \\\\ \nh_{\\Delta }(r)=2K\\left[ h_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda\n_{2})-h_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] +\\Lambda\n_{\\Delta }\\ \\sigma \\delta (r-\\sigma ).\\text{ \\ }%\n\\end{array}%\n\\right.  \\label{f2}\n\\end{equation}\n\nii) For the $D$\\textit{-coefficients}, we get\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{D}^{0}(r)=2K\\left[ c_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda _{2})+%\n\\frac{1}{2}c_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] \\\n+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma )\\text{ \\ } \\\\ \nh_{D}^{0}(r)=2K\\left[ h_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda _{2})+%\n\\frac{1}{2}h_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] \\\n+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma ).\\text{ }%\n\\end{array}%\n\\right.  \\label{f3}\n\\end{equation}%\nFinally, from $c_{D}^{0}(r)$ and $h_{D}^{0}(r)$ we can calculate $c_{D}(r)$\nand $h_{D}(r)$, as described by Eqs. (\\ref{c12}) and (\\ref{c11}),\nrespectively.\n\nIn short, a) our PY-OL solution -- $\\left\\{ c_{0},c_{\\Delta },c_{D}\\right\\} $\nand $\\left\\{ h_{0},h_{\\Delta },h_{D}\\right\\} $ -- satisfies both the PY\nclosures and the core conditions; b) all\\textit{\\ }coefficients contain a\nsurface adhesive $\\delta -$term; c) $\\left\\{ h_{0},h_{\\Delta },h_{D}\\right\\} \n$ all exhibit a step discontinuity at $r=2\\sigma $.\n\n\\bigskip\n\n\\section{EVALUATION OF THE PARAMETERS $K$, $\\Lambda _{1}$ AND $\\Lambda _{2}$}\n\nThe calculation of the Baxter functions $q_{m}s$ ($m=0,1,2$) requires the\nevaluation of $K,$ $\\Lambda _{1}$, and $\\Lambda _{2}$, for a given set of $%\n\\alpha ,\\eta $ and $t$ values, a task that we address next.\n\n\\bigskip\n\n\\subsection{Exact expressions}\n\nFour equations are needed to find the three quantities $\\Lambda\n_{m}=q_{m}(\\sigma ^{-})\/\\sigma ^{2}$ $(m=0,1,2)$, as well as the parameter $%\nK\\left( \\eta ,t,\\alpha \\right) $. We stress that the \\textit{almost fully\nanalytical} determination of these unknown parameters was lacking in Ref.\n[32] and represents an important part of the present work. Our detailed\nanalysis is given in Appendix B, and we quote here the main results.\n\ni) For $\\Lambda _{0}$, the same PY equation found by Baxter for isotropic\nSHS \\cite{Baxter68,Baxter71} \n\\begin{equation}\n12\\eta t\\ \\Lambda _{0}^{2}-\\left( 1+\\frac{12\\eta }{1-\\eta }t\\right) \\Lambda\n_{0}+y_{\\sigma }^{\\mathrm{HS}}(\\eta )t=0.  \\label{b5}\n\\end{equation}%\nOnly the smaller of the two real solutions (when they exist) is physically\nsignificant \\cite{Baxter68,Baxter71}, and reads\n\n\\begin{equation}\n\\Lambda _{0}=y_{0}^{\\mathrm{PY}}(\\sigma )t=\\frac{y_{\\sigma }^{\\mathrm{HS}%\n}(\\eta )t}{\\frac{1}{2}\\left[ 1+\\frac{12\\eta }{1-\\eta }t+\\sqrt{\\left( 1+\\frac{%\n12\\eta }{1-\\eta }t\\right) ^{2}-48\\eta \\ y_{\\sigma }^{\\mathrm{HS}}(\\eta )\\\nt^{2}}\\right] },  \\label{p1}\n\\end{equation}\n\nii) For $\\Lambda _{1}$ and $\\Lambda _{2}$, two other quadratic equations,\ni.e. \\ \\ \\ \\ \n\\begin{equation}\n12\\eta _{m}t\\ \\Lambda _{m}^{2}-\\left( 1+\\frac{12\\eta _{m}}{1-\\eta _{m}}%\nt\\right) \\Lambda _{m}+h_{\\sigma }^{\\mathrm{HS}}(\\eta _{m})t=-\\mathcal{P}%\n\\text{\\quad ~~}(m=1,2).  \\label{p1b}\n\\end{equation}\n\niii) The fourth equation is the following linear relationship between $%\n\\Lambda _{1}$ and $\\Lambda _{2}$%\n\\begin{equation}\n\\frac{12\\eta _{2}\\ \\Lambda _{2}\\ }{1-\\eta _{2}}-\\frac{12\\eta _{1}\\ \\Lambda\n_{1}\\ }{1-\\eta _{1}}=\\frac{\\eta _{2}\\left( 4-\\eta _{2}\\right) \\ }{\\left(\n1-\\eta _{2}\\right) ^{2}}-\\frac{\\eta _{1}\\left( 4-\\eta _{1}\\right) \\ }{\\left(\n1-\\eta _{1}\\right) ^{2}},  \\label{p1c}\n\\end{equation}%\n$\\ $which stems from the condition $a_{2}=a_{1}$.\n\nThe analysis of Appendix B gives\n\n\\begin{equation}\n\\Lambda _{2}\\left( \\eta _{1},\\eta _{2},t,\\alpha \\right) =\\Lambda _{1}\\left(\n\\eta _{2},\\eta _{1},t,\\alpha \\right)\n\\end{equation}%\nwith\n\n\\begin{equation}\n\\Lambda _{m}=\\Lambda +\\Lambda _{m}^{\\mathrm{ex}}\\text{ \\ \\ \\ \\ \\ \\ \\ }(m=1,2)\n\\label{p2a}\n\\end{equation}%\n\\begin{equation}\n\\Lambda =\\frac{1}{3}+\\frac{1}{4}\\left( \\frac{\\eta _{1}}{1-\\eta _{1}}+\\frac{%\n\\eta _{2}}{1-\\eta _{2}}\\right) =\\frac{1}{3}+\\allowbreak \\frac{x(1+4x)}{%\n4\\left( 1+x\\right) \\left( 1-2x\\right) }  \\label{p2b}\n\\end{equation}%\n\\begin{equation}\n\\Lambda _{1}^{\\mathrm{ex}}=\\frac{\\eta _{2}}{4\\left( 1-\\eta _{2}\\right) }%\nW_{0}^{\\mathrm{ex}},\\qquad \\Lambda _{2}^{\\mathrm{ex}}=\\frac{\\eta _{1}}{%\n4\\left( 1-\\eta _{1}\\right) }W_{0}^{\\mathrm{ex}},  \\label{p2c}\n\\end{equation}%\nwhere we have introduced $\\eta _{1}=-x$, $\\eta _{2}=2x$ \\ ( $x\\equiv K\\eta $\n), and $W_{0}^{\\mathrm{ex}}$ is defined in Appendix B. All these quantites\nare analytic functions of $x=K\\eta $. Thus, to complete the solution, we\nneed an equation for $K$, which can be written as\n\n\\begin{equation}\nK=\\alpha t\\ \\mathcal{K},\\text{ }\\ \\ \\ \\ \\ \\text{with \\ \\ \\ }\\ \\mathcal{K}%\n\\text{ }=\\frac{y_{0}^{\\mathrm{PY}}(\\sigma )}{Z(\\eta _{1},\\eta _{2},t)},\n\\label{p3}\n\\end{equation}%\n\\begin{equation}\nZ=\\frac{3}{2}\\left( \\Lambda _{1}+\\Lambda _{2}\\right) -3\\left\\{ \\frac{1}{2}%\n\\sum_{m=1}^{2}\\ \\left[ 12\\eta _{m}\\ \\Lambda _{m}^{2}-\\frac{12\\eta\n_{m}\\Lambda _{m}}{1-\\eta _{m}}+h_{\\sigma }^{\\mathrm{HS}}(\\eta _{m})\\right] +%\n\\frac{K_{\\mathrm{reg}}}{K}\\right\\} t  \\label{p4}\n\\end{equation}%\nand $\\lim_{\\eta \\rightarrow 0}Z(\\eta _{1},\\eta _{2},t)=1$. Insertion of\nfound expressions for $\\Lambda _{1},$ $\\Lambda _{2}$ and $K_{\\mathrm{reg}}$\n(see Appendix B) into Eq. (\\ref{p3}) yields a single equation for $K$ that\nwe have solved numerically, although some further analytic simplifications\nare probably possible.\n\nOur solution is then almost fully analytical, as only the final equation for \n$K$ is left to be solved numerically.\n\n\\subsection{Approximate expressions}\n\nFor practical use we next derive very accurate analytical approximations to $%\nK$, $\\Lambda _{1}$ and $\\Lambda _{2}$, which provide an useful tool for\nfully analytical calculations. Since in all cases of our interest we always\nfind $x=K\\eta \\ll 1$, a serie expansion leads to:\n\n\\begin{equation}\nW_{0}^{\\mathrm{ex}}=\\frac{2}{3}\\allowbreak \\left( 1+5x\\right) t+\\mathcal{O}%\n\\left( x^{2}\\right) ,\n\\end{equation}%\nand, consequently,\n\n\\begin{equation}\n\\Lambda _{1}^{\\mathrm{ex}}=\\frac{x\\left( 1+5x\\right) }{3\\left( 1-2x\\right) }%\nt+\\mathcal{O}\\left( x^{3}\\right) ,\\qquad \\Lambda _{2}^{\\mathrm{ex}}=-\\frac{%\nx\\left( 1+5x\\right) }{6\\left( 1+x\\right) }t+\\mathcal{O}\\left( x^{3}\\right) .\n\\label{r5}\n\\end{equation}%\nSimilarly we can expand $Z$ in Eq. (\\ref{p4}) as\n\n\\begin{equation}\nZ(x,t)=1+z_{1}(t)x+z_{2}(t)x^{2}+O\\left( x^{3}\\right) ,\n\\end{equation}%\nwith%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nz_{1}(t)=\\frac{1}{4}\\left( 3+11t\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nz_{2}(t)=\\frac{1}{4}\\left( 15+61t-4t^{2}\\right) .%\n\\end{array}%\n\\right.  \\label{r6}\n\\end{equation}%\nInsertion of this result into Eq. (\\ref{p3}) yields a cubic equation for $K,$%\n\\begin{equation*}\nz_{2}(t)\\eta ^{2}K^{3}+z_{1}(t)\\eta K^{2}+K-\\alpha t\\ y_{0}^{\\mathrm{PY}%\n}(\\sigma )=0,\n\\end{equation*}%\nwhich, again with the help of Eq. (\\ref{p3}), is equivalent to a cubic\nequation for $Z$ \n\\begin{equation}\nZ^{3}-Z^{2}+z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta \\right]\nZ+z_{2}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta \\right] ^{2}=0.\n\\label{r7}\n\\end{equation}%\nThe physically acceptable solution then reads%\n\\begin{equation}\nZ(\\eta ,t)=\\frac{1}{3}\\left( 1+\\sqrt[3]{\\mathcal{B}+\\sqrt{\\mathcal{B}^{2}-%\n\\mathcal{C}^{3}}}+\\sqrt[3]{\\mathcal{B}-\\sqrt{\\mathcal{B}^{2}-\\mathcal{C}^{3}}%\n}\\right) ,  \\label{r8}\n\\end{equation}%\nwhere\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\mathcal{B}=1+\\frac{9}{2}z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma\n)\\eta \\right] +\\frac{27}{2}z_{2}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}%\n}(\\sigma )\\eta \\right] ^{2} \\\\ \n\\mathcal{C}=1+3z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta %\n\\right] .\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{r9}\n\\end{equation}%\nIn conclusion, our approximate analytic solution for $K$, $\\Lambda _{1}$ and \n$\\Lambda _{2}$ includes three simple steps: i) calculate $K$ by using Eqs. (%\n\\ref{p3}), (\\ref{r8})-(\\ref{r9}), (\\ref{r6}); ii) evaluate $x=K\\eta $; iii)\nsolve for $\\Lambda _{1}$ and $\\Lambda _{2}$ by means of Eqs. (\\ref{p2b}) and\n(\\ref{r5}).\n\n\\bigskip\n\n\\subsection{Numerical comparison}\n\nIn order to assess the precision of previous approximations, we have\ncalculated $K$, $\\Lambda _{1}$ and $\\Lambda _{2}$ by two methods: i) solving\nnumerically Eqs. (\\ref{s2}), and ii) using our analytic approximations.\nAfter fixing $\\alpha =1\/2,$ we have increased the adhesion strength (or\ndecreased the temperature) from $t=0$ (HS limit) up to $t=0.8$, for some\nrepresentative values of the volume fraction ($\\eta =0.01$, $0.1$, $0.2$ and \n$0.4)$. The maximum value of $t$ corresponds to $\\tau =1\/(12t)\\simeq 0.1$,\nwhich lies close to the critical temperature of the isotropic SHS fluid. On\nthe other hand, $\\eta =0.01$ has been chosen to illustrate the fact that, as \n$\\eta \\rightarrow 0$, the parameter $K$ tends to $\\alpha t$. The linear\ndependence of $K$ on $t$ in this case is clearly visible in the top panel of\nFigure 2.\n\nIn Figures 2 and 3 the exact and approximate results for $K$, $\\Lambda _{1}$\nand $\\Lambda _{2}$ are compared. The agreement is excellent: at $\\eta =0.1$, \n$0.2$ and $0.4$, the relative error on $K$ does not exceed $0.1\\%$, $0.4\\%$\nand $1\\%$, respectively, while the maximum of the absolute relative errors\non $\\Lambda _{1}$ and $\\Lambda _{2}$ always remain less than $0.05$, $0.2$\nand $0.6~\\%$ in the three above-mentioned cases. It is worth noting that, as \n$\\eta $ increases, the variations of $\\Lambda _{1}$ and $\\Lambda _{2}$ are\nalways relatively small; on the contrary, $K$ experiences a marked change,\nwith a progressive lowering of the relevant curve.\n\n\\bigskip\n\n\\section{Some illustrative results on the local orientational structure}\n\nArmed with the knowledge of the analytic expression for the $q_{m}s$ a rapid\nnumerical calculation of the three harmonic coefficients $\\left\\{\nh_{0},h_{\\Delta },h_{D}\\right\\} $ appearing in\n\n\\begin{equation}\ng^{\\mathrm{PY-OL}}(1,2)=1+h_{0}(r)+h_{\\Delta }(r)\\Delta (1,2)+h_{D}(r)D(1,2).\n\\label{g1}\n\\end{equation}%\ncan be easily obtained as follows. From the second Baxter IE ($\\ref{ie2b}$),\none can generate $h(r)$ directly from $q(r)$, avoiding the passage through $%\nc(r)$. From $\\left\\{ q_{0},q_{1},q_{2}\\right\\} $ one first obtains $\\left\\{\nh_{0},h_{1},h_{2}\\right\\} $, by applying a slight extension of Perram's\nnumerical method \\cite{Perram75} and then derive $\\left\\{ h_{0},h_{\\Delta\n},h_{D}\\right\\} $, according to the above-mentioned recipes.\n\nThe main aim of the present paper was to present the necessary mathematical\nmachinery to investigate thermophysical properties. We now illustrate the\ninterest of the model by reporting some preliminary numerical results on the\norientational dependence of $g^{\\mathrm{PY-OL}}(1,2) $ -- i.e. on the local\norientational structure -- as a consequence of the anisotropic adhesion. A\nmore detailed analysis will be reported in a forthcoming paper.\n\nConsider the configuration depicted in Figure 4. Let a generic particle $1$\nbe fixed at a position $\\mathbf{r}_{1\\text{ }}$in the fluid with orientation \n$\\mathbf{u}_{1\\text{ }}$, and consider another particle $2$ located along\nthe straigth half-line which originates from the center of $1$ and with\ndirection $\\mathbf{u}_{1\\text{ }}$. This second particle has then a fixed\ndistance $r$ from $1$, but can assume all possible orientations $\\mathbf{u}%\n_{2\\text{ }}$, which -- by axial symmetry -- can be described by a single\npolar angle $\\theta \\equiv \\theta _{2}$ (i.e., the angle between $\\mathbf{u}%\n_{1\\text{ }}$and $\\mathbf{u}_{2\\text{ }}$) with respect to the\nintermolecular reference frame. Within this geometry, we have $\\left( \\theta\n_{1},\\varphi _{1}\\right) =(0,0)$ and $\\varphi _{2}=0$, obtaining $\\Delta\n(1,2)=\\cos \\theta $, $D(1,2)=2\\cos \\theta $. Consequently, $%\ng(1,2)=g(r,\\theta _{1},\\varphi _{1},\\theta _{2},\\varphi _{2})$ reduces to \n\\begin{equation}\ng(r,\\theta )=g_{0}(r)+\\left[ h_{\\Delta }(r)+2h_{D}(r)\\right] \\cos \\theta ,\n\\label{eq6}\n\\end{equation}%\nwhere $\\theta \\equiv \\theta _{2}$, and $g_{0}(r)=1+h_{0}(r)$ is the radial\ndistribution function of the reference isotropic SHS fluid.\n\nClearly, $g(r,\\theta )$ is proportional to the probability of finding, at a\ndistance $r$ from a given molecule $1$, a molecule $2$ having a \\textit{%\nrelative} orientation $\\theta $. We consider the three most significant\nvalues of this angle: i) $\\theta =0$, which corresponds to the `parallel'\nconfiguration of $\\mathbf{u}_{1}$ and $\\mathbf{u}_{2}$; ii) $\\theta =\\pi \/2$%\n, for the `orthogonal' configuration; and $\\theta =\\pi $, for the two\n`antiparallel' (head-to-head and tail-to-tail) configurations. From Eq. (\\ref%\n{eq6}) it follows that%\n\\begin{equation}\n\\begin{array}{c}\ng^{\\mathrm{par}}(r)=g(r,0)=g_{0}(r)+\\left[ h_{\\Delta }(r)+2h_{D}(r)\\right] ,%\n\\text{ \\ \\ \\ } \\\\ \ng^{\\mathrm{ortho}}(r)=g(r,\\pi \/2)=g_{0}(r),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \ng^{\\mathrm{antipar}}(r)=g(r,\\pi )=g_{0}(r)-\\left[ h_{\\Delta }(r)+2h_{D}(r)%\n\\right] .\\text{ \\ }%\n\\end{array}\n\\label{g2}\n\\end{equation}%\nNote that $g^{\\mathrm{ortho}}(r)$ coincides with the isotropic result $%\ng_{0}(r)$.\n\nIn Figures 5 we depict the above sections through the three-dimensional\nsurface corresponding to $g(r,\\theta )$, i.e., $g^{\\mathrm{par}}(r)$, $g^{%\n\\mathrm{ortho}}(r)$ and $g^{\\mathrm{antipar}}(r)$, for $\\eta =0.3$ with $%\nt=0.2$ and $t=0.6$, respectively, at the highest asymmetry value admissible\nin the present model, i.e. $\\alpha =1\/2$. The most significant features from\nthese plots are: i) $g^{\\mathrm{antipar}}(\\sigma ^{+})>g^{\\mathrm{par}%\n}(\\sigma ^{+})$; ii) for $r>2\\sigma $ $g^{\\mathrm{antipar}}(r)\\approx g^{%\n\\mathrm{par}}(r)\\approx g_{0}(r)$, i.e., the anisotropic adhesion seems to\naffect only the first coordination layer, $\\sigma <r<2\\sigma $, around each\nparticle.\n\nThe interpretation of these results is the following. In view of i) we see\nthat the parallel configuration is less probable than the antiparallel one\nat contact. Such a finding, together with ii), means that chain formation\ncharacteristic of polymerization is inhibited by the short-ranged\nanisotropic adhesion exploited here. This strictly contrasts with the case\nof long-ranged DHS fluids, where it is believed \\cite{Camp00,Tlusty00} that\nchaining phenomena might preempt the gas-liquid transition. This specific\nfeature of the present model is extremely interesting and we plan a\nthroughout investigation on this topic in a future publication.\n\n\\bigskip\n\n\\section{Phase stability}\n\nIn view of previous findings, a very natural question is whether the\naddition of our anisotropic sticky term to the potential changes phase\nstability and phase transition curves with respect to the corresponding\nisotropic case. We believe the answer to be positive. This is strongly\nsuggested by results obtained for similar anisotropic models, such as hard\nspheres with `sticky points'\\ \\cite%\n{Sciortino05,Bianchi06,Michele06,Lomakin99,Starr03,Zhang04,Glotzer04a,Glotzer04b,Sciortino07}\nor `sticky patches'\\ \\cite%\n{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07}.\n\nWe now briefly comment on this issue. Within our formalism, this problem of\nstability can be conveniently analyzed using standard formalism\ndevised to this aim \\cite{Gray84,Stecki81,Chen92,Klapp97}.\n\nWe start from the stability condition with respect to small but arbitrary\nfluctuations of the one-particle density $\\rho(1)$ from the equilibrium configuration,\ndenoted as 'eq' \\cite{Stecki81,Chen92,Klapp97}, \n\n\\begin{eqnarray}\n\\int d(1) \\int d(2) \\left [\\frac{\\delta(1,2)}{\\rho(1)} - c\\left(1,2\\right) \n\\right]_{eq} \\delta \\rho(1) \\delta \\rho(2) > 0.\n\\label{stability}\n\\end{eqnarray}\n\nHere $d(i)$ stands for $d \\mathbf{r}_i ~d \\Omega_i$, $i=1,2$, and we assume the\nequilibrium one-particle density to be $\\rho\/4\\pi$ \\cite{Stecki81,Chen92,Klapp97}. \n\nWe expand the fluctuations both in Fourier modes and in spherical harmonics\n\\cite{Gray84}\n\n\\begin{eqnarray}\n\\delta \\rho \\left(j\\right) \\equiv \\delta \\rho \\left(\\mathbf{r}_j,\\Omega_j\\right)\n&=& \\int \\frac{d \\mathbf{k}}{\\left(2\\pi\\right)^3} ~\\mathrm{e}^{\\mathrm{i} \\mathbf{k} \\cdot\n\\mathbf{r}_j} \\sum_{l=0}^{+\\infty} \\sum_{m=-l}^{+l} \\delta \\tilde{\\rho} \\left(\\mathbf{k}\n\\right) Y_{lm} \\left(\\Omega_j\\right). \n\\label{expansion}\n\\end{eqnarray}\n\nUsing the orthogonality relation \\cite{Gray84}\n\n\\begin{eqnarray}\n\\int d \\Omega ~Y_{lm}^{*} \\left(\\Omega\\right) Y_{l'm'}\\left(\\Omega\\right) &=&\n\\delta_{l l'} \\delta_{m m'},\n\\label{orthogonality}\n\\end{eqnarray}\n\nstandard manipulations \\cite{Klapp97} show that condition (\\ref{stability}) can\nbe recast into the form\n\n\\begin{eqnarray}\n\\sum_{l_{1},l_{2}=0}^{+\\infty} \\sum_{m_{1}=-l_{1}}^{+l_{1}} \\sum_{m_{2}=-l_{2}}^{+l_{2}}\n\\int \\frac{d \\mathbf{k}}{\\left(2 \\pi\\right)^3} ~ \\delta \\tilde{\\rho}_{l_{1} m_{1}} \n\\left(\\mathbf{k} \\right) \\delta \\tilde{\\rho}_{l_{1} m_{1}}^{*} \n\\left(\\mathbf{k} \\right) \\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &>0&,\n\\label{stability2}\n\\end{eqnarray}\n\nwhere the matrix elements $\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right)$\nare given by\n\n\\begin{eqnarray} \\label{matrix}\n\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &=& \\left(-1\\right)^{m_{1}}\n\\frac{4 \\pi}{\\rho} \\delta_{l_{1} l_{2}} \\delta_{m_{1},-m_{2}} -\n\\int d \\Omega_1 \\int d ~ \\Omega_{2} Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) \nY_{l_{2} m_{2}} \\left(\\Omega_2\\right) \\\\ \\nonumber\n&\\times&  \\int d\\mathbf{r} ~\\mathrm{e}^{\\mathrm{i} \\mathbf{k}\n\\cdot \\mathbf{r}} c\\left(\\mathbf{r},\\Omega_1,\\Omega_2\\right).\n\\end{eqnarray}\n\nThe problem of the stability has been reported to the character \nof the eigenvalues of matrix (\\ref{matrix}). This turns out to be particularly\nsimple in our case. Using the results (\\ref{wertheim_int}) it is easy to\nsee that\n\n\\begin{eqnarray}\n\\int d \\mathbf{r}~ \\mathrm{e}^{\\mathrm{i} \\mathbf{k} \\cdot \\mathbf{r}}\nc\\left(\\mathbf{r},\\Omega_1,\\Omega_2 \\right) &=& \\tilde{c}_{0} \\left(k\\right)\n+\\tilde{c}_{\\Delta} \\left(k\\right) \\Delta\\left(\\Omega_1,\\Omega_2\\right) +\n\\overline{c}_{D} \\left(k\\right) D\\left(\\Omega_1,\\Omega_2,\\Omega_k\\right).\n\\label{integral_c}\n\\end{eqnarray}\n\nInsertion of Eq.(\\ref{integral_c}) into Eq.(\\ref{matrix}) leads to\n\n\\begin{eqnarray} \\label{matrix2}\n\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &=& \n\\left(-1\\right)^{m_{1}}\n\\frac{4 \\pi}{\\rho} \\delta_{l_{1} l_{2}} \\delta_{m_{1},-m_{2}} \\\\ \\nonumber\n&-& \\left[\n\\tilde{c}_{0}\\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(0)}\n+\\tilde{c}_{\\Delta} \\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(\\Delta)}\n+\\tilde{c}_{D} \\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(D)}\n\\right],\n\\end{eqnarray}\n\n\\noindent where we have introduced the following integrals, which can be evaluated\nin the intermolecular frame, using standard properties of the\nspherical harmonics \\cite{Gray84}\n\n\\begin{eqnarray} \\label{integrals}\nI_{l_{1} m_{1} l_{2} m_{2}}^{(0)} &\\equiv& \\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) = 4 \\pi \\delta_{l_{1} 0,} \\delta_{l_{2},0} \\delta_{m_{1} 0}\n\\delta_{m_{2}} \\delta_{0} \\\\ \\nonumber\nI_{l_{1} m_{1} l_{2} m_{2}}^{(\\Delta)} &\\equiv& \\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) ~\\Delta\\left(\\Omega_1,\\Omega_2\\right)= \\frac{4}{3} \\pi \\delta_{l_{1} 1} \n\\delta_{l_{2},1} \\delta_{m_{1} 0} \\delta_{m_{2},0} \\\\ \\nonumber\nI_{l_{1} m_{1} l_{2} m_{2}}^{(D)} \\left(\\cos \\theta \\right) &\\equiv& \n\\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) D\\left(\\Omega_1,\\Omega_2,\\Omega_k\\right) \\\\ \\nonumber \n&=& \\frac{4}{3} \\pi \\delta_{l_{1},1} \n\\delta_{l_{2},1} \\delta_{m_{1}, 0}\n\\delta_{m_{2},0} ~2 ~P_{2}\\left(\\cos \\theta \\right) \n\\end{eqnarray}\n\nand where $P_2(x)=(3 x^2 -1)\/2$ is the second Legendre polynomial.\n\nHence, the matrix (\\ref{matrix}) is diagonal and the relevant terms\nare\n\n\\begin{eqnarray}\n\\tilde{A}_{0000} \\left(k\\right) &=& 4 \\pi\n\\left[ \\frac{1}{\\rho}-\\tilde{c}_{0} \\left(k\\right) \\right],\n\\label{element00}\n\\end{eqnarray}\nwhose positiveness is recognized as the isotropic stability condition,\nand\n\\begin{eqnarray}\n\\tilde{A}_{1010}\\left(\\mathbf{k}\\right) &=& 4 \\pi \\left\\{\n\\frac{1}{\\rho} -\\frac{1}{3} \\left[\\tilde{c}_{\\Delta}\\left(k\\right)\n+ 2 P_{2} \\left(\\cos \\theta\\right) \\overline{c}_{D}\\left(k\\right)\n\\right] \\right\\}.\n\\label{element11}\n\\end{eqnarray} \n\n\\noindent All remaining diagonal terms have the form $\\tilde{A}_{l0l0}=\n4 \\pi\/\\rho>0$. \n\nIn order to test for possible angular instabilities, we consider\nthe limit $k \\to 0$ of Eq.~(\\ref{element11}) namely\n\n\\begin{eqnarray}\n\\tilde{A}_{1010}\\left(0\\right) &=& \\frac{4 \\pi}{\\rho} \\left\\{\n1 -\\frac{\\rho}{3} \\left[\\tilde{c}_{\\Delta}\\left(0\\right)\n+ 2 P_{2} \\left(\\cos \\theta\\right) \\overline{c}_{D}\\left(0\\right)\n\\right] \\right\\}.\n\\label{element11k0}\n\\end{eqnarray} \n\\noindent\nThis can be quickly computed with the aid of Eqs.~(\\ref{so11}), (\\ref{bf1}),\nthe fact that $\\bar{c}_{D}(0)=\\tilde{c}_{D}^{0}(0)$ and the identity\n(\\ref{so10}). We find\n\n\\begin{eqnarray}\n\\label{element11_res}\n\\tilde{A}_{1010}\\left(0\\right) &=& \\frac{4 \\pi}{\\rho} a_1^2,\n\\end{eqnarray}\nwhich is independent of the angle $\\theta$. This value is found to be always\npositive as $a_1>0$ (see Fig.\\ref{fig6}).\nWithin this first-order approximation, therefore the only instability\nin the system stems from the isotropic compressibility. \nThe reason for this can be clearly traced back to the first-order\napproximation to the angular dependence of the correlation functions. If\nquadratic terms in $\\Delta $ and $D$ were included into the series expansion\nfor correlation functions, the particular combination leading to a cancellation\nof the angular dependence in the stability matrix \n$\\tilde{A}_{l_{1}m_{1}l_{2}m_{2}}\\left(0\\right)$ would not occur, leading to\na different result. \n\nThis fact is consistent with the more general statement that, in any\napproximate theory, thermodynamics usually requires a higher degree of\ntheoretical accuracy than the one sufficient for obtaining significant\nstructural data. Conceptually, the need of distinguishing structural results\nfrom thermodynamical ones is rather common. For instance, in statistical\nmechanics of liquids it is known that approximating the model potential only\nwith its repulsive part (for instance, the hard sphere term) can account for\nall essential features of the structure, but yields unsatisfactory\nthermodynamics. On the other hand, the present paper refers to a \\textit{%\nsimplified} statistical-mechanical tool, i.e. the OZ equation within our\nPY-OL closure, which has been explicitly selected to allow an analytical\nsolution. Our results however indicate that the first-order expansion used\nin the PY-OL closure can give reasonable information about structure, but\nnot on thermodynamics, where a higher level of sophistication is required.\n\n\\section{Concluding remarks}\n\nIn this paper we have discussed an anisotropic variation of the original\nBaxter model of hard spheres with surface adhesion. In addition to the HS\npotential, molecules of the fluid interact via an isotropic sticky\nattraction plus an additional anisotropic sticky correction, whose strength\ndepends on the orientations of the particles in dipolar way. By varying the\nvalue of a parameter $\\alpha $, the anisotropy degree can be changed.\nConsequenly, the strength of the total sticky potential can vary from twice\nthe isotropic one down to the limit of no adhesion (HS limit). These\nparticles may be regarded as having two non-uniform, hemispherical,\n`dipolar-like patches', thus providing a link with uniformly adhesive\npatches \\cite{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07}.\n\nWe have obtained a full analytic solution of the molecular OZ equation,\nwithin the PY-OL approximation, by using Wertheim's technique \\cite%\n{Wertheim71}. Our PY-OL approximation should be tested against exact\ncomputer simulations, in order to assess its reliability. Nevertheless, we\nmay reasonably expect the results to be reliable even at experimentally\nsignificant densities, notwithstanding the truncation of the higher-orders\nterms in the angular expansion. Only one equation, for the parameter $K$,\nhas to be solved numerically. In additon, we have provided analytic\napproximations to $K$, $\\Lambda _{1}$ and $\\Lambda _{2}$ so accurate that,\nin practice, the whole solution can really be regarded as fully analytical.\nFrom this point of view, the present paper complements the above-mentioned\nprevious work by Blum \\textit{et al. }\\cite{Blum90}.\n\nWe have also seen that thermophysical properties require a more detailed\ntreatment of the angular part than the PY-OL closure. Nonetheless, \neven within the PY-OL oversimplified framework, our findings\nare suggestive of a dependence of the fluid-fluid coexistence line on\nanisotropy.\n\nOur analysis envisions a number of interesting perspectives, already hinted\nby the preliminary numerical results reported here. It would be very\ninteresting to compare the structural and thermodynamical properties of this\nmodel with those stemming from truly dipolar hard spheres \\cite{Stecki81,\nChen92,Klapp97}.\nThe possibility of local orientational ordering can be assessed by computing\nthe pair correlation function $g(1,2)$ for the most significant\ninterparticle orientations. We have shown that this task can be easily\nperformed within our scheme. This should provide important information about\npossible chain formation and its subtle interplay with the location of the\nfluid-fluid transition line. The latter bears a particular interest in view\nof the fact that computer simulations on DHS are notoriously difficult and\ntheir predictions regarding the location of such a transition line have\nproven so far unconclusive \\cite{Frenkel02}. The long-range nature of DHS\ninteractions may in fact promote polymerization preempting the usual\nliquid-gas transition \\cite{Tlusty00}. Our preliminary results on the\npresent model strongly suggest that this is not the case for sufficiently\nshort-ranged interactions, thus allowing the location of such a transition\nline to be studied as a function of the anisotropy degree of the model. Our\nsticky interactions have only attractive adhesion, the only repulsive part\nbeing that pertinent to hard spheres, whereas the DHS potential is both\nattractive and repulsive, depending on the orientations.\n\nFinally, information about the structural ordering in the present model\nwould neatly complement those obtained by us in a recent parallel study on a\nSHS fluid with one or two uniform circular patches \\cite{Fantoni07}. Work\nalong this line is in progress and will be reported elsewhere.\n\n\\acknowledgments \nWe acknowledge financial support from PRIN 2005027330. It is our pleasure to\nthank Giorgio Pastore and Mark Miller for enlighting discussions on the\nsubject.\n\n\\bigskip ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nMillisecond pulsars are commonly believed to be descendants of normal neutron stars that have been spun-up and recycled back as radio pulsars by acquiring angular momentum from their companion during the low-mass X-ray binary (LMXB) phase \\citep{ACR82,RS82}.\n\nThere are about $\\sim$20 high confidence nuclear or accretion powered (see Table~1) millisecond X-ray pulsars (MSXPs) which are thought to be the progenitors of millisecond radio pulsars (MSRPs) \\citep{WK98}. These MSXPs may become observable in radio wavelengths once accretion ceases, or the column density of the plasma from the fossil disk around the neutron star becomes thin enough to allow vacuum gap formation that leads to the production of coherent radio emission. Towards the end of the secular LMXB evolution, as accretion rates fall below a critical value above which detection presumably may be hampered due to absorption or dispersion \\citep{TBE94}, the neutron star can re-appear as a MSRP.\n\nAlthough the connection between LMXBs and MSRPs has been significantly strengthened after the discovery of quasi-periodic kHz oscillations and X-ray pulsations in some transient X-ray sources \\citep{WK98, MSS02, GCM02, GMM05}, no radio pulsations from MSXPs have been detected so far \\citep{BBP03}. \n\nAt the end of the recycling process the neutron star will reach an equilibrium period \\citep{BH91} which is approximated by the Keplerian orbital period at the Alfven radius \\citep{GL92}:\n\n\\begin{equation}\nP_{eq} \\sim 1.9\\, ms\\, B_{9}^{6\/7} \\biggl (\\frac{M}{1.4 \\,M_{\\sun}}\\biggr)^{-5\/7} \\biggl(\\frac{\\dot{m}}{\\dot{M}_{Edd}}\\biggr)^{-3\/7} R^{16\/7}_{6}\n\\end{equation}\nwhere $B_{9}$ and $R_{6}$ are the neutron star surface magnetic dipole field and radius in units of $10^{9}$ G and $10^{6}$ cm respectively. The Eddington limited accretion rate $\\dot{M}_{Edd}$ for a neutron star typically is $\\sim10^{-8}\\,M_{\\sun}\\,yr^{-1}$ above which the radiation pressure generated by accretion will stop the accretion flow. This equilibrium period combined with the dominant mechanism for energy loss delineates the subsequent kinematics of the spun-up millisecond pulsar. The magnetic dipole model then implies a ``spin-up region'' ($\\dot{P}\\, \\sim\\, P_{0}^{4\/3} $) \\citep[see][]{ACW99} on which the recycled neutron stars will be reborn as MSRPs. At the end of the active phase, MSXPs accreting with $\\dot{m}$ and  spinning with $P_{eq}$, presumably transition into a MSRPs with an initial spin period of $P_{0} \\sim P_{eq}$.\n\nIn the standard spin-down model, the MSRP evolution is driven by pure magnetic dipole radiation, i.e. braking index $n=3$ in vacuum \\citep[see][]{MT77, LK04}.  Alternative energy loss mechanisms such as multipole radiation or gravitational wave emission, especially during the initial phases of the reborn millisecond pulsars, have been suggested by several authors  \\citep{K91,B98} but have yet to be observationally corroborated. Advanced Laser Interferometer Gravitational Wave Observatory (LIGO) will be able to probe the frequency space at which millisecond pulsars are expected to radiate gravitational waves, thereby putting stringent constraints on the micro physics of millisecond pulsars.\n\nThe advances in radio observations, increased sky coverage with deep exposures of current surveys combined with robust post-bayesian statistical techniques that incorporate minimal assumptions, give us unprecedented predictive power on the joint period-spindown ($P-\\dot{P}$) and implied magnetic field ($B$) distributions.\n\nIn this {\\it Letter}, we attempt to go beyond phenomenological arguments and test whether MSXPs can produce the characteristics of the observed MSRPs within the framework of the standard model \\citep[and references therein]{BH91}.\n\t\n\t\\section{\\label{sec:dist}The Joint Period - Spindown ($P-\\dot{P}$) Distribution}\n\t\t\t\t\t\t\t\nThe evolution of millisecond pulsars can be consistently described in terms of {\\bf i)} the equilibrium period distribution ($D$) of MSXPs at the end of the LMXB evolution {\\bf ii)} the mass accretion rates ($\\dot{M}$) of the progenitor population during the recycling process  {\\bf iii)} Galactic birth rates ($R$), and {\\bf iv)} the dominant energy loss mechanism after the onset of radio emission.  \n\n\t\t\t\\subsection{\\label{sec:stat}Statistics}\n\nWe devise a semi-analytical evolution function $\\mathcal{E}$ to parametrize the evolution of millisecond pulsars after the accretion phase, which can be described in closed form as:\n\\begin{eqnarray}\n\\displaystyle\\sum_{i=0}^{r} \\mathcal{E}(D_{i},\\dot{M}_{i},R_{i}\\, | \\, \\alpha_{i}^{k},\\beta_{i}^{k})  \\xrightarrow{n=3} \\mathcal{PDF}(P,\\dot{P})  \\label{stat.eq} \n\\end{eqnarray}\nwhere $\\mathcal{PDF}$ is the probability distribution function. The shape parameters $\\alpha$ and $\\beta$ define the distributions (i.e., $D,\\dot{M}, R$ for k=1,2,3) for the Beta functions\\footnote{Beta functions are commonly preferred in Bayesian statistics as the least restrictive and most flexible prior distributions. It can take the form of an uninformative (e.g. uniform) prior, a monotonic line, concave, convex, unimodal (e.g. normal) or any extreme combinations of these shapes.}\\citep{EHP00} inferred from observations at each Monte-Carlo realization ``r''. \n\nThe evolution function $\\mathcal{E}$ is built by randomly choosing initiation seeds from a period distribution $D$, which is then convolved via the standard model to consequently sample the $P-\\dot{P}$ parameter space. For the observed MSXPs, the period distribution which seeds will be randomly chosen from is the observed $P_{MSXP}$ distribution (table~1). We uniquely construct a ``relaxed multidimensional Kolmogorov-Smirnov (K-S) filter'' (fig.~\\ref{fig:probdist}) to check population consistencies by calculating the 2D K-S \\citep{FF87} probabilities ($P_{2DK-S}$) between observed MSRPs and the synthetic population that is formed by these properly evolved progenitor seeds. The filtering is reiterated for each realization to obtain synthetic populations with consistent distributions as:\n\\begin{eqnarray}\nD\\,(\\alpha_{i}^{1},\\beta_{i}^{1})  \\xrightarrow{filter}  D_{i} \\label{statD.eq} \\\\\n\\dot{M}(\\alpha_{i}^{2},\\beta_{i}^{2})   \\xrightarrow{filter}  \\dot{M}_{i}   \\label{statM.eq} \\\\\nR\\,(\\alpha_{i}^{3},\\beta_{i}^{3})  \\xrightarrow{filter}  R_{i}  \\label{statR.eq}\n\\end{eqnarray}\nwhich is then used to construct the $\\mathcal{PDF}$ in Equation ~\\protect\\ref{stat.eq}.\n\nNominally any $P_{2DK-S} > 0.2$ value would imply consistent populations in a 2D K-S test. By allowing $0.005 < P_{2DK-S} < 0.2$ with lower fractions (see fig.~\\ref{fig:probdist}), we oversample outliers to compensate for possible statistical fluctuations and contaminations. A peak sampling rate around the nominal acceptance value of $P_{2DK-S}\\sim0.2$ is the most optimal scheme that prevents strong biases due to over or under-sampling. The main goal for oversampling outliers and relaxing the K-S filter is to test whether the standard model can at least marginally produce very fast millisecond pulsars with relatively high magnetic fields like PSR B1937+21.\n\nThe predictive significance of the $P-\\dot{P}$  distribution for the probability map (fig~\\protect\\ref{fig:MSPs}) is obtained from a Monte-Carlo run with $r=10^{7}$ valid realizations that produce consistent synthetic samples. Whilst sampling the $P-\\dot{P}$ space, no assumptions were made regarding the progenitor period distribution ($D$), the accretion ($\\dot{M}$), or the Galactic birth ($R$) rates. The filter  (eq.~\\protect\\ref{statD.eq}, \\protect\\ref{statM.eq}, \\protect\\ref{statR.eq}) is implicitly driven by the observed MSRPs.\n\t\nFig \\protect\\ref{fig:MSPs} shows the expected $P-\\dot{P}$ distribution for the standard model assuming that MSRPs have evolved from a progenitor population similar to the observed MSXPs. We do not include MSRPs in globular clusters because the $P-\\dot{P}$ values in these cases may not necessarily be the sole imprint of the binary evolution, but can be significantly changed by possible gravitational interactions due to the crowded field. To explore the extend of the effects of an unevenly sampled progenitor population, we also show the region in the $P-\\dot{P}$ space that is sensitive to alternative $P_{MSXP}$ distributions. The probability map is overlaid with the observed MSRPs. \n\n\t\\section{\\label{sec:dis}Discussion and Conclusions}\n\nThe discovery of millisecond pulsations from neutron stars in LMXBs has substantiated the theoretical prediction that links MSRPs and LMXBs. Since then, the recycling process that produces MSRPs on a spin-up region from LMXBs, followed by spin-down due to dipole radiation has been conceived as the ``standard evolution'' of millisecond pulsars. However, the question whether all observed MSRPs could be produced within this framework has not been quantitatively addressed until now.\n\nThe standard evolutionary process produces millisecond pulsars with periods ($P$) and spin-downs ($\\dot{P}$) that are not entirely independent. The possible $P-\\dot{P}$ values that MSRPs can attain are {\\it jointly} constrained by the equilibrium period distribution ($D$) of the progenitor population, the mass accretion rates ($\\dot{M}$) during the recycling process and the dominant energy loss mechanism after the onset of radio emission. \n\nIn order to test whether the observed MSRPs can be reconciled with a single coherent progenitor population that evolves via magnetic dipole braking after the spin-up process, we have produced the predictive joint $P-\\dot{P}$ distribution of MSRPs for the standard model. We did not put restrictions on any of the parameters that drive the evolution. Acceptable $D,\\dot{M}$ and $R$ values were implicitly filtered.  We have relaxed the K-S filter (see fig.~\\ref{fig:probdist}) in order to oversample outliers and see whether it is even remotely feasible to produce young millisecond pulsars, like PSR B1937+21 or J0218+4232, that have higher B fields. The color contours in Figure~\\ref{fig:MSPs} represent the $P-\\dot{P}$ densities for MSRPs that are direct descendants of observed MSXPs (i.e. initial spin periods $P_{0}\\sim P_{MSXP}$). \n\nThe standard evolutionary model is able to successfully produce the general demographics of older MSRPs. It fails, however, to predict the younger and fastest MSRP sub-population that have higher $B$ fields.\n\nAccretion rates that MSRPs have experienced during their accretion phase deduced from observed $P-\\dot{P}$ values, combined with the observed MSXP period distribution ($D \\equiv  P_{MSXP}$) produces mostly older MSRPs, including MSRPs with spin-down ages $\\tau_{c} > 10^{10}$ yrs. Figure ~\\ref{fig:MSPs} shows clearly that the apparent enigma of millisecond pulsars with spin-down ages older than the age of the Galaxy is mainly a manifestation of very low accretion rates during the late stages of the LMXB evolution. \n\nOn the other hand, no physically motivated $P_{MSXP}$ distribution has been able to produce the whole MSRP population consistently. The observed period distribution of MSXPs is likely to be under-sampled due to observational selection effects. It is also possible that some neutron stars in LMXBs simply do not produce observable pulses. In order to understand how the predicted $P-\\dot{P}$ distribution is affected by different MSXP period distributions, we have estimated the whole extend of the $P-\\dot{P}$ region that is sensitive to the prior. The values that may be produced for different $P_{MSXP}$ distributions are shown by the shaded areas in Figure ~\\ref{fig:MSPs}. No MSXP period distribution could mimic the observed relative ratios of young\/old pulsars with high B fields. The fraction of the observed young\/old MSRPs with high $B$ fields is higher than what the standard model predicts by several orders of magnitude. This may further be exacerbated by strong selection effects that limit our ability to observe very fast millisecond pulsars \\citep{HRS07}. The choice of a standard K-S test instead of the relaxed 2D K-S only increases the statistical significance. Hence, we argue that young millisecond pulsars with higher magnetic fields (e.g. PSR B1937+21) are inconsistent with the standard model. \n\nTherefore, it is tempting to suggest that the fastest spinning millisecond pulsars, in particular PSR B1937+21, may originate from a different evolutionary channel.  While it appears that ordinary magnetic-dipole spin down from a source population similar to the observed MSXPs is adequate to explain the great majority of observed MSRPs, the low final accretion rates that are required cannot be reconciled with the high accretion rates needed to produce the fastest, youngest pulsars. We believe that it is necessary to posit the existence of a separate class of progenitors, most likely with a different distribution of magnetic fields, accretion rates and equilibrium spin periods, presumably among the LMXBs that have not been revealed as MSXPs.  Understanding this additional channel is clearly critical to developing a natural solution to the long lasting ``birth rate problem'' \\citep[see, e.g.][]{KN88}.\n\nIt is also possible that the standard evolutionary model fails at another point. For example, if MSRPs during some portion of their evolution lose energy through a dominant mechanism other than magnetic dipole radiation (e.g. multipole radiation, gravitational wave or neutrino emission), then the evolution of pulsars through the $P-\\dot{P}$ diagram could be complex. \n\nA combination of the above mentioned factors (i.e. alternative progenitors and subsequent non-standard radiation) are then likely to play a role in millisecond pulsar evolution. A MSXP period distribution that has sharp multimodal features coupled with non-standard energy loss mechanisms may be able to reconcile for the joint $P-\\dot{P}$ distribution of millisecond pulsars.\n\n\\acknowledgements \nThe research presented here has made use of the August 2008 version of the ATNF Pulsar Catalogue \\citep{MHT93}. The authors acknowledge NSF grant AST-0506453.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}