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+{"text":"\\section{Introduction}\n\nMany complex fluids display shear banding, in which a state of\ninitially homogeneous shear flow gives way to the formation of\ncoexisting bands of differing shear rate, with layer normals in the\nflow-gradient direction. For recent reviews,\nsee~\\cite{Olmsted2008,Manneville2008a,Fielding2014,Divoux2015}.\nFollowing its early observation in wormlike micellar surfactant\nsolutions~\\cite{Britton1997}, over the past two decades shear banding\nhas been seen in virtually all the major classes of complex fluids and\nsoft solids. Examples include microgels~\\cite{Divoux2010},\nclays~\\cite{Martin2012}, emulsions~\\cite{Coussot2002}\nfoams~\\cite{Rodts2005}, lamellar surfactant\nphases~\\cite{Salmonetal2003a}, triblock\ncopolymers~\\cite{Berretetal2001a,Mannevilleetal2007a}, star\npolymers~\\cite{Rogers2008}, and -- subject to ongoing\ncontroversy~\\cite{Wangetal2003a,Wangetal2006c,Wangetal2008a,Li2013,Wang2014,Li2015,Wang2011,Wangetal2008a} -- \nlinear polymers.\n\nPrior to about 2010, the majority of studies of shear banding focused\non conditions of a steadily applied shear flow. The criterion for the\npresence of steady state banding in this case is well known: that the\nunderlying homogeneous constitutive curve of shear stress as a\nfunction of shear rate has a regime of negative slope. (In some cases\nof strong concentration coupling shear banding can arise even for a\nmonotonic constitutive curve~\\cite{Fielding2003a}, but we\ndo not consider that case here.)  Such a regime is predicted by the\noriginal tube theory of Doi and Edwards for non-breakable\npolymers~\\cite{DoiEdwards}, and by the reptation-reaction model of\nwormlike micellar surfactants~\\cite{Cates1990}.  It is\nstraightforward to show that a state of initially homogeneous shear\nflow is linearly unstable, in this regime of negative constitutive\nslope, to the formation of shear bands~\\cite{Yerushalmi1970}.  The\ncomposite steady state flow curve of shear stress as a function of\nshear rate then displays a characteristically flat plateau regime, in\nwhich shear bands are observed.\n\nFrom an experimental viewpoint, the evidence for steady state shear\nbanding under a steadily applied shear flow is now overwhelming in the case\nof wormlike micelles. For reviews,\nsee~\\cite{Cates2007,Berret2005}. For linear unbreakable\npolymers the issue remains controversial, as recently reviewed in\nRef.~\\cite{Snijkers2015}. In particular the original\nDoi-Edwards model did not account for a process known as convective\nconstraint release\n(CCR)~\\cite{Marrucci1996,Ianniruberto2014a,Ianniruberto2014}.\nSince CCR (which we describe below) was proposed, there has been an\nongoing debate about its efficacy in potentially eliminating the\nregime of negative constitutive slope and restoring a monotonic\nconstitutive curve, thereby eliminating steady state banding.\nHowever, a non-monotonic constitutive curve and associated steady\nstate shear banding has been seen in molecular dynamics simulations of\npolymers~\\cite{Likhtmanetal2012}, for long enough chain lengths.  It\nis important to note, though, that the polydispersity that is often\npresent in practice in unbreakable polymers also tends to restore\nmonotonicity.\n\nBesides the conditions of steady state flow just described, many flows\nof practical importance involve a strong time dependence.  In view of\nthis, a natural question to ask is whether shear banding might also\narise in these time-dependent flows and, if so, under what conditions.\nOver the past decade, a body of experimental data has accumulated to\nindicate that it does indeed occur: in shear\nstartup~\\cite{Divoux2010, Divoux2011a, Wangetal2009a,\n  Huetal2007a, Wangetal2008a, Martin2012}, following a step\nstrain (in practice a rapid strain ramp)~\\cite{Wangetal2010a,\n  Boukany2009a, Wangetal2006a, Fang2011, Wangetal2007a,\n  Archer1995, Wangetal2008c}, and following a step\nstress~\\cite{Gibaud2010,Divoux2011,\n  Wangetal2009a,Huetal2007a, Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010}.\n\n\nConsistent with this growing body of experimental evidence,\ntheoretical\nconsiderations~\\cite{Moorcroft2013,Moorcroft2014,Moorcroft2011,Adamsetal2011,Manningetal2009a}\nalso suggest that shear banding might arise rather generically in\nflows with a sufficiently strong time-dependence, even in fluids that\nhave a monotonically increasing constitutive curve and so do not\ndisplay steady state banding under conditions of a continuously\napplied shear.  Indeed, the calculations to date suggest that the set\nof fluids that show banding in steady state is only a subset of those\nthat exhibit banding in time-dependent flows. In view of this,\nalthough the question concerning the existence or otherwise of steady\nstate shear banding in polymers remains an important one, the\nresolution of that controversy is likely to be of less practical\nimportance to the broader issue of whether shear banding arises more\ngenerally in time-dependent flows.\n\nIn the last five years progress has been made in establishing\ntheoretically, separately for each of the time-dependent flow\nprotocols listed above (shear startup, step strain and step stress), a\nfluid-universal criterion~\\cite{Moorcroft2013} for the onset of\nshear banding, based on the shape of the time-dependent rheological\nresponse function for the particular protocol in question. We now\nbriefly review these criteria as backdrop to understanding the results\nthat follow below for shear banding in large amplitude oscillatory\nshear (LAOS).\n\nIn shear startup (the switch-on at some time $t=0$ of a constant shear\nrate $\\dot{\\gamma}$), the onset of banding is closely associated with the\npresence of an\novershoot~\\cite{Moorcroft2013,Moorcroft2014,Moorcroft2011,Adamsetal2011,Manningetal2009a}\nin the startup signal of stress as a function of time (or equivalently\nas a function of strain), as it evolves towards its eventual steady\nstate on the material's flow curve. This concept builds on the early\ninsight of Ref.~\\cite{Marrucci1983}. The resulting bands may, or may\nnot, then persist to steady state, according to whether or not the\nunderlying constitutive curve of stress as a function of strain rate\nis non-monotonic. This tendency of a startup overshoot to trigger\nbanding was predicted on the basis of fluid-universal analytical\ncalculations in Ref.~\\cite{Moorcroft2013}, and has been\nconfirmed in numerical simulations of polymeric fluids (polymer solutions, polymer melts and\nwormlike micelles)~\\cite{Moorcroft2014,Adamsetal2011},\npolymer glasses~\\cite{Fielding2013} and soft glassy materials\n(dense emulsions, microgels, foams, {\\it\n  etc.})~\\cite{Moorcroft2011,Manningetal2009a,Jagla2010}.\nIt is consistent with experimental observations in wormlike micellar\nsurfactants~\\cite{Hu2008, Wangetal2008c},\npolymers~\\cite{Wangetal2008a, Wangetal2008b, Wangetal2009a,\n  Wangetal2009b, Boukany2009a, Wangetal2003a, Huetal2007a,\n  Wangetal2009d}, carbopol\ngels~\\cite{Divoux2011a,Divoux2010} and Laponite\nclay suspensions~\\cite{Martin2012}.\n\nFollowing the imposition of a step stress in a previously undeformed\nsample, the onset of shear banding is closely associated with the\nexistence of a regime of simultaneous upward slope and upward\ncurvature in the time-differentiated creep response curve of shear\nrate as a function of\ntime~\\cite{Moorcroft2013,Fielding2014}.  This criterion\nwas also predicted on the basis of fluid-universal analytical\ncalculations in Ref.~\\cite{Moorcroft2013}, and has been\nconfirmed in numerical simulations of polymeric\nfluids~\\cite{Moorcroft2014} and soft glassy\nmaterials~\\cite{Fielding2014}.  It is consistent with\nexperimental observations in polymers~\\cite{Wangetal2009a,Huetal2007a,Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010, Wangetal2009d}, carbopol\nmicrogels~\\cite{Divoux2011} and carbon black\nsuspensions~\\cite{Gibaud2010}.\n\nIn the shear startup and step stress experiments just described, the\ntime-dependence is inherently transient in nature: after (typically)\nseveral strain units, the system evolves to its eventual steady state\non the material's flow curve.  In any such protocol, for a fluid with\na monotonic constitutive curve that precludes steady state banding,\nany observation of banding is predicted to be limited to this regime\nof time-dependence following the inception of the flow.  That poses an\nobvious technical challenge to experimentalists: of imaging the flow\nwith sufficient time-resolution to detect these transient bands.  This\nis particularly true for a polymeric fluid with a relatively fast\nrelaxation spectrum.  For soft glassy materials, in contrast, the\ndynamics are typically much slower and any bands associated with the\nonset of flow, though technically transient, may persist for a\nsufficiently long time to be mistaken for the material's ultimate\nsteady state response for any practical\npurpose~\\cite{Moorcroft2011,Fielding2014}.\n\nIn the past decade, the rheological community has devoted considerable\nattention to the of study large amplitude oscillatory shear (LAOS).\nFor a recent review, see Ref.~\\cite{Hyun2011}. In this\nprotocol, the applied flow has the form of a sustained oscillation and\nis therefore perpetually time-dependent, in contrast to the transient\ntime-dependence of the shear startup and step stress protocols just\ndescribed.  But by analogy with the predictions of transient shear\nbanding in shear startup and step stress, a sustained oscillatory flow\nmight (in certain regimes that we shall discuss) be expected to\nrepeatedly show banding at certain phases of the cycle, or even to\nshow sustained banding round the whole cycle.  Importantly, again by\nanalogy with our knowledge of shear startup and step stress, this\neffect need not be limited to fluids with a non-monotonic constitutive\ncurve that show steady state banding in a continuously applied shear\nflow, but might instead arise as a natural consequence of the\ntime-dependence inherent to the oscillation.\n\nIndeed, a particularly attractive feature of LAOS is that the severity\nof the flow's time-dependence, relative to the fluid's intrinsic\ncharacteristic relaxation timescale $\\tau$, can be tuned by varying\nthe frequency $\\omega$ of the applied oscillation. A series of LAOS\nexperiments can thereby explore the full range between steady state\nbehaviour in the limit $\\omega\\to 0$, where the oscillation\neffectively corresponds to a repeated series of quasi-static sweeps up\nand down the flow curve, and strongly time-dependent behaviour for\n$\\omega > 1\/\\tau$. A fluid with a non-monotonic underlying\nconstitutive curve that admits steady state banding is then clearly\nexpected to exhibit banding in the limit of $\\omega\\to 0$, as the\nshear rate quasi-statically transits the plateau in the steady state\nflow curve. In contrast, a monotonic constitutive curve precludes\nbanding for $\\omega\\to 0$.  Crucially, though, as noted above, the\nabsence of banding in steady state conditions does not rule out the\npossibility of banding in flows with a strong enough time-dependence,\n$\\omega \\gtrapprox O(1\/\\tau)$.\n\nIndeed, intuitively, a square-wave caricature of a large amplitude\noscillatory shear strain (LAOStrain) experiment points to a perpetual\nswitching between a shear startup like process in the forward\ndirection, followed by `reverse startup' in the opposite direction.\nAny regime in which these startup-like events are associated with an\novershoot in the associated curve of stress as a function of strain\nthen strongly suggests the possibility of shear banding during those\nquasi-startup parts of the cycle, by analogy with the criterion for\nbanding in a true shear startup from rest.  In the same spirit, a\nsquare-wave caricature of a large amplitude oscillatory shear stress\n(LAOStress) experiment indicates a perpetually repeated series of step\nstress events, jumping between positive and negative stress values,\nand so admitting the possibility of shear banding if the criterion for\nbanding following a step stress is met.\n\nIn practice, of course, LAOS is more complicated than the caricatures\njust described and the criteria for banding in shear startup and step\nstress might only be expected to apply in certain limiting regimes.\nNonetheless, in what follows we shall show that many of our results\nfor banding in LAOStrain and LAOStress can, to a large extent, be\nunderstood within the framework of these existing criteria for the\nsimpler time-dependent protocols.\n\nExperimentally, shear banding has indeed been observed in LAOS: in\npolymer solutions~\\cite{Wangetal2006c}, dense\ncolloids~\\cite{Cohen2006}, and also in wormlike micellar\nsurfactants that are known to shear band in steady\nstate~\\cite{Dimitriou2012,KateGurnon2012,Gurnon2014a}.\n\nFrom a theoretical viewpoint, several approaches to the interpretation\nof LAOS data have been put forward in the\nliterature~\\cite{Hyun2011}. These include Fourier transform\nrheology~\\cite{Wilhelm2002}; measures for quantifying\nLissajous-Bowditch curves (defined below) in their elastic\nrepresentation of stress versus strain, or viscous representation of\nstress versus strain rate~\\cite{Tee1975}; a decomposition\ninto characteristic sine, square and triangular wave prototypical\nresponse functions~\\cite{Klein2007,Klein2008};\ndecomposition into elastic and viscous stress contributions using\nsymmetry arguments~\\cite{Cho2005}; Chebyshev series\nexpansions of these elastic and viscous\ncontributions~\\cite{Ewoldt2008a}; and interpretations of the\nLAOS cycle in terms of a sequence of physical\nprocesses~\\cite{Rogers2012,Rogers2011}.\n\nHowever, many of these existing theoretical studies assume either\nexplicitly or implicitly that the flow remains homogeneous, and\nthereby fail to take account of the possibility of shear banding. An\nearly exception can be found in\nRefs.~\\cite{Zhou2010,Zhou2008}, which studied a\nmodel of wormlike micellar surfactants with a non-monotonic\nconstitutive curve in LAOStrain.  Another exception is in the paper of\nAdams and Olmsted~\\cite{Adams2009}, which recognised that\nshear banding can arise even in the absence of any non-monotonicity in\nthe underlying constitutive curve.\n\nThe work that follows here builds on the remarkable insight of these\nearly papers, in carrying out a detailed numerical study of shear\nbanding in LAOStrain and LAOStress within the Rolie-poly\nmodel~\\cite{Grahametal2003a} of polymers and wormlike\nmicellar surfactant solutions.  Consistent with the above discussion,\nin LAOStrain we observe banding at low frequencies $\\omega\\to 0$ and\nsufficiently high strain rate amplitudes $\\dot{\\gamma}\\gtrapprox 1\/\\tau$ in\nfluids for which the underlying constitutive curve of shear stress as\na function of shear rate is non-monotonic. At higher frequencies\n$\\omega=O(1\/\\tau)$ and for sufficiently high strain amplitudes\n$\\gamma\\gtrapprox 1$ we instead see `elastic' shear banding associated\nwith an overshoot in the elastic curve of stress as a function of\nstrain, in close analogy with the elastic banding predicted in a fast\nshear startup\nexperiment~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011,Adams2009}.\nImportantly, we show that this elastic banding arises robustly even in\na wide range of model parameter space for which the underlying\nconstitutive curve is monotonic, precluding steady state banding.\n\nIn LAOStress we observe banding in fluids that shear thin sufficiently\nstrongly to have either a negatively, or weakly positively, sloping\nregion in the underlying constitutive curve. We emphasise again that\nfluids in the latter category do not display steady state banding, and\ntherefore that, for such fluids, the banding predicted in LAOStress is\na direct result of the time-dependence of the applied flow. In this\ncase the banding is triggered in each half cycle as the stress magnitude\ntransits in an upward direction the region of weak slope and the\nstrain rate magnitude increases dramatically such that the material effectively\nyields. This is strongly reminiscent of the transient banding\ndiscussed previously in step\nstress~\\cite{Moorcroft2013,Moorcroft2014}.\n\nWhile it would be interesting to interpret our findings within one (or\nmore) of the various mathematical methodologies for analysing LAOS\ndiscussed above (and in particular to consider the implications of\nbanding for the presence of higher harmonics in the output rheological\ntime series), in the present manuscript we focus instead on the\nphysical understanding that can be gained by considering the shapes of\nthe signals of stress versus strain or strain rate (in LAOStrain) and\nstrain rate versus time (in LAOStress).  In that sense, this work is\nclosest in spirit to the sequence of physical processes (SPP) approach\nof Refs.~\\cite{Rogers2012,Rogers2011} (which did\nnot, however, explicitly consider heterogeneous response).  In\nparticular, we seek to interpret the emergence of shear banding in\nLAOS on the basis of the existing criteria for the onset of banding in\nthe simpler time-dependent protocols of shear startup and step\nstress~\\cite{Moorcroft2013}.\n\nThe paper is structured as follows. In Sec.~\\ref{sec:models} we\nintroduce the model, flow geometry and protocols to be considered.\nSec.~\\ref{sec:methods} outlines the calculational methods that we\nshall use.  Sec.~\\ref{sec:recap} contains a summary of previously\nderived linear instability criteria for shear banding in steady shear,\nfast shear startup and step shear stress protocols, with the aim of providing a\nbackdrop to understanding shear banding in oscillatory protocols.  In\nSecs.~\\ref{sec:LAOStrain} and~\\ref{sec:LAOStress} we present our\nresults for LAOStrain and LAOStress respectively, and discuss their\npotential experimental verification.  Finally\nSec.~\\ref{sec:conclusions} contains our conclusions and an outlook for\nfuture work.\n\n\n\\section{Model, flow geometry and protocols}\n\\label{sec:models}\n\nWe write the stress $\\tens{\\Sigma}(\\tens{r},t)$ at any time $t$ in a\nfluid element at position $\\tens{r}$ as the sum of a viscoelastic\ncontribution $\\tens{\\sigma}(\\tens{r},t)$ from the polymer chains or\nwormlike micelles, a Newtonian contribution characterised by a\nviscosity $\\eta$, and an isotropic contribution with pressure\n$p(\\tens{r},t)$:\n\\begin{equation}\n\\tens{\\Sigma} = \\tens{\\sigma} + 2 \\eta \\tens{D} - p\\tens{I}.\n\\label{eqn: total_stress_tensor}\n\\end{equation}\nThe Newtonian stress $2 \\eta \\tens{D}(\\tens{r},t)$ may arise from the\npresence of a true solvent, and from any polymeric\ndegrees of freedom considered fast enough not to be ascribed their own\nviscoelastic dynamics. The symmetric strain rate tensor $\\tens{D} =\n\\frac{1}{2}(\\tens{K} + \\tens{K}^T)$ where $K_{\\alpha\\beta} =\n\\partial_{\\beta}v_{\\alpha}$ and $\\tens{v}(\\tens{r},t)$ is the fluid\nvelocity field.  \n\nWe consider the zero Reynolds number limit of creeping flow, in which\nthe condition of local force balance requires the stress field\n$\\tens{\\Sigma}(\\tens{r},t)$ to be divergence free:\n\\begin{equation}\n\\vecv{\\nabla}\\cdot\\,\\tens{\\Sigma} = 0.\n\\label{eqn: force_balance}\n\\end{equation}\nThe pressure field $p(\\tens{r},t)$ is determined by enforcing that the\nflow remains incompressible:\n\\begin{equation}\n\\label{eqn: incomp}\n\\vecv{\\nabla}\\cdot\\vecv{v} = 0.\n\\end{equation}\n\nThe viscoelastic stress is then written in terms of a constant elastic\nmodulus $G$ and a tensor $\\visc(\\tens{r},t)$ characterising the\nconformation of the polymer chains or wormlike micelles,\n$\\tens{\\sigma} = G\\, (\\visc - \\tens{I})$.  We take the dynamics of\n$\\visc$ to be governed by the Rolie-poly (RP)\nmodel~\\cite{Grahametal2003a} with\n\\begin{widetext}\n\\begin{eqnarray}\n\\partial_t{\\visc}+\\tens{v}\\cdot\\nabla\\tens{\\visc} &=& \\tens{K} \\cdot \\visc + \\visc \\cdot \\tens{K}^T - \\frac{1}{\\tau_d}\\left(\\visc - \\tens{I}\\right) \n- \\frac{2(1-A)}{\\tau_R}\\left[\\, \\visc + \\beta A^{-2\\delta}\\left(\\visc - \\tens{I}\\right) \\right] + D\\nabla^2\\visc,\n\\label{eqn: rolie-poly_tensor}\n\\end{eqnarray}\n\\end{widetext}\nin which $A = \\sqrt{3\/T\\,}$ with trace $T = \\text{tr}\\,\\tens{\\visc}$.\nThis RP model is a single mode simplification of the GLAMM model\n\\cite{GLAMM}, which provides a microscopically derived stochastic\nequation for the dynamics of a test chain (or micelle) in its mean\nfield tube of entanglements with other chains.  The timescale $\\tau_d$\nsets the characteristic time on which a chain escapes its tube by\nmeans of 1D curvilinear diffusion along the tube's contour, known as\nreptation, allowing the molecular orientation to refresh itself. The\nRouse timescale $\\tau_R$ sets the shorter time on which chain stretch,\nas characterised by $T = \\text{tr}\\,\\tens{\\visc}$, relaxes.  The ratio\n$\\tau_d\/\\tau_R=3Z$, where $Z$ is the number of entanglements per chain.\nThe parameters $\\beta$ and $\\delta$ govern a phenomenon known as\nconvective constraint\nrelease~\\cite{Marrucci1996,Ianniruberto2014a,Ianniruberto2014}\n(CCR), in which the relaxation of the stretch of a test chain has the\neffect of also relaxing entanglement points, thereby facilitating the\nrelaxation of tube orientation.  The diffusive term $D\\nabla^2\\visc$\nadded to the right hand side of Eqn.~\\ref{eqn: rolie-poly_tensor} is\nrequired to account for the slightly diffuse nature of the interface\nbetween shear bands~\\cite{Luetal2000a}: without it the shear\nrate would be discontinuous across the interface, which is unphysical.\n\nUsing this model we will consider shear flow between infinite flat\nparallel plates at $y = \\{0,L\\}$, with the top plate moving in the\n$\\vecv{\\hat{x}}$ direction at speed $\\overline{\\dot{\\gamma}}(t) L$. We assume\ntranslational invariance in the flow direction $\\vecv{\\hat{x}}$ and\nvorticity direction $\\vecv{\\hat{z}}$ such that the fluid velocity can\nbe written as $\\vecv{v} = v(y,t)\\vecv{\\hat{x}}$. The local shear rate\nat any position $y$ is then given by\n\\begin{equation}\n\\dot{\\gamma}(y,t) = \\partial_{y}v(y,t),\n\\end{equation}\nand the spatially averaged shear rate\n\\begin{equation}\n\\overline{\\dot{\\gamma}}(t) = \\frac{1}{L}\\int_{0}^{L} \\dot{\\gamma}(y,t)dy.\n\\end{equation}\nSuch a flow automatically satisfies the constraint of\nincompressibility, Eqn.~\\ref{eqn: incomp}.  The force balance\ncondition, Eqn.~\\ref{eqn: force_balance}, further demands that the\ntotal shear stress is uniform across the cell, in the planar flow\nsituation considered here, giving $\\partial_{y}\\Sigma_{xy} =0$. The\nviscoelastic and Newtonian contributions may, however, each depend on\nspace provided their sum remains uniform:\n\\begin{equation}\n\\Sigma_{xy}(t) = GW_{xy}(y,t) + \\eta \\dot{\\gamma}(y,t).\n\\label{eqn: shear_stress}\n\\end{equation}\n\nFor such a flow, the RP model can be written componentwise as\n\\begin{widetext}\n\\begin{eqnarray}\n\\dot{W}_{xy}  &=& \\dot{\\gamma} W_{yy} - \\frac{W_{xy}}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}(1+ \\beta A)W_{xy} + D\\partial_y^2 W_{xy}, \\nonumber\\\\\n\\dot{W}_{yy}  &=& - \\frac{W_{yy}-1}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}\\left[W_{yy}+ \\beta A(W_{yy}-1)\\right]+ D\\partial_y^2W_{yy},\\nonumber\\\\\n\\dot{T}     &=&  2\\dot{\\gamma}W_{xy}  - \\frac{T-3}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}\\left[T + \\beta A(T - 3)\\right]+ D\\partial_y^2 T.\\quad \\quad\n\\label{eqn: sRP_components}\n\\end{eqnarray}\n(The other components of $\\tens{W}$ decouple to form a separate\nequation set, with trivial dynamics.)  In the limit of fast chain\nstretch relaxation $\\tau_R \\to 0$ we obtain the simpler\n`non-stretching' RP model in which the trace $T=3$ and\n\\begin{eqnarray}\n\\dot{W}_{xy} &=& \\dot{\\gamma} \\left[W_{yy} - \\frac{2}{3} (1+\\beta)W_{xy}^2\\right]\\;\\;\\;\\;\\;\\;\\; -\\frac{1}{\\tau_d}W_{xy},+ D\\partial_y^2 W_{xy}\\nonumber\\\\\n\\dot{W}_{yy} &=& \\frac{2}{3}\\dot{\\gamma}\\left[\\betaW_{xy}-(1+\\beta)W_{xy}W_{yy} \\right] - \\frac{1}{\\tau_d}(W_{yy}-1)+ D\\partial_y^2W_{yy}. \\quad \\quad\n\\label{eqn: nRP_components}\n\\end{eqnarray}\n\\end{widetext}\nFor convenient shorthand we shall refer to this simpler non-stretching\nform as the nRP model. We refer to the full `stretching' model of\nEqns.~\\ref{eqn: sRP_components} as the sRP model.\n\nFor boundary conditions at the walls of the flow cell we assume no\nslip and no permeation for the fluid velocity, and zero-gradient\n$\\partial_y W_{\\alpha\\beta}=0$ for every component $\\alpha\\beta$ of\nthe polymeric conformation tensor.\n\nIn what follows we consider the behaviour of the Rolie-poly model in the following two flow protocols:\n\n\\begin{itemize}\n\n\\item LAOStrain, with an imposed strain \n\\begin{equation}\n\\gamma(t)=\\gamma_0\\sin(\\omega t),\n\\end{equation}\nto which corresponds the strain rate\n\\begin{equation}\n\\dot{\\gamma}(t)=\\gamma_0\\omega\\cos(\\omega t)=\\dot{\\gamma}_0\\cos(\\omega t).\n\\end{equation}\n\n\\item LAOStress, with an imposed stress\n\\begin{equation}\n\\Sigma(t)=\\Sigma_0\\sin(\\omega t).\n\\end{equation}\n\n\\end{itemize}\n\nThe model, flow geometry and protocol just described are characterised\nby the following parameters: the polymer modulus $G$, the reptation\ntimescale $\\tau_d$, the stretch relaxation timescale $\\tau_R$, the CCR\nparameters $\\beta$ and $\\delta$, the stress diffusivity $D$, the\nsolvent viscosity $\\eta$, the gap size $L$, the frequency $\\omega$ and\nthe amplitude $\\gamma_0$ (for LAOStrain) or $\\Sigma_0$ (for\nLAOStress). We are free to choose units of mass, length and time,\nthereby reducing the list by three: we work in units of length in\nwhich the gap size $L = 1$, of time in which the reptation time $\\tau_d\n= 1$ and of mass (or actually stress) in which the polymer modulus\n$G=1$.  We then set the value of the diffusion constant $D$ such that\nthe interface between the bands has a typical width $\\ell =\n\\sqrt{D\\tau_d}=2\\times 10^{-2}L$, much smaller than the gap size. This\nis the physically relevant regime for the macroscopic flow cells of\ninterest here, and we expect the results we report to be robust to\nreducing $l$ further.  Following Ref.~\\cite{Grahametal2003a} we set\n$\\delta = -\\frac{1}{2}$.\n\nAdimensional quantities remaining to be explored are then the model\nparameters $\\eta$, $\\beta$ and (for the sRP model only) $\\tau_R$; and\nthe protocol parameters $\\omega$ and $\\gamma_0$ or $\\Sigma_0$. For\neach set of model parameters we explore the whole plane of feasibly\naccessible values of protocol parameters $\\omega$ and $\\gamma_0$ or\n$\\Sigma_0$.\n\nAmong the model parameters the CCR parameter has the range $0 \\leq\n\\beta \\leq 1$. Within this there is no current consensus as to its\nprecise value, and we shall therefore explore widely the full range\n$0\\to 1$. For the fluids of interest here the Newtonian viscosity is\ntypically much smaller than the zero shear viscosity of the\nviscoelastic component, giving $\\eta\\ll 1$ in our units. Based on a\nsurvey of the experimental data, a range of $10^{-7}$ to $10^{-3}$ was\nsuggested by Graham et al. in Ref.~\\cite{Graham2013}\nConsistentwith comments made in Ref.~\\cite{Agimelen2013} we find values\nless than $10^{-5}$ unfeasible to explore numerically, due to a\nresulting large separation of timescales between $\\tau_d$ and $\\eta\/G$.\nTherefore we adopt typical values $\\eta=10^{-4}$ and $10^{-5}$.  Given\nthat that the susceptibility to shear banding increases with\ndecreasing $\\eta$, we note that the levels of banding reported in what\nfollows are likely, if anything, to be an underestimate of what might\nbe observed experimentally. We return in our concluding remarks to\ndiscuss this issue further.\n\nWe explore a wide range of values of the stretch relaxation time\n$\\tau_R$, or equivalently of the degree of entanglement\n$Z=\\tau_d\/3\\tau_R$: we consider $Z=1$ to $350$ for the sRP model (and\nnote that the nRP model has $Z\\to\\infty$ by definition).\nExperimentally, values of $Z$ in the range of $50$ appear commonplace\nand $100$ towards the upper end of what might currently be used\nexperimentally in nonlinear rheological studies.  One of the\nobjectives of this work is to provide a roadmap of values of $Z$ and\n$\\beta$ in which shear banding is expected to be observed, for typical\nsmall values of $\\eta$, in a sequence of LAOS protocols that scan\namplitude and frequency space.\n\n\\section{Calculation methods} \n\\label{sec:methods}\n\nIn this section we outline the theoretical methods to be used\nthroughout the paper. In order to develop a generalised framework\nencompassing both the nRP and sRP models, we combine all the relevant\ndynamical variables (for any given model) into a state vector\n$\\vecv{s}$, with $\\vecv{s}=(W_{xy},W_{yy})^T$ for the nRP model and\n$\\vecv{s}=(W_{xy},W_{yy},T)^T$ for the sRP model.  Alongside this we\ndefine a projection vector $\\vecv{p}$ of corresponding dimensionality\n$d$, with $\\vecv{p}=(1,0)$ for the nRP model and $\\vecv{p}=(1,0,0)$\nfor sRP.\n\nThe total shear stress $\\Sigma_{xy}=\\Sigma$, from which we drop the\n$xy$ subscript for notational brevity, is then given by\n\\begin{equation}\n\\label{eqn: governing_eqn_force}\n\\Sigma(t) = G\\vecv{p} \\cdot \\vecv{s}(y,t) + \\eta \\dot{\\gamma}(y,t),\n\\end{equation}\nand the viscoelastic constitutive equation has the generalised form\n\\begin{equation}\n\\partial_{t\\,}\\vecv{s}(y,t) = \\vecv{Q}(\\vecv{s},\\dot{\\gamma}) + D\\partial_y^2\\vecv{s}.\n\\label{eqn: governing_eqn_diffusive}\n\\end{equation}\nThe dimensionality and functional form of $\\vecv{Q}$ then specify the\nparticular constitutive model. In this way our generalised notation in\nfact encompasses not only the nRP model (for which $d=2$) and sRP\nmodel (for which $d=3$) but many more besides, including the Johnson\nSegalman, Giesekus and Oldroyd B models~\\cite{Larson1988}.\n\n\\subsection{Homogeneous base state}\n\\label{sec:base}\n\nFor any given applied flow our approach will be first to calculate the\nfluid's response within the simplifying assumption that the\ndeformation must remain homogeneous across the cell. While this is an\nartificial (and indeed incorrect) constraint in any regime where shear\nbanding is expected, it nonetheless forms an important starting point\nfor understanding the mechanism by which shear banding sets in.  (We\nalso note that most papers in the literature make this assumption\nthroughout, thereby disallowing any possibility of shear banding\naltogether.)\n\nWithin this assumption of homogeneous flow, the response of the system\nfollows as the solution to the set of ordinary differential equations\n\\begin{equation}\n\\label{eqn: governing_eqn_force_local}\n\\base{\\Sigma}(t) = G\\vecv{p} \\cdot \\base{\\vecv{s}}(t) + \\eta \\base{\\dot{\\gamma}}(t),\n\\end{equation}\nand\n\\begin{equation} \n\\dot{\\base{\\vecv{s}}}(t) = \\vecv{Q}(\\base{\\vecv{s}},\\base{\\dot{\\gamma}}).\n\\label{eqn: governing_eqn_diffusive_local}\n\\end{equation}\nIn these either $\\base{\\dot{\\gamma}}(t)$ or $\\base{\\Sigma}(t)$ is imposed, in\nLAOStrain and LAOStress respectively, and the other dynamical\nquantities are calculated numerically using an explicit Euler\nalgorithm~\\cite{NumRecipes}. We use the `hat' notation to denote\nthat the state being considered is homogeneous.\n\n\\subsection{Linear stability analysis}\n\\label{sec:lsa}\n\nHaving calculated the behaviour of the fluid within the assumption\nthat the flow remains homogeneous, we now proceed to consider whether\nthis homogeneous `base state' flow will, at any point during an applied\noscillatory protocol, be unstable to the formation of shear bands. To\ndo so we add to the base state, for which we continue to use the hat\nnotation, heterogeneous perturbations of (initially) small amplitude:\n\\begin{eqnarray}\n\\Sigma(t) &=& \\base{\\Sigma}(t),\\nonumber\\\\\n\\dot{\\gamma}(y,t)&=& \\base{\\dot{\\gamma}}(t) + \\sum_{n=1}^\\infty \\delta \\dot{\\gamma}_n(t) \\cos(n\\pi y\/L),\\nonumber\\\\\n\\vecv{s}(y,t) &=& \\vecv{\\base{s}}(t) + \\sum_{n=1}^\\infty \\delta\\vecv{s}_n(t) \\cos(n\\pi y\/L).\n\\label{eqn: LSA}\n\\end{eqnarray}\nNote that the total stress $\\Sigma$ is not subject to heterogeneous\nperturbations because the constraint of force balance decrees that it\nmust remain uniform across the gap, at least in a planar shear cell.\nSubstituting Eqns.~\\ref{eqn: LSA} into Eqns.~\\ref{eqn:\n  governing_eqn_force} and~\\ref{eqn: governing_eqn_diffusive}, and\nexpanding in successive powers of the magnitude of the small\nperturbations $\\delta{\\dot{\\gamma}_n},\\vecv{\\delta s_n}$, we recover at\nzeroth order Eqns.~\\ref{eqn: governing_eqn_force_local} and~\\ref{eqn:\n  governing_eqn_diffusive_local} for the dynamics of the base state. At first order the heterogeneous perturbations obey\n\\begin{eqnarray}\n\\label{eqn: perturbation}\n0&=&G\\tens{p}\\cdot \\delta\\vecv{s}_n(t)+\\eta\\delta\\dot{\\gamma}_n(t),\\nonumber\\\\\n\\dot{\\delta\\vecv{s}}_n &=& \\tens{M}(t) \\cdot \\delta\\vecv{s}_n + \\tens{q}\\delta{\\dot{\\gamma}}_n,\n\\end{eqnarray}\nin which  $\\tens{M} =\n\\partial_{\\vecv{s}\\,}\\vecv{Q}|_{\\vecv{\\base{s}},\\base{\\dot{\\gamma}}}-\\tens{\\delta}D(n\\pi\/L)^2$\nand $\\vecv{q}\n= \\partial_{\\dot{\\gamma}}\\vecv{Q}|_{\\vecv{\\base{s}},\\base{\\dot{\\gamma}}}$. Combining\nthese gives\n\\begin{equation}\n\\label{eqn: one}\n\\dot{\\delta\\vecv{s}}_n = \\tens{P}(t) \\cdot \\delta\\vecv{s}_n,\n\\end{equation}\nwith\n\\begin{equation}\n\\tens{P}(t) = \\tens{M}(t) - \\frac{G}{\\eta}\\vecv{q}(t)\\, \\vecv{p}.\n\\label{eqn: two}\n\\end{equation}\nIn any regime where the heterogeneity remains small, terms of second\norder and above can be neglected.\n\nTo determine whether at any time $t$ during an imposed oscillatory\nflow the heterogeneous perturbations\n$\\delta{\\dot{\\gamma}}_n,\\delta\\vecv{s}_n(t)$ have positive rate of growth,\nindicating linear instability of the underlying homogeneous base state\nto the onset of shear banding, we consider first of all the\ninstantaneous sign of the eigenvalue $\\lambda(t)$ of $\\tens{P}(t)$ that\nhas the largest real part. A positive value of $\\lambda(t)$ is clearly\nsuggestive that heterogeneous perturbations will be instantaneously\ngrowing at that time $t$. We note, however, that the concept of a\ntime-dependent eigenvalue must be treated with caution.  In view of\nthis we cross check predictions made on the basis of the eigenvalue by\nalso directly numerically integrating the linearised Eqns.~\\ref{eqn:\n  one} using an explicit Euler algorithm. This allows us to determine\nunambiguously whether the heterogeneous perturbations will be at any\ninstant growing (taking the system towards a banded state) or decaying\n(restoring a homogeneous state), at the level of this linear\ncalculation.\n\nIn these linear stability calculations we neglect the diffusive term\nin the viscoelastic constitutive equation, setting $D=0$. Reinstating\nit would simply transform any eigenvalue $\\lambda \\to \\lambda_n=\\lambda -\nD n^2\\pi^2\/L^2$ and provide a mechanism whereby any heterogeneity with\na wavelength of order the microscopic lengthscale $l$, or below,\ndiffusively decays. Accordingly the results of this linear calculation\nonly properly capture the dynamics of any heterogeneous perturbations\nthat have macroscopically large wavelengths, which are the ones of\ninterest in determining the initial formation of shear bands starting\nfrom a homogeneous base state.\n\nAs a measure of the degree of flow heterogeneity at any time $t$ in\nthis linear calculation, we shall report in our results sections below\n$\\delta\\dot{\\gamma} (t)$ normalised by the amplitude of the imposed\noscillation $\\dot{\\gamma}_0$ in LAOStrain, or by $1+|\\dot{\\gamma}(t)|$ in LAOStress,\nwhere $\\dot{\\gamma}(t)$ is the instantaneous value of the shear rate. (We\nfind numerically that bands tend to form in LAOStress when\n$|\\dot{\\gamma}(t)|\\gg 1$. The additional 1 in the normalisation is used\nsimply to prevent the divergence of this measure when $\\dot{\\gamma}(t)$ passes\nthrough 0 in each half cycle.) Note that we no longer need to specify\nthe mode number $n$ for $\\delta\\dot{\\gamma}$, because within the assumption\n$D=0$ just described, we are confining our attention to the limit of\nlong wavelength modes only and noting them all to have the same\ndynamics, to within small corrections set by $D$.\n\n\\subsection{Full nonlinear simulation}\n\\label{sec:nonlinear}\n\nWhile the linear analysis just described provides a calculationally\nconvenient method for determining whether shear banding will arise in\nany given oscillatory measurement, enabling us to quickly build up an\noverall roadmap of parameter space, it cannot predict the detailed\ndynamics of the shear bands once the amplitude of heterogeneity has\ngrown sufficiently large that nonlinear effects are no longer\nnegligible. Therefore in what follows we shall also perform full\nnonlinear simulations of the model's spatio-temporal dynamics by\ndirectly integrating the full model Eqns.~\\ref{eqn:\n  governing_eqn_force} and~\\ref{eqn: governing_eqn_diffusive} using a\nCrank-Nicolson algorithm \\cite{NumRecipes}, with the system's\nstate discretised on a grid of $J$ values of the spatial coordinate\n$y$, checked in all cases for convergence with respect to increasing\nthe number of grid points.\n\nAs a measure of the degree of shear banding at any time $t$ in this\nnonlinear calculation we report the difference between the maximum and\nminimum values of the shear rate across the cell:\n\\begin{equation}\n\\Delta_{\\dot{\\gamma}}(t) = \\frac{1}{N}\\Big[|\\dot{\\gamma}_{\\rm max}(t) - \\dot{\\gamma}_{\\rm min}(t)|\\Big],\n\\label{eqn: dob}\n\\end{equation}\nagain normalised depending upon the employed protocol, by $N$, where $N$ is the amplitude of the imposed oscillation $\\dot{\\gamma}_0$ in LAOStrain, and $1+|\\dot{\\gamma}(t)|$ in LAOStress.\n\n\\subsection{Seeding the heterogeneity}\n\\label{sec:seed}\n\nWhen integrating the model equations to determine the time evolution\nof any flow heterogeneity, whether linearised or in their full\nnonlinear form, we must also specify the way in which whatever\nheterogeneous perturbations that are the precursor to the formation of\nshear bands are seeded initially. Candidates include any residual\nheterogeneity left in the fluid by the initial procedure of sample\npreparation; imperfections in the alignment of the rheometer plates;\ntrue thermal noise with an amplitude set by $k_{\\rm B}T$; and\nrheometer curvature in cone-and-plate or cylindrical Couette devices.\nWe consider in particular the last of these because it is likely to be\nthe dominant source of heterogeneity in commonly used flow cells,\nwhich typically have a curvature of about $10\\%$.\n\nWhile modelling the full effects of curvature is a complicated task,\nits dominant consequence can be captured simply by including a slight\nheterogeneity in the total stress field. (The assumption made above of\na uniform stress across the gap only holds in an idealised planar\ndevice.) Accordingly we set $\\Sigma(t)\\to\\Sigma(t)\\left[1+q\n  h(y)\\right]$ where $q$ sets the amplitude of the curvature and\n$h(y)$ is a function with an amplitude of $O(1)$ that prescribes its\nspatial dependence. The detailed form of $h(y)$ will differ from\ndevice to device: for example in a cylindrical Couette it is known to\nhave a $1\/r^2$ dependence, where $r$ is the radial coordinate.\nHowever, the aim here is not to model any particular device geometry\nin detail, but simply to capture the dominant effect of curvature in\nseeding the flow heterogeneity. Accordingly we set\n$h(y)=\\cos(\\pi\/L)$ which is the lowest Fourier mode to fit into the\nsimulation cell while still obeying the boundary conditions at the\nwalls.\n\n\\section{Shear banding in other time dependent protocols}\n\\label{sec:recap}\n\nAs a preamble to presenting our results for shear banding in\noscillatory flow protocols in the next two sections below, we first\nbriefly collect together criteria derived in previous work for linear\ninstability to the formation of shear bands in simpler time-dependent\nprotocols: slow shear rate sweep, fast shear startup, and step stress.\n\n\\subsection{Slow shear rate sweep}\n\\label{sec:recapSweep}\n\nA common experimental protocol consists of slowly sweeping the shear\nrate $\\dot{\\gamma}$ upwards (or downwards) in order to measure a fluid's\n(quasi) steady state flow curve. In this protocol the criterion for\nlinear instability to the onset of shear banding, given a base state\nof initially homogeneous shear flow, has long been known to\nbe~\\cite{Yerushalmi1970}\n\\begin{equation}\n\\label{eqn:criterionSteady}\n\\frac{\\partial\\Sigma}{\\partial\\dot{\\gamma}} < 0.\n\\end{equation}\n\n\\subsection{Fast shear startup}\n\\label{sec:recapStartup}\n\nAnother common experimental protocol consists of taking a sample of\nfluid that is initially at rest and with any residual stresses well\nrelaxed, then suddenly jumping the strain rate from zero to some\nconstant value such that $\\dot{\\gamma}(t)=\\dot{\\gamma}_0\\Theta(t)$, where\n$\\Theta(t)$ is the Heaviside function. Commonly measured in response\nto this applied flow is the time-dependent stress signal $\\Sigma(t)$\nas it evolves towards its eventual steady state value, for that\nparticular applied shear rate, on the fluid's flow curve. This\nevolution typically has the form of an initial elastic regime with\n$\\Sigma\\approx G\\gamma$ while the strain $\\gamma$ remains small,\nfollowed by an overshoot in the stress at a strain of $O(1)$, then a\ndecline to the final steady state stress on the flow curve. In\nRef.~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011}\nwe gave evidence that the presence of an overshoot in this stress\nstartup signal is generically indicative of a strong tendency to form\nshear bands, at least transiently. These bands may, or may not, then\npersist for as long as the shear remains applied, according to whether\nor not the underlying constitutive curve of stress as a function of\nstrain rate is non-monotonic.\n\nSuch behaviour is to be expected intuitively. Consider a shear startup\nrun performed at a high enough strain rate that the material's\nresponse is initially elastic, with the stress startup signal\ndepending only on the accumulated strain $\\gamma=\\dot{\\gamma} t$ and not\nseparately on the strain rate $\\dot{\\gamma}$. The decline in stress following\nan overshoot in the stress startup signal corresponds to a negative\nderivative\n\\begin{equation}\n\\label{eqn:criterionSimpleElastic}\n\\frac{\\partial \\Sigma}{\\partial\\gamma}<0.\n\\end{equation}\nThis clearly has the same form as~(\\ref{eqn:criterionSteady}) above,\nwith the strain rate now replaced by the strain.  As such it is the\ncriterion that we might intuitively expect for the onset of strain\nbands in a nonlinear elastic solid, following the early intuition of\nRef.~\\cite{Marrucci1983}\n\nIn close analogy to this intuitive expectation, for a complex fluid\nsubject to a fast, elastically dominated startup the criterion for the\nonset of banding was shown in Ref.~\\cite{Moorcroft2013} to be\nthat the stress signal $\\Sigma(\\gamma=\\dot{\\gamma} t)$ of the initially\nhomogeneous startup flow obeys\n\\begin{equation} \n\\label{eqn:criterionStartup}\n-\\textrm{tr}\\tens{M} \\frac{\\partial \\Sigma}{\\partial\\gamma} +\n\\dot{\\gamma}\\frac{\\partial^2\\Sigma}{\\partial\\gamma^2} < 0,\n\\end{equation}\nwhere $\\textrm{tr}\\tens{M}<0$ in this startup protocol.  This result\nholds exactly for any model whose equations are of the generalised\nform in Sec.~\\ref{sec:methods} above, and have only two relevant\ndynamical variables, $d=2$. (Recall that for the nRP model these two\nvariables are the shear stress $W_{xy}$ and one component of normal\nstress $W_{yy}$, in units in which the polymer modulus $G=1$.)  The\ncriterion~(\\ref{eqn:criterionStartup}) closely resembles the simpler\nform~(\\ref{eqn:criterionSimpleElastic}) motivated intuitively above,\nwith an additional term informed by the curvature in the signal of\nstress as a function of strain. The effect of this additional term is\nto trigger the onset of banding just {\\em before} overshoot, as the\nstress startup signal starts to curve downwards from its initial\nregime of linear elastic response.\n\nWhat this criterion tells us is that the presence of an overshoot in the\nstress signal of an underlying base state of initially homogeneous shear startup acts as a causative trigger for the formation of shear bands. A common misconception is that instead it is the onset of shear\nbanding that causes the stress drop.  While it is true that the onset\nof banding may reduce the stress further compared to that expected on\nthe basis of a homogeneous calculation, we emphasise that the\ndirection of mechanistic causality here is that the stress drop\nfollowing overshoot causes shear banding and not (primarily) vice\nversa.\n\nWith criterion (\\ref{eqn:criterionStartup}) in mind, theorists should\nbe alert that any model predicting startup stress overshoot in a\ncalculation in which the flow is artificially constrained to remain\nhomogeneous is likely to further predict the formation of shear bands\nin a full heterogeneous calculation that allows bands to form. Likewise\nexperimentalists should be alert that any observations of stress\novershoot in shear startup is strongly suggestive of the presence of\nbanding in the material's flow profile.\n\nIn Ref.~\\cite{Moorcroft2014} the analytically derived\ncriterion~(\\ref{eqn:criterionStartup}) was confirmed numerically for\nfast shear startup in the nRP model, where it should indeed apply\nexactly due to the presence of just $d=2$ relevant dynamical variables\n$W_{xy}$ and $W_{yy}$ in that model. It was also shown to apply to\ngood approximation in the sRP model, for which $d=3$, for strain rates\nlower than the inverse stretch relaxation time (where the dynamics of\nthe sRP model indeed well approximate those of the nRP model).\n\nBanding associated with startup stress overshoot has also been\ndemonstrated in several numerical studies of soft glassy materials\n(SGMs)~\\cite{Moorcroft2011,Fielding2014,Manningetal2009a}.\n(The term SGM is used to describe a broad class of materials including\nfoams, emulsions, colloids, surfactant onion phases and microgels, all\nof which show structural disorder, metastability, a yield stress, and\noften also rheological ageing below the yield stress.)  In these soft\nglasses, however, it should be noted that the decrease in stress\nfollowing the startup overshoot arises from increasing plasticity\nrather than falling elasticity. This makes it more difficult to derive\nan analytical criterion analogous to~(\\ref{eqn:criterionStartup}).\nAccordingly the theoretical evidence for shear banding following\nstartup overshoot in these soft glasses, while very convincing,\nremains primarily numerical to date.\n\nConsistent with these theoretical predictions, experimental\nobservations of banding associated with startup stress overshoot are\nwidespread: in wormlike micellar surfactants~\\cite{Hu2008,\n  Wangetal2008c}, polymers~\\cite{Wangetal2008a, Wangetal2008b,\n  Wangetal2009a, Wangetal2009b, Boukany2009a, Wangetal2003a,\n  Huetal2007a, Wangetal2009d}, carbopol\ngels~\\cite{Divoux2011a,Divoux2010} and Laponite clay\nsuspensions~\\cite{Martin2012}. Nonetheless, we also note other\n  studies of polymer solutions~\\cite{Li2015} where stress overshoot is\n  seen without observable banding. It would be particularly\n  interesting to see further experimental work on polymeric fluids to\n  delineate more fully the regimes, for example of entanglement number\n  and degree of polydispersity, in which banding arises with\n  sufficient amplitude to be observed experimentally.\n\n\n\n\\subsection{Step stress}\n\\label{sec:recapCreep}\n\nBesides the strain-controlled protocols just described, a fluid's\nrheological behaviour can also be probed under conditions of imposed\nstress.  In a step stress experiment, an initially well relaxed fluid\nis suddenly subject to the switch-on of a shear stress $\\Sigma_0$ that\nis held constant thereafter, such that $\\Sigma(t)=\\Theta(t)\\Sigma_0$.\nCommonly measured in response to this applied stress is the material's\ncreep curve, $\\gamma(t)$, or the temporal derivative of this,\n$\\dot{\\gamma}(t)$. In Ref.~\\cite{Moorcroft2013} the criterion for\nlinear instability to the formation of shear bands, starting from a\nstate of initially homogeneous creep shear response, was shown to be\nthat\n\\begin{equation}\n\\frac{\\partial^2\\dot{\\gamma}}{\\partial t^2}\/\\frac{\\partial\\dot{\\gamma}}{\\partial t}>0.\n\\end{equation}\nThis tells us that shear banding should be expected in any step stress\nexperiment in which the differentiated creep response curve\nsimultaneously curves upwards and slopes upwards. (Indeed it should\nalso be expected in any experiment where that response function\nsimultaneously curves downwards and slopes downwards, though we do not\nknow of any instances of such behaviour.) This prediction has been\nconfirmed numerically in the Rolie-poly model of polymers and wormlike\nmicelles~\\cite{Moorcroft2014}, as well as in the soft glassy\nrheology model of foams, dense emulsions, microgels, {\\it\n  etc}~\\cite{Fielding2014}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure1.eps}\n\\caption{LAOStrain: sketch of regions of shear rate amplitude and\n  frequency space in which we expect limiting low frequency `viscous'\n  and high frequency `elastic' behaviours, and regimes of linear and\n  nonlinear response. LAOStrain runs at the locations marked $X_L$ and\n  $X_H$ are explored in Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} for\n  the nRP model with non-monotonic and monotonic underlying\n  constitutive curve respectively.}\n\\label{fig:sketch}\n\\end{figure}\n\nExperimentally, shear banding associated with a simultaneously\nupwardly curving and upwardly sloping differentiated creep response\ncurve has indeed been seen in in entangled\npolymers~\\cite{Wangetal2009a,Huetal2007a,Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010, Wangetal2009d}, carbopol\nmicrogels~\\cite{Divoux2011} and carbon black\nsuspensions~\\cite{Gibaud2010}.\n\n\\section{Large amplitude oscillatory strain}\n\\label{sec:LAOStrain}\n\n\nWe now consider shear banding in the time-dependent strain-imposed\noscillatory protocol of LAOStrain. Here a sample of fluid, initially\nwell relaxed at time $t=0$, is subject for times $t>0$ to a strain of\nthe form\n\\begin{equation}\n\\gamma(t)=\\gamma_0\\sin(\\omega t),\n\\end{equation}\nto which corresponds the strain rate\n\\begin{equation}\n\\dot{\\gamma}(t)=\\gamma_0\\omega\\cos(\\omega t)=\\dot{\\gamma}_0\\cos(\\omega t).\n\\end{equation}\nAfter an initial transient, once many cycles have been executed, the\nresponse of the system is expected to attain a state that is\ntime-translationally invariant from cycle to cycle, $t\\to\nt+2\\pi\/\\omega$. All the results presented below are in this long-time\nregime, usually for the $N=20$th cycle after the flow\ncommenced. The dependence of the stress on the cycle number was carefully studied in wormlike micelles in Ref.~\\cite{Fujii2015}.\n\nTo characterise any given applied LAOStrain we must clearly specify\ntwo quantities: the strain amplitude and the frequency\n$(\\gamma_0,\\omega)$, or alternatively the strain rate amplitude and\nthe frequency $(\\dot{\\gamma}_0,\\omega)$, where $\\dot{\\gamma}_0=\\gamma_0\\omega$. In\nwhat follows we usually choose the latter pairing $(\\dot{\\gamma}_0,\\omega)$.\nAny given LAOStrain experiment is then represented by its location in\nthat plane of $\\dot{\\gamma}_0$ and $\\omega$. See Fig.~\\ref{fig:sketch}.\n\nIn any experiment where the applied strain rate remains small,\n$\\dot{\\gamma}_0 \\ll 1$, a regime of linear response is expected.  (Recall\nthat in dimensional form this condition corresponds to $\\dot{\\gamma}_0\\tau_d\n\\ll 1$.)  But even in an experiment where the strain rate does not\nremain small, linear response can nonetheless still be expected if the\noverall applied strain remains small, $\\gamma_0 \\ll 1$. Accordingly,\nlinear response should obtain in the region below the long-dashed line\nmarked in Fig.~\\ref{fig:sketch}. Because shear banding is an\ninherently nonlinear phenomenon, we expect the interesting region of\nthis $(\\dot{\\gamma}_0,\\omega)$ plane from our viewpoint to be in the\nnonlinear regime, above the long-dashed line, and we focus our\nattention mostly on this in what follows.\n\n\n\nBesides considering whether any given applied LAOStrain will result in\nlinear or nonlinear response, also relevant is the characteristic\ntimescale $1\/\\omega$ of the oscillation compared to the fluid's\nintrinsic terminal relaxation timescale $\\tau_d=1$. For low\nfrequencies $\\omega\\ll 1$, to the left of the leftmost dotted line in\nFig.~\\ref{fig:sketch}, we expect the material's reconfiguration\ndynamics to keep pace with the applied deformation. This will lead to\nquasi steady state response in which the stress slowly sweeps up\nand down the steady state flow curve as the shear rate varies through\na cycle. In contrast for high frequencies $\\omega\\gg 1$, to the right\nof the rightmost dotted line in Fig.~\\ref{fig:sketch}, the material's\nrelaxation dynamics cannot keep pace with the applied deformation and\nwe expect elastic-like response.\n\nWe illustrate these two limiting regimes by studying the response of\nthe nRP model to an imposed LAOStrain at each of the two locations\nmarked $X_L$ and $X_H$ in Fig.~\\ref{fig:sketch}.  For simplicity, for\nthe moment, we artificially constrain the flow to remain homogeneous\nand confine ourselves to calculating the uniform `base state' as\noutlined in Sec.~\\ref{sec:base}. The results are shown in\nFig.~\\ref{fig:nonmon} for the nRP model with parameters for which the\nunderlying constitutive curve is non-monotonic, such that (in any\nheterogeneous calculation) the fluid would show shear banding under\nconditions of steady applied shear.  Fig.~\\ref{fig:mon} shows results\nwith model parameters for which the constitutive curve is monotonic,\nsuch that no banding would be expected in steady shear flow.\n\n\nThe left panels of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} contain\nresults for the low frequency oscillation marked $X_L$ in\nFig.~\\ref{fig:sketch}. Here we choose to plot the stress response\n$\\Sigma(t)$ in a Lissajous-Bowditch figure as a parametric function of\nthe time-varying imposed strain rate $\\dot{\\gamma}(t)$, consistent with the\nexpectation of fluid-like response in this low-frequency regime.\n(Throughout the paper we shall describe such a plot of stress versus strain rate as being in the\n`viscous' representation.) As can be seen, in each case the fluid\nindeed tracks up and down its (quasi) steady state homogeneous\nconstitutive curve $\\Sigma(\\dot{\\gamma})$ in the range $-\\dot{\\gamma}_0 < \\dot{\\gamma} <\n\\dot{\\gamma}_0$.  For any set of model parameters, several of these LAOStrain\nresponse curves $\\Sigma(\\dot{\\gamma})$ collected together for different\n$\\dot{\\gamma}_0$ and low frequency $\\omega$ would all collapse onto this\nmaster constitutive curve.\n\nAlso shown by the colour scale in the left panels of\nFigs.~\\ref{fig:nonmon} and~\\ref{fig:mon} is the eigenvalue as\nintroduced in Sec.~\\ref{sec:lsa}. Recall that a positive eigenvalue at\nany point in the cycle strongly suggests that the homogeneous base\nstate is linearly unstable to the development of shear banding at that\npoint in the cycle.  (In any region where this scale shows black the\neigenvalue is either negative, or so weakly positive as to cause only\nnegligible banding growth.) As expected, a regime of instability is\nindeed seen in Fig.~\\ref{fig:nonmon}, in the region\nwhere the constitutive curve has negative slope,\n\\begin{equation}\n\\frac{\\partial\\Sigma}{\\partial\\dot{\\gamma}} <0.\n\\end{equation}\nFor a fluid with a monotonic constitutive curve, no instability is\nobserved at this low frequency (Fig.~\\ref{fig:mon}, left).\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure2.eps}\n\\caption{LAOStrain in the nRP model with a non-monotonic underlying\n  constitutive curve. Model parameters $\\beta=0.4$, $\\eta=10^{-5}$.\n  {\\bf Left:} Viscous Lissajous-Bowditch\n  figure shows stress $\\Sigma$ versus strain rate $\\dot{\\gamma}$ for an\n  imposed frequency and strain rate $(\\omega,\\dot{\\gamma}_0)=(0.001,50.0)$\n  marked as $X_L$ in the low frequency regime of\n  Fig.~\\ref{fig:sketch}.  {\\bf Right:} Elastic Lissajous-Bowditch figure shows\n  stress $\\Sigma$ versus strain $\\gamma$ for an imposed frequency and\n  strain rate $(\\omega,\\dot{\\gamma}_0)=(31.6,200.0)$ marked as $X_H$ in the\n  high frequency regime of Fig.~\\ref{fig:sketch}.  Colourscale shows\n  eigenvalue.}\n\\label{fig:nonmon}\n\\end{figure}\n\n\n\n\nThe corresponding results for the high frequency run marked $X_H$ in\nFig.~\\ref{fig:sketch} are shown in the right panels of\nFigs.~\\ref{fig:nonmon} and~\\ref{fig:mon}. Here we choose to plot the\nstress response $\\Sigma(t)$ in a Lissajous-Bowditch figure as a\nparametric function of the time-varying strain $\\gamma(t)$, in the\nso-called `elastic' representation. Indeed, just as in the low\nfrequency regime the material behaved as a viscous fluid with the\nstress response falling onto the steady state master constitutive\ncurve in the viscous representation $\\Sigma(\\dot{\\gamma})$, for a high\nfrequency cycle we might instead expect a regime of elastic response\nin which only the accumulated strain is important, and not\n(separately) the strain rate, giving a master response curve of stress\nversus strain, $\\Sigma(\\gamma)$.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure3.eps}\n\\caption{As in Fig.~\\ref{fig:nonmon}, but for a value of the CCR\n  parameter $\\beta=1.0$, for which the underlying homogeneous\n  constitutive curve is monotonic.}\n\\label{fig:mon}\n\\end{figure}\n\nWe might further have intuitively expected this curve to be the same\nas that obtained in a fast shear startup from rest, with (in the\npositive strain part of the cycle) elastic response $\\Sigma\\approx G\\gamma$\nat low strain $\\gamma\\ll 1$, followed by stress overshoot at a typical\nstrain $\\gamma=+O(1)$, then decline towards a constant stress at\nlarger strains (with the symmetric curve in the negative-strain part\nof the cycle, such that $\\Sigma\\to-\\Sigma$ for $\\gamma\\to-\\gamma$). In\nother words, in LAOStrain at high frequency we might have expected the\nsystem to continuously explore its elastic shear startup curve\n$\\Sigma(\\gamma)$ between $\\gamma=-\\gamma_0$ and $\\gamma=+\\gamma_0$.\n\nHowever, this intuition is not met in a straightforward way. In the\nright panels of Fig.~\\ref{fig:nonmon} and~\\ref{fig:mon} we observe\ninstead an open cycle that is explored in a clockwise sense as time\nproceeds through an oscillation: the stress transits the upper part of\nthe loop (from bottom left to top right) in the forward part of the\ncycle as the strain increases from left to right, and the\nsymmetry-related lower part of the loop in the backward part, where\nthe strain decreases from right to left.\n\nThis can be understood as follows. For any LAOStrain run at high\nfrequency $\\omega \\gg 1$ but in the linear regime with strain\namplitude $\\gamma_0\\ll 1$, we do indeed find the stress response to\nfall onto a closed master curve $\\Sigma(\\gamma)$, which also\ncorresponds to that obtained in a fast stress startup from rest, with\nlinear elastic response $\\Sigma\\approx G\\gamma$. (Data not shown.) In\ncontrast, for amplitudes $\\gamma_0 > 1$ the system only explores this\nstartup-from-rest curve in the first half of the {\\em first} cycle\nafter the inception of flow. (This has the usual form, with elastic\nresponse for small strains, stress overshoot at a strain\n$\\gamma=O(1)$, then decline to a constant stress.)  In the second half\nof the cycle, when the strain rate reverses and the strain decreases,\nthe stress response departs from the startup-from-rest curve. With\nhindsight this is in fact obvious: as this backward shear part of the\ncycle commences the initial condition is not that of a well-relaxed\nfluid, but one that has just suffered a large forward strain.\n\n\n\n\nThe same is true for the next forward half cycle: its initial\ncondition is that of a fluid that has just suffered a large negative\nstrain, corresponding to the lower left point in the right panel of\nFigs.~\\ref{fig:nonmon} or~\\ref{fig:mon}. Starting from that initial\ncondition the stress evolution nonetheless thereafter resembles that\nof a fast startup, with an initial fast rise followed by an overshoot\nthen decline to constant stress, before doing the same in reverse\n(with a symmetry-related `negative overshoot') during the next half\ncycle, giving the open curves as described. Associated with this\novershoot in each half cycle is a positive eigenvalue indicating\ninstability to the onset of shear banding. Importantly, we note that\nthis arises even in the case of a monotonic underlying constitutive\ncurve (Fig.~\\ref{fig:mon}, right), and therefore even in a fluid that would\nnot display steady state banding under a steadily applied shear flow.\nIt is the counterpart for LAOStrain of the `elastic' banding triggered\nby stress overshoot in a fast shear startup from rest, as explored\npreviously in\nRef.~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011}.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure4.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  nRP model with a non-monotonic constitutive curve. Each point in this\n  $\\dot{\\gamma}_0,\\omega$ plane corresponds to a particular LAOStrain run\n  with strain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding that occurs at any point in the\n  cycle, after many cycles.  Model parameters: $\\beta=0.4$,\n  $\\eta=10^{-5}$.  Cell curvature $q=10^{-4}$. Crosses indicate the\n  grid of values of $\\dot{\\gamma}_{0}$ and $\\omega$ in Pipkin diagram of\n  Fig.~\\ref{fig:PipkinNonMon}.}\n\\label{fig:pinPointNonMon}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure5.eps}\n\\caption{As in Fig.~\\ref{fig:pinPointNonMon}, but with a CCR parameter\n  $\\beta=1.0$, for which the fluid has a monotonic underlying\n  constitutive curve.  Crosses indicate the grid of values of\n  $\\dot{\\gamma}_{0}$ and $\\omega$ used in the Pipkin diagram of\n  Fig.~\\ref{fig:PipkinMon}.}\n\\label{fig:pinPointMon}\n\\end{figure}\n\n\\begin{figure}[tp!]\n\\includegraphics[width=9cm]{figure6.eps}\n\\caption{Lissajous-Bowditch curves in LAOStrain for the nRP model with\n  a non-monotonic constitutive curve. Results are shown in the elastic\n  representation in (a), and the viscous representation in\n  (b). Columns of fixed frequency $\\omega$ and rows of fixed\n  strain-rate amplitude $\\dot{\\gamma}_0$ are labeled at the top and\n  right-hand side. Colourscale shows the time-dependent degree of\n  shear banding.  Model parameters: $\\beta=0.4, \\eta=10^{-5}$. Cell\n  curvature: $q=10^{-4}$. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by\n  the thicker box is shown in Fig.~\\ref{fig:portraitNonMon}.}\n\\label{fig:PipkinNonMon}\n\\end{figure}\n\n\\begin{figure}[tp!]\n\\includegraphics[width=9cm]{figure7.eps}\n\\caption{As in Fig.~\\ref{fig:PipkinNonMon} but for a value of the CCR\n  parameter $\\beta=1.0$, for which the fluid's underlying constitutive\n  curve is monotonic. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by the\n  thicker box is shown in Fig.~\\ref{fig:portraitMon}.}\n\\label{fig:PipkinMon}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8cm]{figure8.eps}\n\\caption{LAOStrain in the nRP model with a non-monotonic constitutive\n  curve. Strain rate amplitude $\\dot{\\gamma}_{0}=10.0$ and frequency\n  $\\omega=3.16$. Model parameters $\\beta=0.4, \\eta=10^{-5}$.  Cell\n  curvature $q=10^{-4}$. Number of numerical grid points $J=1024$.\n  {\\bf Left:} stress response in the elastic representation. Solid\n  black and red-dashed line: calculation in which the flow is\n  constrained to be homogeneous.  Red-dashed region indicates a\n  positive eigenvalue showing instability to the onset of shear\n  banding. Green dot-dashed line: stress response in a full nonlinear\n  simulation that allows banding. {\\bf Right:} Velocity profiles\n  corresponding to stages in the cycle indicated by matching symbols\n  in the left panel. Each profile is normalised by the speed of the\n  moving plate.}\n\\label{fig:portraitNonMon}\n\\end{figure}\n\nIndeed, following the calculation first set out in\nRef.~\\cite{Moorcroft2013}, it is straightforward to show that\nthe condition for a linear instability to banding in this `elastic'\nhigh frequency regime $\\omega\\gg 1$ is the same as in fast shear\nstartup:\n\\begin{equation} \n\\label{eqn:criterionElastic}\n-\\textrm{tr}\\tens{M} \\frac{\\partial \\Sigma}{\\partial\\gamma} +\n\\dot{\\gamma}\\frac{\\partial^2\\Sigma}{\\partial\\gamma^2} < 0.\n\\end{equation}\nAs already discussed, this gives a window of instability setting in\njust before the stress overshoot (or negative overshoot) in each half\ncycle in the right panels of Fig.~\\ref{fig:nonmon} and~\\ref{fig:mon}\ndue to (in the positive $\\dot{\\gamma}$ part of the cycle in which the stress\ntransits from bottom left to top right) the negatively sloping and\ncurving $\\Sigma(\\gamma)$. An analogous statement applies in the other\npart of the cycle, with the appropriate sign reversals. Note that\nthese overshoots are sufficiently weak as to be difficult to resolve\nby eye on the scale of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon}.\n\nInterestingly,~(\\ref{eqn:criterionElastic}) also predicts a region of\n(weaker) instability immediately after the reversal of strain in each\nhalf cycle, as also seen in the right panels of Figs.~\\ref{fig:nonmon}\nand~\\ref{fig:mon}.  Analytical considerations show that this\nadditional regime of instability is not driven by any negative slope or\ncurvature in $\\Sigma(\\gamma)$, but instead arises from a change in\nsign of $\\textrm{\\tens{M}}$.  This instability has no counterpart that\nwe know of in shear startup. Its associated eigenvector is dominated\nby the normal stress component $W_{yy}$ rather than the strain rate or\nshear stress. Heterogeneity in this quantity could feasibly be\naccessed in birefringence experiments.\n\nThe results of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} can be\nsummarised as follows. At low frequencies the system sweeps slowly up\nand down the underlying constitutive curve $\\Sigma(\\dot{\\gamma})$ as the\nshear rate varies through the cycle. If that curve is non-monotonic,\nthis homogeneous base state is unstable to shear banding in the region\nof negative constitutive slope, $d\\Sigma\/d\\dot{\\gamma}<0$. At high\nfrequencies the system instead executes a process reminiscent of\nelastic shear startup in each half cycle, but with an initial\ncondition corresponding to the state left by the shear of opposite\nsense in the previous half cycle. Associated with this is a stress\novershoot in each half cycle, giving instability to elastic shear\nbanding. Crucially, this elastic instability occurs whether or not the\nunderlying constitutive curve is non-monotonic or monotonic, and\ntherefore whether or not the fluid would shear band in steady shear.\n\nFrom a practical experimental viewpoint it is important to note that,\nwhereas in a single shear startup run these `elastic' strain bands\nwould form transiently then heal back to homogeneous flow (unless the\nsample has a non-monotonic underlying constitutive curve and so also\nbands in steady state), in LAOStrain the banding will recur in each\nhalf cycle and so be potentially easier to access experimentally.\nAny time-averaging measurement should of course only take data in the\nforward part of each cycle, to avoid averaging to zero over the\ncycle.\n\nHaving explored in Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} the\ntendency to form shear bands in two particular LAOStrain runs (one in\nthe limit of low frequency, $X_L$ in Fig.~\\ref{fig:sketch}, and one in\nthe limit of high frequency, $X_H$), we now explore the full\n$(\\dot{\\gamma}_0,\\omega)$ plane of Fig.~\\ref{fig:sketch} by showing in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon} colour maps of\nthe extent of shear banding across this plane.  Recall that each\npinpoint in this plane corresponds to a single LAOS experiment with\nstrain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  To build up\nthese colour maps, we sweep over a grid of 100x100\nvalues of $\\dot{\\gamma}_0,\\omega$ and execute a LAOStrain run at each point.\nSolving the model's full nonlinear dynamics on such a dense grid would\nbe unfeasibly time-consuming computationally. Therefore at each\n$\\dot{\\gamma}_0,\\omega$ we instead integrate the linearised equations set out\nin Sec.~\\ref{sec:lsa}. In each such run we record the degree of\nbanding $\\delta\\dot{\\gamma}$, maximised over the cycle after many cycles.  It\nis this quantity, normalised by the maximum strain rate amplitude $\\dot{\\gamma}_{0}$, that is represented by the colourscale in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}.\n\nFig.~\\ref{fig:pinPointNonMon} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is\nnon-monotonic. As expected, significant banding (bright\/yellow region)\nis observed even in the limit of low frequency $\\omega\\to 0$ for\nstrain rate amplitudes $\\dot{\\gamma}_0$ exceeding the onset of negative slope\nin the underlying constitutive curve. This region of banding is the\ndirect (and relatively trivial) analogue of banding in a slow strain\nrate sweep along the steady state flow curve.\nFig.~\\ref{fig:pinPointMon} shows results for the nRP model with\nparameters such that the underlying constitutive curve is monotonic.\nHere steady state banding is absent in the limit $\\omega\\to 0$. In\nboth Fig.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}, however,\nsignificant banding is observed at high frequencies for a strain\namplitude $\\gamma_0>1 $: this is the elastic banding associated with\nthe stress overshoot in each half cycle, described in detail above for\nthe $(\\dot{\\gamma}_0,\\omega)$ values denoted by $X_H$ in\nFig.~\\ref{fig:sketch}.\n\nIt is important to emphasise, therefore, that even a fluid with a\npurely monotonic constitutive curve, which does not shear band in\nsteady flow, is still nonetheless capable of showing strong shear\nbanding in a time-dependent protocol of high enough frequency\n(Fig.~\\ref{fig:pinPointMon}). Also important to note is that for a\nfluid with a non-monotonic constitutive curve the region of steady\nstate `viscous' banding at low frequencies crosses over smoothly to\nthat of `elastic' banding as the frequency increases\n(Fig.~\\ref{fig:pinPointNonMon}). \n\nCorresponding to the degree of banding in the shear rate\n$\\delta\\dot{\\gamma}$, as plotted in Figs.~\\ref{fig:pinPointNonMon}\nand~\\ref{fig:pinPointMon}, is an equivalent degree of banding $G\\delta\nW_{xy}=-\\eta\\delta\\dot{\\gamma}$ (to within small corrections of order the\ncell curvature, $q$) in the shear component of the polymeric\nconformation tensor. This follows trivially by imposing force balance\nat zero Reynolds number. Counterpart maps for the degree of banding in\nthe component $\\delta W_{yy}$ of the polymeric conformation tensor can\nalso be built up. These reveal closely similar regions of banding to\nthose shown in Figs.~\\ref{fig:pinPointNonMon}\nand~\\ref{fig:pinPointMon}. (Data not shown.) Experimentally,\nheterogeneities in the polymeric conformation tensor can be accessed\nby flow birefringence.\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8cm]{figure9.eps}\n\\caption{As in Fig.~\\ref{fig:portraitNonMon} but for the nRP model with\n  a CCR parameter $\\beta=1.0$ for which the underlying homogeneous\n  constitutive curve is monotonic, and for a LAOStrain with strain\n  rate amplitude $\\gamma_{0}=56.2$ and frequency\n  $\\omega=10.0$. Number of numerical grid points $J=512$.}\n\\label{fig:portraitMon}\n\\end{figure}\n\n\nAs noted above, to build up such comprehensive roadmaps as in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon} in a\ncomputationally efficient way, we omitted all nonlinear effects and\nintegrated instead the linearised equations of Sec.~\\ref{sec:lsa}.\nThese are only strictly valid in any regime where the amplitude of the\nheterogeneity remains small. In omitting nonlinear effects, they tend\nto overestimate the degree of banding in any regime of sustained\npositive eigenvalue, in predicting the heterogeneity to grow\nexponentially without bound, whereas in practice it would be cutoff by\nnonlinear effects. We now remedy this shortcoming by exploring the\nmodel's full nonlinear spatiotemporal dynamics. To do so within\nfeasible computational run times, we focus on a restricted grid of\nvalues in the $\\dot{\\gamma}_0,\\omega$ plane, marked by crosses in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}.\n\nThe results are shown in Fig.~\\ref{fig:PipkinNonMon} for the nRP model\nwith model parameters for which the underlying constitutive curve is\nnon-monotonic.  At low frequencies the results tend towards the\nlimiting fluid-like behaviour discussed above, in which the the stress\nslowly tracks up and down the steady state flow curve $\\Sigma(\\dot{\\gamma})$.\n(Progression towards this limit can be seen by following the top row\nof panels in Fig.~\\ref{fig:PipkinNonMon}b) to the left.)  Viscous\nbanding is seen for sufficiently high strain rate amplitudes $\\dot{\\gamma}_0$\ndue to the negatively sloping underlying homogeneous constitutive\ncurve.  At high frequencies the response tends instead towards the\nlimiting elastic-like behaviour discussed above. For large enough\nstrain amplitudes the stress then shows an open cycle $\\Sigma(\\gamma)$\nas a function of strain, with an overshoot in each half cycle that\ntriggers the formation of `elastic' banding.  (Progression towards\nthis limit is seen by following the top row of panels in\nFig.~\\ref{fig:PipkinNonMon}a to the right.)\n\nOvershoots in the elastic Lissajous-Bowditch curve of stress as a\nfunction of strain have been identified in earlier\nwork~\\cite{Ewoldt2009loops} as leading to self-intersection of the\ncorresponding viscous Lissajous-Bowditch curve of stress as a function\nof strain-rate. Such an effect is clearly seen here in the Rolie-poly\nmodel: see for example the runs highlighted by the thicker boxes in\nFig.~\\ref{fig:PipkinNonMon} and in Fig.~\\ref{fig:PipkinMon}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure10.eps}\n\\caption{Effect of CCR parameter $\\beta$ and entanglement number $Z$\n  (and so of chain stretch relaxation time $\\tau_R=\\tau_d\/3Z$) on shear\n  banding in LAOStrain.  (Recall that the non-stretching version of\n  the model has $\\tau_R\\to 0$ and so $Z\\to\\infty$.)  Empty circles: no\n  observable banding.  Hatched circles: observable banding, typically\n  $\\Delta_{\\dot{\\gamma}}\/\\dot{\\gamma}_0 \\approx 10\\%-100\\%$. Filled circles: significant\n  banding $\\Delta_{\\dot{\\gamma}}\/\\dot{\\gamma}_0 \\ge 100\\%$.  For hatched and filled\n  symbols we used the criterion that banding of the typical magnitude\n  stated is apparent in a region spanning at least half a decade by\n  half a decade in the plane of $\\dot{\\gamma}_0,\\omega$, by examining maps as\n  in Fig.~\\ref{fig:stretchPinPoint} in by eye. The square shows the\n  parameter values explored in detail in\n  Fig.~\\ref{fig:stretchPinPoint}.}\n\\label{fig:stretchMaster}\n\\end{figure}\n\nFor intermediate frequencies the stress is a complicated function of\nboth strain rate and also, separately, the strain. The three\ndimensional curve $(\\Sigma,\\dot{\\gamma},\\gamma)$ is then best shown as two\nseparate projections: first in the elastic representation of the\n$\\Sigma,\\gamma$ plane (Fig.~\\ref{fig:PipkinNonMon}a), and second, in\nthe viscous representation of the $(\\Sigma,\\dot{\\gamma})$ plane\n(Fig.~\\ref{fig:PipkinNonMon}b).  Collections of these\nLissajous-Bowditch curves on a grid of $(\\dot{\\gamma}_0,\\omega)$ values as in\nFig.~\\ref{fig:PipkinNonMon} are called Pipkin diagrams.\n\n\\begin{figure}[tbp]\n   \\includegraphics[width=10.0cm]{figure11.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  sRP model with a monotonic constitutive curve. Each point in this\n  $\\dot{\\gamma}_0,\\omega$ plane corresponds to a particular LAOStrain run\n  with strain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding at any point in the cycle, after\n  many cycles.  Model parameters: $\\beta=0.7$, $Z=75$ (and so\n  $\\tau_R=0.0044$), $\\eta=10^{-5}$.  Cell curvature $q=2\\times\n  10^{-3}$. Note the different colour scale from Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon}. The model's full nonlinear dynamics for the ($\\dot{\\gamma}_{0}, \\omega$) value marked by the cross are explored in Fig.~\\ref{fig:stretchPortrait}.}\n\\label{fig:stretchPinPoint}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure12.eps}\n\\caption{sRP model with a monotonic constitutive curve in LAOStrain of\n  strain rate amplitude $\\dot{\\gamma}_{0}=20.0$ and frequency $\\omega=8.0$.\n  Model parameters $\\beta=0.7, Z=75, \\eta=10^{-5}$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} stress response in the elastic representation. Solid black\n  and red-dashed line: calculation in which the flow is constrained to\n  be homogeneous.  Red-dashed region indicates a positive eigenvalue\n  showing instability to the onset of shear banding. Green dot-dashed\n  line: stress response in a full nonlinear simulation that allows\n  banding (almost indistinguishable from homogeneous signal in this\n  case.) {\\bf Right:} Velocity profiles corresponding to stages in the\n  cycle indicated by matching symbols in left panel.}\n\\label{fig:stretchPortrait}\n\\end{figure}\n\nFor the particular LAOStrain run highlighted by a thicker box in\nFig.~\\ref{fig:PipkinNonMon}, a detailed portrait of the system's\ndynamics is shown in Fig.~\\ref{fig:portraitNonMon}. Here the stress is\nshown in the elastic representation, as a function of strain $\\gamma$\n(left hand panel). Two curves are shown here. The first shows the\nstress signal in a calculation in which the flow is artificially\nconstrained to remain homogeneous. A linear instability analysis for\nthe dynamics of small heterogeneous perturbations about this\ntime-evolving homogeneous base state then reveals instability towards\nthe formation of shear bands (a positive eigenvalue) in the portion of\nthat curve shown as a red dashed line. A full nonlinear simulation\nthen reveals the formation of shear bands, and leads to a stress\nsignal (green dot-dashed line) that deviates from that of the\nhomogeneously constrained calculation, in particular in having a much\nmore precipitous stress drop due to the formation of bands.\n\nThe associated velocity profiles at four points round the part of the\ncycle with increasing strain are shown in the right hand panel. Before\nthe stress overshoot, no banding is apparent (black circles). The\novershoot then triggers strong shear banding (red squares), with most\nof the shear concentrated in a thin band at the left hand edge of the\ncell. Interestingly, the shear in the right hand part of the cell is\nin the opposite sense to the overall applied shear. This is consistent\nwith the fact that the stress is a decreasing function of strain in\nthis regime: the material is being unloaded, and an elastic-like\nmaterial being unloaded will shear backwards. As the overall applied\nstrain increases towards the end of the window of instability, the\nflow heterogeneity gradually decays away. This process repeats in each\nhalf cycle (with the obvious sign reversals in the part of the cycle\nin which the strain is decreasing).\n\nThe corresponding Pipkin diagram for a fluid with a monotonic\nconstitutive curve (Fig.~\\ref{fig:PipkinMon}) likewise confirms its\ncounterpart linear diagram in Fig.~\\ref{fig:pinPointMon}. Here\n`viscous' banding is absent at low frequencies, because the fluid is\nnot capable of steady state banding. Crucially, however, a strong\neffect of elastic banding is still seen at high frequencies.  A\ndetailed portrait of the system's dynamics in this elastic regime, for\nthe strain rate amplitude and frequency marked by the thicker box in\nFig.~\\ref{fig:PipkinMon}, is shown in Fig.~\\ref{fig:portraitMon}. As\ncan be seen, it shows similar features to those just described in\nFig.~\\ref{fig:portraitNonMon}. We emphasise, then, that even polymeric\nfluids that do not band under conditions of steady shear are still\ncapable of showing strong banding in a time-dependent protocol at high\nenough frequency. This important prediction is consistent with the\nearly insight of Adams and Olmsted in Ref.~\\cite{Adams2009}.\n\nSo far we have presented results for the nRP model, which assumes an\ninfinitely fast rate of chain stretch relaxation compared to the rate\nof reptation, such that the ratio $\\tau_R\/\\tau_d\\to 0$. This corresponds\nto assuming that the polymer chains are very highly entangled, with a\nnumber of entanglements per chain $Z=\\tau_d\/3\\tau_R\\to\\infty$. We now\nconsider the robustness of these results to reduced entanglement\nnumbers, and accordingly increased chain stretch relaxation time\n$\\tau_R$ (in units of $\\tau_d$). \n\nThe results are summarised in Fig.~\\ref{fig:stretchMaster}, which\nshows the regions of the plane of the CCR parameter $\\beta$ and\nentanglement number $Z$ in which significant banding (filled circles),\nobservable banding (hatched circles), and no banding (open circles)\noccur.  (Recall that results presented for the nRP model above pertain\nto the values $\\beta=0.4,1.0$ in the limit $Z\\to\\infty$.) As can be\nseen, by reducing the number of entanglements per chain the effect of\nshear banding is reduced and eventually eliminated.  However it is\nimportant to note that, depending on the value of $\\beta$, significant\nbanding is still observed for experimentally commonly used\nentanglement numbers, typically in the range $20-100$. Furthermore, significant banding is seen in a large region of the ($\\beta, Z$) plane for which the material's underlying constitutive curve is monotonic. As discussed\nabove, there is no current consensus as to the value of the CCR\nparameter $\\beta$ in the range $0<\\beta<1$. Using the routemap\nprovided in Fig.~\\ref{fig:stretchMaster}, a study of shear banding in\nLAOStrain experiments could provide one way to obtain a more accurate\nestimate of the value of this parameter.\n\nFor the pairing of $\\beta$ and $Z$ values marked by the square in\nFig.~\\ref{fig:stretchMaster}, we show in\nFig.~\\ref{fig:stretchPinPoint} a colour map of the degree of banding\nexpected in LAOStrain in the space of strain rate amplitude $\\dot{\\gamma}_0$\nand frequency $\\omega$. This figure, which is for the sRP model,\n  is the counterpart of the earlier Figs.~\\ref{fig:pinPointNonMon}\n  and~\\ref{fig:pinPointMon} discussed above for the nRP model, with\nthe degree of banding calculated for computational\nefficiency within the assumption of linearised dynamics. Recall that\neach pinpoint in this plane corresponds to a single LAOStrain run with\napplied strain rate $\\dot{\\gamma}(t)=\\dot{\\gamma}_0\\cos(\\omega t)$.\n\nConsistent with the underlying constitutive curve being monotonic for\nthese parameters, `viscous' banding is absence in the limit of low\nfrequencies $\\omega\\to 0$.  However significant banding is still\nobserved for runs with strain rate amplitudes $O(10-100)$ and\nfrequencies $O(1-10)$. This is the counterpart of the `elastic'\nbanding reported above in the nRP model, though the effect of finite\nchain stretch in the sRP model is to moderate the degree of banding. A\ndetailed portrait of the model's nonlinear banding dynamics at a\nstrain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$ marked by the\ncross in Fig.~\\ref{fig:stretchPinPoint} is shown in\nFig.~\\ref{fig:stretchPortrait}. Significant shear banding associated with the stress overshoot is apparent in each half cycle.\n\n\\section{Large amplitude oscillatory stress}\n\\label{sec:LAOStress}\n\nWe now consider the time-dependent stress-imposed oscillatory protocol\nof LAOStress. Here the sample is subject for times $t>0$ to a stress\nof the form\n\\begin{equation}\n\\Sigma(t)=\\Sigma_0\\sin(\\omega t),\n\\end{equation}\ncharacterised by the frequency $\\omega$ and stress amplitude\n$\\Sigma_0$. As for the case of LAOStrain above, all the results\npresented below are in the long-time regime, once many ($N=20$) cycles\nhave been executed and the response of the system has settled to be\ntime-translationally invariant from cycle to cycle, $t\\to\nt+2\\pi\/\\omega$.\n\nIn Sec.~\\ref{sec:recapCreep} we reviewed existing work demonstrating\nthe tendency to form shear bands in a {\\rm step} stress experiment.\nHere an initially well relaxed sample is suddenly subject at time\n$t=0$ to the switch-on of a shear stress of amplitude $\\Sigma_0$,\nwhich is held constant for all subsequent times.  The criterion for an\nunderlying base state of initially homogeneous creep response to\nbecome linearly unstable to the formation of shear bands is then that\nthe time-differentiated creep response curve $\\dot{\\gamma}(t)$\nobeys~\\cite{Moorcroft2013}:\n\\begin{equation}\n\\label{eqn:critCreep}\n\\frac{\\partial^2\\dot{\\gamma}}{\\partial t^2}\/\\frac{\\partial\\dot{\\gamma}}{\\partial t}>0.\n\\end{equation}\nTherefore, shear banding is expected in any regime where the\ntime-differentiated creep curve simultaneously slopes up and curves\nupwards; or instead simultaneously slopes down and curves downwards.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure13.eps}\n\\caption{LAOStress in the nRP model with a non-monotonic constitutive\n  curve. Model parameters: $\\beta=0.1$, $\\eta=10^{-4}$. Frequency\n  $\\omega=0.01$ and stress amplitude $\\Sigma_0=0.7$.  {\\bf Left:}\n  stress versus strain rate (shown on a log scale) in the positive\n  stress part of the cycle. Colour scale shows eigenvalue, with\n  negative values also shown as black. Green dashed line: underlying\n  constitutive curve. {\\bf Right:} corresponding stress versus time\n  plot.}\n\\label{fig:eigenNonMon}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure14.eps}\n\\caption{As in Fig.~\\ref{fig:eigenNonMon} but at a higher imposed\n  frequency $\\omega=1.0$ and for a value of the CCR $\\beta=0.9$, for\n  which the nRP model has a monotonic underlying constitutive curve.\n  {\\bf Right:} corresponding stress versus time plot.}\n\\label{fig:eigenMon}\n\\end{figure}\n\n\n\nHaving been derived within the assumption of an imposed stress that is\nconstant in time, criterion (\\ref{eqn:critCreep}) would not {\\it a\n  priori} be expected to hold for the case of LAOStress. Nonetheless\nit might reasonably be expected to apply, to good approximation, in any\nregime of a LAOStress experiment where a separation of timescales\narises such that the shear rate $\\dot{\\gamma}(t)$ evolves on a much shorter\ntimescale than the stress. In this case, from the viewpoint of the\nstrain rate signal, the stress appears constant in comparison and the\nconstant-stress criterion (\\ref{eqn:critCreep}) is expected to hold.\nIndeed, in what follows we shall show that many of our results for\nLAOStress can be understood on the basis of this simple piece of\nintuition.\n\nWe start in Fig.~\\ref{fig:eigenNonMon} by considering the nRP model in\na parameter regime for which the underlying constitutive curve is\nnon-monotonic (see the dotted line in the left panel), such that shear\nbanding would be expected under conditions of a steadily applied shear\nrate.  With the backdrop of this constitutive curve we consider a\nLAOStress run at low frequency $\\omega\\to 0$. For definiteness we will\nfocus on the part of the cycle where the stress is positive, but\nanalogous remarks will apply to the other half of the cycle, with\nappropriate changes of sign.\n\nConsider first the regime in which the stress is slowly increased from\n$0$ towards its maximum value $\\Sigma_0$. In this part of the cycle,\nat the low frequencies of interest here, we expect the system to\ninitially follow the high viscosity branch of the constitutive curve.\nIn any experiment for which the final stress $\\Sigma_0$ exceeds the\nlocal maximum in the constitutive curve, the system must at some stage\nduring this increasing-stress part of the cycle transit from the high\nto low viscosity branch of the constitutive curve. This transition is\nindeed seen in Fig.~\\ref{fig:eigenNonMon}: it occurs via ``top\njumping'' from the stress maximum across to the low viscosity branch.\nConversely, on the downward part of the sweep as the stress decreases\nfrom its maximum value $\\Sigma_0$, the system initially follows the\nlow viscosity branch until it eventually jumps back to the high\nviscosity branch. (We return in our concluding remarks to discuss the\npossible effect of thermal nucleation events, which are not included\nin these simulations, on these process of jumping between the two\nbranches of the constitutive curve.)\n\n\n\nThe corresponding signal of strain rate versus time during this slow\nup-then-down stress oscillation is shown in the right panel of\nFig.~\\ref{fig:eigenNonMon}. As can be seen, the regimes where the\nshear rate transits between the two different branches of the\nconstitutive curve occur over relatively short time intervals. (The\nduration of this process is informed by the short timescale $\\eta\/G$,\nwhereas the stress evolves on the much longer timescale $2\\pi\/\\omega$.)\nThis separation of timescales renders the stress signal approximately\nconstant in comparison to the fast evolution of the strain rate.\nCriterion (\\ref{eqn:critCreep}) might therefore be expected to apply\nin this regime of transition, at least to good approximation.\n\n\n\n\nFurthermore, during the transition from the high to low viscosity\nbranch we see a regime in which the shear rate signal simultaneously\nslopes up and curves up as a function of time: criterion\n(\\ref{eqn:critCreep}) not only applies but is also met, and we\ntherefore expect an instability to banding.  Plotting, by means of a\ncolourscale, the eigenvalue as defined in Sec.~\\ref{sec:lsa}, we find\nthat it is indeed positive. Likewise, during the rapid transition back\nfrom the low to high viscosity branch, we find a regime in which the\nshear rate signal simultaneously slopes down and curves down. As seen\nfrom the colourscale, the eigenvalue is also positive in this regime\n(although more weakly than during the upward transition). In what\nfollows, we will confirm the expectation of shear band formation\nduring these times of positive eigenvalue by simulating the model's\nfull nonlinear spatiotemporal dynamics.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.5cm]{figure15.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  nRP model with a non-monotonic constitutive curve. Each point in\n  this $\\Sigma_0,\\omega$ plane corresponds to a particular LAOStress\n  run with stress amplitude $\\Sigma_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding that occurs at any point in the\n  cycle, after many cycles.  Model parameters: $\\beta=0.4$,\n  $\\eta=10^{-4}$.  Cell curvature $q=2\\times 10^{-3}$. Crosses\n  indicate the grid of values of $\\Sigma_{0}$ and $\\omega$ in\n  Fig.~\\ref{fig:PipkinNonMon2}.}\n\\label{fig:pinPointNonMon2}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.5cm]{figure16.eps}\n\\caption{ As in Fig.~\\ref{fig:pinPointNonMon}, but with a CCR\n  parameter $\\beta=0.9$, for which the fluid has a monotonic\n  underlying constitutive curve.  Crosses indicate the grid of values\n  of $\\Sigma_{0}$ and $\\omega$ used in the Pipkin diagram of\n  Fig.~\\ref{fig:PipkinMon2}.}\n\\label{fig:pinPointMon2}\n\\end{figure}\n\nThese processes of rapid transition between different branches of the\nconstitutive curve are of course not expected in a LAOStress\nexperiment at low frequency for a fluid with a monotonic constitutive\ncurve. In this case, for a LAOStress run in the limit $\\omega\\to 0$\nthe system instead sweeps quasi-statically along the monotonic\nconstitutive curve, with no associated banding.  As in the case of\nLAOStrain, however, it is crucial to realise that the absence of\nbanding in an experiment at zero frequency does not preclude the\npossibility of banding in a time-dependent protocol at finite\nfrequency, even in a fluid with a monotonic constitutive curve.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure17.eps}\n\\caption{Lissajous-Bowditch curves in LAOStress for the nRP model with\n  a non-monotonic constitutive curve. Results are shown as shear-rate\n  versus time in (a), and in the viscous representation of stress\n  versus strain rate in (b). Columns of fixed frequency $\\omega$ and\n  rows of fixed strain-rate amplitude $\\dot{\\gamma}_0$ are labeled at the top\n  and right-hand side. Colourscale shows the time-dependent degree of\n  shear banding.  Model parameters: $\\beta=0.4, \\eta=10^{-4}, l=0.02$.\n  Cell curvature: $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. A detailed portrait of the run\n  outlined by the thicker box is shown in\n  Fig.~\\ref{fig:portraitNonMon2}.}\n\\label{fig:PipkinNonMon2}\n\\end{figure}\n\n\n\\begin{figure}[tp!]\n\\includegraphics[width=8.5cm]{figure18.eps}\n\\caption{As in Fig.~\\ref{fig:PipkinNonMon2} but for a value of the CCR\n  parameter $\\beta=0.9$, for which the fluid's underlying constitutive\n  curve is monotonic. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by the thicker box is shown in Fig.~\\ref{fig:portraitMon2}.}\n\\label{fig:PipkinMon2}\n\\end{figure}\n\nWith that in mind, we show in Fig.~\\ref{fig:eigenMon} the counterpart\nof Fig.~\\ref{fig:eigenNonMon}, but now for the nRP model with a\nmonotonic constitutive curve subject to a LAOStress run at a finite\nfrequency $\\omega=1$, of order the fluid's reciprocal stress\nrelaxation timescale.  The key to understanding the emergent dynamics\nin this case is the existence in the underlying zero-frequency\nconstitutive curve (shown by a dotted line in the left panel) of a\nregion in which the stress is a relatively flat (though still\nincreasing) function of the strain rate. As the system transits this\nregion during the increasing stress part of a finite-frequency stress\ncycle, we again observe a regime of quite sudden progression from low\nto high strain rate. This is seen in the left to right transition in\nthe stress versus strain rate representation in the left panel of\nFig.~\\ref{fig:eigenMon}, and (correspondingly) in the rapid increase\nof strain rate versus time in the right panel.\n\n\n\nDuring this regime of rapid transit we again have conditions in which\nthe strain rate evolves rapidly compared to the stress, such that the\nconstant-stress criterion (\\ref{eqn:critCreep}) should apply to good\napproximation. Furthermore, during the first part of the transition,\nthe strain rate signal simultaneously slopes and curves upward as a\nfunction of time.  The eigenvalue is therefore positive, indicating\nlinear instability of an initially homogeneous base state to the\nformation of shear bands. We will again confirm this prediction by\nsimulating the model's full nonlinear spatiotemporal dynamics below.\n\nIn the context of Figs.~\\ref{fig:eigenNonMon} and~\\ref{fig:eigenMon}\nwe have discussed the dynamics of the nRP model with a non-monotonic\nand monotonic constitutive curve respectively, focusing in each case\non one particular value of the imposed frequency $\\omega$ and stress\namplitude $\\Sigma_0$. We now consider the full plane of\n$(\\Sigma_0,\\omega)$ by showing in Figs.~\\ref{fig:pinPointNonMon2}\nand~\\ref{fig:pinPointMon2} colour maps of the extent of banding across\nit.  Recall that each point in this plane corresponds to a single LAOS\nexperiment with stress amplitude $\\Sigma_0$ and frequency $\\omega$.\nTo build up these maps we sweep over a grid of 20x20 values of\n$\\Sigma_0,\\omega$ and execute at each point a LAOStress run,\nintegrating the model's linearised equations set out in\nSec.~\\ref{sec:lsa}. We then represent the degree of banding\n$\\delta\\dot{\\gamma}$, maximised over the cycle after many cycles, by the\ncolourscale.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure19.eps}\n\\caption{ LAOStress in the nRP model with a non-monotonic constitutive\n  curve. Stress amplitude $\\Sigma_{0}=0.8$ and frequency $\\omega=1.0$.\n  Model parameters $\\beta=0.4, \\eta=10^{-4}, l=0.02$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} strain rate response as a function of time, focusing on the\n  region in which the system transits from the high to low viscosity\n  branch of the constitutive curve. Solid black and red-dashed line:\n  calculation in which the flow is constrained to be homogeneous.\n  Red-dashed region indicates a positive eigenvalue showing\n  instability to the onset of shear banding. Green dot-dashed line:\n  stress response in a full nonlinear simulation that allows banding.\n  {\\bf Right:} Velocity profiles corresponding to stages in the cycle\n  indicated by matching symbols in the left panel.}\n\\label{fig:portraitNonMon2}\n\\end{figure}\n\n\n\nFig.~\\ref{fig:pinPointNonMon2} shows results with model parameters for\nwhich the underlying constitutive curve is non-monotonic. As expected,\nfor stress amplitudes $\\Sigma_0$ exceeding the local maximum in the\nunderlying constitutive curve, significant banding is observed even in\nthe limit of low frequency $\\omega\\to 0$. This is associated with the\nprocesses of jumping between the two different branches of the\nconstitutive curve discussed above.  \n\nFig.~\\ref{fig:pinPointMon2} shows results for the nRP model with a\nmonotonic underlying constitutive curve. Here steady state banding is\nabsent in the limit $\\omega\\to 0$, as expected.  However, significant\nbanding is still nonetheless observed at frequencies of order the\nreciprocal reptation time, for imposed stress amplitudes exceeding the\nregion of weak slope in the constitutive curve, consistent with our\ndiscussion of Fig.~\\ref{fig:eigenMon} above.\n\nTo obtain the comprehensive roadmaps of\nFigs.~\\ref{fig:pinPointNonMon2} and~\\ref{fig:pinPointMon2} in a\ncomputationally efficient way, we discarded any nonlinear effects and\nintegrated the linearised model equations set out in\nSec.~\\ref{sec:lsa}. However, these linearised equations tend to\noverestimate the degree of banding in any regime of sustained positive\neigenvalue.  Therefore in Figs.~\\ref{fig:PipkinNonMon2}\nand~\\ref{fig:PipkinMon2} we now simulate the model's full nonlinear\nspatiotemporal dynamics, restricting ourselves for computational\nefficiency to grids of 4x4 values of $\\Sigma_0,\\omega$ as marked by\ncrosses in Figs.~\\ref{fig:pinPointNonMon2} and~\\ref{fig:pinPointMon2}.\n\n\nFig.~\\ref{fig:PipkinNonMon2} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is\nnon-monotonic.  At low frequencies the results tend towards the\nlimiting behaviour discussed above, in which the stress slowly tracks\nup and down the steady state flow curve $\\Sigma(\\dot{\\gamma})$ in between\nregimes of sudden transition between the two branches of the curve,\nduring which shear bands form. This is most pronounced in the case of\nthe jump between the high and low viscosity branches during the upward\nsweep.  Banding on the downward sweep is only apparent in a relatively\nmore limited region of $\\Sigma_0,\\dot{\\gamma}$ space, consistent with the\ntransition of $\\dot{\\gamma}_0$ being more modest in this part of the cycle\nduring which the stress decreases.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure20.eps}\n\\caption{As in Fig.~\\ref{fig:portraitNonMon2} but for the nRP model\n  with a CCR parameter $\\beta=0.9$ for which the underlying\n  homogeneous constitutive curve is monotonic. Number of numerical\n  grid points $J=512$.}\n\\label{fig:portraitMon2}\n\\end{figure}\n\nFor the particular run highlighted by the thicker box in\nFig.~\\ref{fig:PipkinNonMon2}, a detailed portrait of the system's\ndynamics is shown in Fig.~\\ref{fig:portraitNonMon2}. The left panel\nshows the strain rate signal as a function of time, zoomed on the\nregion in which the strain rate makes its transit from the high to low\nviscosity branch of the constitutive curve. The black and red-dashed\nline show the results of a calculation in which the flow is\nartificially constrained to remain homogeneous. The red-dashed region\nindicates the regime in which the criterion (\\ref{eqn:critCreep}) for\nlinear instability to the formation of shear bands is met, which also\ncorresponds to the regime in which the strain rate signal\nsimultaneously slopes up and curves upwards.\n\n\n\nIn a simulation that properly takes account of flow heterogeneity,\nshear bands indeed develop during this regime where the criterion is\nmet, then decay again once the strain rate signal curves down and\nstability is restored. This sequence can be seen in the velocity\nprofiles in the right hand panel. The stress signal associated with\nthis run that allows bands to form is shown by the green dot-dashed\nline in the left panel, and is only barely distinguishable from that\nof the run in which the flow is constrained to remain homogeneous.\n\nFig.~\\ref{fig:PipkinMon2} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is monotonic,\nwith the grid of $(\\Sigma_0,\\omega)$ values that it explores marked by\ncrosses in Fig.~\\ref{fig:pinPointMon2}. True top-jumping events are\nabsent here, and no shear banding arises in the limit of zero\nfrequency. As discussed above, however, a similar rapid transition\nfrom low to high shear rate is seen in runs at a frequency $O(1)$, as\nthe stress transits the region of weak slope in the constitutive curve\nduring the increasing-stress part of the cycle.  Associated with this\ntransit is a pronounced tendency to form shear bands.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure21.eps}\n\\caption{Effect of CCR parameter $\\beta$ and entanglement number $Z$\n  (and so of chain stretch relaxation time $\\tau_R=\\tau_d\/3Z$) on shear\n  banding in LAOStress.  (Recall that the non-stretching version of\n  the model has $\\tau_R\\to 0$ and so $Z\\to\\infty$.)  Empty circles: no\n  observable banding.  Hatched circles: observable banding,\n  $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) = 10\\%-31.6\\%$. Dot-filled circles:\n  significant banding, $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) = 31.6\\% - 100\\%$.\n  Filled circles: strong banding, $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) > 100\\%$.\n  For the hatched, dot-filled and filled symbols we used the criterion\n  that banding of the typical magnitude stated is apparent for any of\n  $\\omega=0.1,0.316$ or $1.0$, given a stress amplitude $\\Sigma_0$\n  exceeding the region of weak slope in the constitutive curve. The\n  square shows the parameter values explored in detail in\n  Fig.~\\ref{fig:stretchPortrait1}. The solvent viscosity $\\eta$ is $3.16\\times10^{-5}$.}\n\\label{fig:stretchMaster2}\n\\end{figure}\n\n\nThis can be seen for the run highlighted by the thicker box in\nFig.~\\ref{fig:PipkinMon2}, of which a detailed portrait is shown in\nFig.~\\ref{fig:portraitMon2}. This shows very similar features to its\ncounterpart for a non-monotonic underlying constitutive curve. In\nparticular, the regime of simultaneous upward slope and upward\ncurvature in the strain rate signal as the stress transits the region\nof weak positive constitutive slope triggers pronounced shear banding.\n\nThese results illustrate again the crucial point: that shear bands can\nform in a protocol with sufficiently strong time-dependence, even in a\nfluid for which the underlying constitutive curve is monotonic such\nthat banding is forbidden in steady state flows.\n\nSo far, we have restricted our discussion of LAOStress to the nRP\nmodel, for which the stretch relaxation time $\\tau_R$ is set to zero\nupfront so that any chain stretch relaxes to zero instantaneously,\nhowever strong the applied flow. The results of these calculations are\nexpected to apply, to good approximation, to experiments performed in\nflow regimes where chain stretch remains small. This typically imposes\nthe restriction $\\dot{\\gamma}\\tau_R\\ll 1$. We now turn to the sRP model to\nconsider the effects of finite chain stretch in experiments where this\nrestriction is not met.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure22.eps}\n\\caption{sRP model with a monotonic constitutive curve in LAOStress of\n  stress amplitude $\\Sigma_{0}=0.8$ and frequency $\\omega=0.1$.\n  Model parameters $\\beta=0.7, Z=100, \\eta=3.16\\times10^{-5}$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} strain rate signal versus time. Solid black\n  and red-dashed line: calculation in which the flow is constrained to\n  be homogeneous.  Red-dashed region indicates a positive eigenvalue\n  showing instability to the onset of shear banding. Green dot-dashed\n  line: stress response in a full nonlinear simulation that allows\n  banding (indistinguishable from homogeneous signal in this\n  case.) {\\bf Right:} Velocity profiles corresponding to stages in the\n  cycle indicated by matching symbols in left panel.}\n\\label{fig:stretchPortrait1}\n\\end{figure}\n\nFig.~\\ref{fig:stretchMaster2} shows the regions of the plane of the CCR\nparameter $\\beta$ and entanglement number $Z$ in which banding can be\nexpected even with chain stretch.  As for the case of LAOStrain above\nwe note that, depending on the value of $\\beta$, significant banding\nis still observed for experimentally commonly used entanglement\nnumbers, typically in the range $20-100$. Furthermore, observable\nbanding is clearly evident over a large region of this plane in which\nthe underlying constitutive curve is monotonic, precluding steady\nstate banding. Again, we hope that this figure might act as a roadmap\nto inform the discussion concerning the value of the CCR parameter\n$\\beta$.\n\n\nFor the pairing of $\\beta$ and $Z$ values marked by the square in\nFig.~\\ref{fig:stretchMaster2}, we show in\nFig.~\\ref{fig:stretchPortrait1} a detailed portrait of the model's nonlinear dynamics at a stress amplitude $\\Sigma_0$ and frequency $\\omega$ for which observable banding occurs.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have studied theoretically the formation of shear bands in large\namplitude oscillatory shear (LAOS) in the Rolie-poly model of polymers and wormlike micellar surfactants, with the particular aims of\nidentifying the regimes of parameter space in which shear banding is\nsignificant, and the mechanisms that trigger its onset.\n\n\nAt low frequencies, the protocol of LAOStrain effectively corresponds\nto a repeating series of quasi-static sweeps up and down the steady\nstate flow curve.  Here, as expected, we see shear banding in those\nregimes of parameter space for which the fluid's underlying\nconstitutive curve is non-monotonic, for strain rate amplitudes large\nenough to enter the banding regime in which the stress is a\ncharacteristically flat function of strain rate.\n\nIn LAOStrain at higher frequencies we report banding not only in the\ncase of a non-monotonic constitutive curve, but also over a large\nregion of parameter space for which the constitutive curve is\nmonotonic and so precludes steady state banding. We emphasise that\nthis is an intrinsically time-dependent banding phenomenon that would\nbe absent under steady state conditions, and we interpret it as the\ncounterpart of the `elastic' banding predicted recently in the context\nof shear startup experiments at high strain rates~\\cite{Moorcroft2013}.\n\nIn LAOStress we report shear banding in those regimes of parameter\nspace for which the underlying constitutive curve is either negatively\nor weakly positively sloping. In this case, the bands form during the\nprocess of yielding associated with the dramatic increase in shear\nrate that arises during that part of the cycle in which the stress magnitude\ntransits the regime of weak constitutive slope in an upward direction.\nAlthough the banding that we observe here is dramatically apparent\nduring those yielding events, these events are nonetheless confined to\na relatively small part of the stress cycle as a whole and would\ntherefore need careful focus to be resolved experimentally. (A\npossible related protocol, more focused on the banding regime,\ncould be to perform a shifted stress oscillation\n$\\Sigma(t)=\\Sigma_{\\rm plat} + \\Delta\\Sigma\\sin(\\omega t)$ where\n$\\Sigma_{\\rm plat}$ is a characteristic stress value in the region of\nweak slope in the constitutive curve and $\\Delta\\Sigma$ smaller in\ncomparison.) \n\nThe dramatic increase in strain rate associated with transiting to the\nhigh shear branch in LAOStress is likely to present practical\nexperimental difficulties in open flow cells such as Couette or\ncone-and-plate. To circumvent this, flow in a closed microfluidic channel provides an attractive alternative to those macroscopic\ngeometries in seeking to access this effect experimentally. \n\nIn each case we have demonstrated that the onset of shear banding can,\nfor the most part, be understood on the basis of previously derived\ncriteria for banding in simpler time-dependent\nprotocols~\\cite{Moorcroft2013}. In particular, the trigger for\nbanding in LAOStrain at low frequencies is that of a negatively\nsloping stress versus strain rate, which has long been recognised as\nthe criterion for banding under conditions of a steady applied shear\nflow. The trigger in LAOStrain at high frequencies is instead that of\nan overshoot in the signal of stress as a function of strain, in close\nanalogy to the criterion for banding onset during a fast shear startup\nrun. The trigger for banding in LAOStress is that of a regime of\nsimultaneous upward slope and upward curvature in the\ntime-differentiated creep response curve of strain rate as a function\nof time. This again is a close counterpart to the criterion for\nbanding following the imposition of a step stress.\n\nFor both LAOStrain and LAOStress we have provided a map of shear\nbanding intensity in the space of entanglement number $Z$ and CCR\nparameter $\\beta$. We hope that this will provide a helpful roadmap\nexperimentalists, and might even help to pin down the value of the CCR\nparameter, for which no consensus currently exists. \n\nWe have also commented that the value of the Newtonian viscosity\n$\\eta$ is typically much smaller than the zero shear viscosity $G\\tau$\nof the viscoelastic contribution, giving $\\eta\\ll 1$ in our units.\n\nExperimental literature suggests a range $\\eta=10^{-7}$ to $10^{-3}$.\nWe have typically used $\\eta=10^{-5}$ or\n$\\eta=10^{-4}$ in our numerics, and noted that the degree of banding\ntends to increase with decreasing $\\eta$. We also noted that the\ntimescale to transit from the high to low viscosity branch during\nyielding in each half cycle in LAOStress decreases with decreasing\n$\\eta\/G$. In view of these facts, a study of time-dependent banding in\nfluids with smaller values of $\\eta$ than those used here might\nwarrant the inclusion of inertia, because the small timescale for the\npropagation of momentum might exceed the short timescale $\\eta\/G$ in\nthose cases\n\nIn all our numerical studies the initial seed triggering the formation\nof shear bands was taken to be the weak curvature that is present in\ncommonly used experimental flow cells. In order to demonstrate the\nprinciple that the banding we report requires only a minimal seed,\nrather than being an artefact of strong cell curvature, all our runs\nhave assumed a curvature that is much smaller than that of most flow\ncells in practice. We also neglected stochastic noise altogether in\nall the results presented here. (We have nonetheless also performed\nruns with small stochastic noise instead of cell curvature and find\nqualitatively all the same effects.)\n\nHowever, one obvious shortcoming to this approach of taking only a very\nsmall initial seed is that it tends to suppress the nucleation events\nthat are, in a real experimental situation, likely to trigger banding\neven before the regime of true linear\ninstability~\\cite{Grand1997}, particularly in low frequency\nruns.  It would therefore be interesting in future work to study the\neffect of a finite temperature with particular regard to the\nnucleation kinetics to which it would give rise.\n\nThe calculations performed in this work all assumed from the outset\nthat spatial structure develops only in the flow gradient direction,\nimposing upfront translational invariance in the flow and vorticity\ndirections. We defer to future work a study of whether, besides the\nbasic shear banding instabilities predicted here, secondary\ninstabilities~\\cite{Fardin2014} of the interface between the\nbands~\\cite{Nghe2010,Fielding2005}\nor of the high shear band itself~\\cite{Fielding2010} will have\ntime to form in any given regime of amplitude and frequency space.\n\nWe have ignored throughout the effects of spatial variations in\n  the concentration field. However, it is well known that in a\n  viscoelastic solution heterogeneities in the flow field, and in\n  particular in the normal stresses, can couple to the dynamics of\n  concentration fluctuations via a positive feedback mechanism that\n  enhances the tendency to form shear\n  bands~\\cite{Milner1993,Schmitt1995,Fielding2003a,Fielding2003b,Fielding2003c}.\n  In the calculations performed here in LAOS we have observed\n  significant differences in the viscoelastic normal stresses between\n  the bands (approaching $50-70\\%$ of the cycle-averged value of the\n  same quantity, at least in the calculations without chain stretch).\n  It would therefore clearly be interesting in future work to consider\n  the effects of concentration coupling on the phenomena reported\n  here.\n\nThroughout we have ignored the possibility of edge fracture,\n  because the one-dimensional calculations performed here lack any\n  free surfaces and are unable to address it. It would clearly be\n  interesting in future work to address the effects of edge fracture\n  with regards the phenomena considered here~\\cite{Skorski2011,\n    Li2013, Li2015}.\n\nAll the calculations performed here have adopted what is\n  essentially a single-mode approach, taking account of just one\n  reptation relaxation timescale $\\tau_d$ and one stretch relaxation\n  timescale $\\tau_R$. It would be interesting in future work to\n  consider the effect of multiple relaxation timescales, which is\n  likely to be an important feature of the dynamics of unbreakable\n  polymers. (In wormlike micelles, in contrast, chain breakage and\n  recombination narrows the relaxation spectrum significantly such\n  that the single-mode approach adopted here is already likely to\n  provide a reasonably full picture.)\n\nWe hope that this work will stimulate further experimental studies of\nshear banding in time-dependent flows of complex fluids, with a\nparticular focus on the concept that banding is likely to arise rather\ngenerically during yielding-like events (following a stress overshoot\nin strain controlled protocols, or during a sudden increase in strain\nrate in stress controlled protocols) even in fluids with a monotonic\nconstitutive curve that precludes steady state banding in a\ncontinuously applied shear. In polymers this could form part of the\nlively ongoing debate concerning the presence or otherwise of shear\nbanding in those materials. In wormlike micelles it would be\ninteresting to see a study of LAOS across the full phase diagram (as\nset out, for example, in Ref.~\\cite{Berret1997}), from\nsamples that band in steady state to those above the dynamical\ncritical point, which don't.\n\n{\\it Acknowledgements} The authors thank Alexei Likhtman, Elliot\nMarsden, Peter Olmsted, Rangarajan Radhakrishnan, Daniel Read and\nDimitris Vlassopoulos for interesting discussions. The research\nleading to these results has received funding from the European\nResearch Council under the European Union's Seventh Framework\nProgramme (FP\/2007-2013), ERC grant agreement number 279365.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMyocardial Contrast Echocardiography (MCE) is a cardiac ultrasound imaging technique that utilizes vessel-bound microbubbles as contrast agents. In contrast to conventional B-mode echocardiography which only captures the structure and motion of the heart, MCE also allows for the assessment of myocardial perfusion through the controlled destruction and replenishment of microbubbles \\cite{wei_quantification_1998}. The additional perfusion information gives it great potential for the detection of coronary artery diseases. However, current perfusion analysis of MCE data mainly relies on human visual assessment which is time consuming and not reproducible \\cite{DBLP:conf\/fimh\/MaSRBL09}. There is generally a lack of automatic computerized algorithms and methods to help clinician perform accurate perfusion quantification \\cite{DBLP:conf\/fimh\/MaSRBL09}. One major challenge is the automatic segmentation of the myocardium before subsequent perfusion analysis can be carried out.\n\nIn this paper, we extend the Random Forests (RF) framework \\cite{DBLP:journals\/ml\/Breiman01} to segment the myocardium in our MCE data. RF is a machine learning technique that has gained increasing use in the medical imaging field for tasks such as segmentation \\cite{lempitsky_random_2009} and organ localization \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}. RF has been successful due to its accuracy and computational efficiency. Promising results of myocardial delineation on 3D B-mode echo has also been demonstrated in \\cite{lempitsky_random_2009}. However, classic RF has two limitations. First, our MCE data exhibit large sources of intensity variations \\cite{tang_quantitative_2011} due to factors such as speckle noise, low signal-to-noise ratio, attenuation artefacts, unclear and missing myocardial borders, presence of structures (papillary muscle) with similar appearance to the myocardium. These intensity variations reduce the discriminative power of the classic RF that utilizes only local intensity features. Second, RF segmentation operates on a pixel basis where the RF classifier predicts a class label for each pixel independently. Structural relationships and contextual dependencies between pixel labels are ignored \\cite{DBLP:conf\/iccv\/KontschiederBBP11,DBLP:conf\/ipmi\/MontilloSWIMC11} which results in segmentation with inconsistent pixel labelling leading to unsmooth boundaries, false detections in the background and holes in the region of interest. To overcome the above two problems, we need to incorporate prior knowledge of the shape of the structure and use additional contextual and structural information to guide the RF segmentation.\n\\\\\n\\indent\nThere are several works which have incorporated local contextual information into the RF framework. Lempitsky et al. \\cite{lempitsky_random_2009} use the pixel coordinates as position features for the RF so that the RF learns the myocardial shape implicitly. Tu et Bai \\cite{DBLP:journals\/pami\/TuB10} introduce the concept of auto-context which can be applied to RF by using the probability map predicted by one RF as features for training a new RF. Montillo et al. \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} extend the auto-context RF by introducing entanglement features that use intermediate probabilities derived from higher levels of a tree to train its deeper levels. Kontschieder et al. \\cite{DBLP:conf\/iccv\/KontschiederBBP11} introduce the structured RF that builds in structural information by using RF that predicts structured class labels for a patch rather than the class label of an individual pixel. Lombaert et al. \\cite{DBLP:conf\/miccai\/LombaertCA15} use spectral representations of shapes to classify surface data.\n\\\\\n\\indent\nThe above works use local contextual information that describes the shape of a structure implicitly. The imposed structural constraint are not strong enough to guide the RF segmentation in noisy regions of MCE data. In this paper, we proposed the Shape Model guided Random Forests (SMRF) which provides a new way to incorporate global contextual information into the RF framework by using a statistical shape model that captures the explicit shape of the entire myocardium. This imposes stronger, more meaningful structural constraints that guide the RF segmentation more accurately. The shape model is learned from a set of training shapes using Principal Component Analysis (PCA) and is originally employed in Active Shape Model (ASM) where the model is constrained so that it can only deform to fit the data in ways similar to the training shapes \\cite{DBLP:journals\/cviu\/CootesTCG95}. However, ASM requires a manual initialization and the final result is sensitive to the position of this initialization. Our SMRF is fully automatic and enjoys both the local discriminative power of the RF and the prior knowledge of global structural information contained in the statistical shape model. The SMRF uses the shape model to guide the RF segmentation in two ways. First, it directly incorporates the shape model into the RF framework by introducing a novel Shape Model (SM) feature which has outperformed the other contextual features and produced a more accurate RF probability map. Second, the shape model is fitted to the probability map to generate a smooth and plausible myocardial boundary that can be used directly for subsequent perfusion analysis.\n\n\\section{Method}\nIn this section, we first review some basic background on statistical shape model and RF. We then introduce the two key aspects of our SMRF---the novel SM feature and the fitting of the shape model.\n\\subsubsection{\\textit{Statistical Shape Model:}}\nA statistical shape model of the myocardium is built from 89 manual annotations using PCA \\cite{DBLP:journals\/cviu\/CootesTCG95}. Each annotation has $N=76$ landmarks comprising 4 key landmarks with 18 landmarks spaced equally in between along the boundary of manual tracing (Fig. \\ref{fig:Model} left). The shape model is represented as:\n\\begin{equation}\\label{eq:Model}\n  \\boldsymbol{x}=\\bar { \\boldsymbol{x} } + \\boldsymbol{Pb}\n\\end{equation}\n\\noindent where $\\boldsymbol{x}$ is a 2$N$-D vector containing the 2D coordinates of the $N$ landmark points, $\\bar{\\boldsymbol{x}}$ is the mean coordinates of all training shapes, $\\boldsymbol{b}$ is a set of $K$ shape parameters and $\\boldsymbol{P}$ contains $K$ eigenvectors with their associated eigenvalues ${ \\lambda  }_{ i }$. $K$ is the number of modes and set to 10 to explain 98\\% of total variance so that fine shape variations are modeled while noise is removed. Values of ${b}_{i}$ are bounded between $\\pm s\\sqrt { \\lambda _{ i } } $ so that only plausible shape similar to the training set is generated (Fig. \\ref{fig:Model} right). Refer to \\cite{DBLP:journals\/cviu\/CootesTCG95} for details on statistical shape model.\n\\subsubsection{\\textit{Random Forests:}}\nMyocardial segmentation can be formulated as a problem of binary classification of image pixels. An RF classifier \\cite{DBLP:journals\/ml\/Breiman01} is developed that predicts the class label (myocardium or background) of a pixel using a set of features. The RF is an ensemble of decision trees. During training, each branch node of a tree learns a pair of feature and threshold that results in the best split of the training pixels into its child nodes. The splitting continues recursively until the maximum tree depth is reached or the number of training pixels in the node falls below a minimum. At this time, a leaf node is created and the class distribution of the training pixels reaching the leaf node is used to predict the class label of unseen test pixels. The average of the predictions from all the trees gives a segmentation probability map. Refer to \\cite{DBLP:journals\/ml\/Breiman01}, \\cite{lempitsky_random_2009} for details on RF.\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.5\\textwidth}\n  \\centering\n  \\includegraphics[height=0.16\\textheight]{Fig1}\n  \\caption{}\n  \\label{fig:Model}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.5\\textwidth}\n  \\centering\n  \\includegraphics[height=0.16\\textheight]{Fig2}\n  \\caption{}\n  \\label{fig:Feature}\n\\end{subfigure}\n\\caption{(a) Left: A manual annotation from training set showing key landmarks (\\textit{red}) and other landmarks in between (\\textit{green}). Right: First two modes of variations of the shape model. (b) Left: Landmarks $\\boldsymbol{x}$ (\\textit{blue dots}) generated randomly by the shape model in (\\ref{eq:Model}). Right: $d_1$($d_2$) is the SM feature value measuring the signed shortest distance from pixel $\\boldsymbol{p_1}$($\\boldsymbol{p_2}$) to the myocardial boundary $B(\\boldsymbol{x})$ (\\textit{blue contour}). $d_1$ is positive and $d_2$ is negative.}\n\\label{fig:Result}\n\\end{figure}\n\\subsubsection{\\textit{Shape Model Feature:}}\nThe classic RF uses local appearance features which are based on surrounding image intensities of the reference pixel \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}. We introduced an additional novel SM feature that is derived from the shape model. The SM feature randomly selects some values for the shape model parameters $\\boldsymbol{b}$ and generates a set of landmarks $\\boldsymbol{x}$ using (\\ref{eq:Model}) (Fig. \\ref{fig:Feature} left). The landmarks can be joined to form a myocardial boundary. Let $B(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb})$ be the myocardial boundary formed by joining the landmarks generated using some values of $\\boldsymbol{b}$. The SM feature value is then given by the signed shortest distance $d$ from the reference pixel $\\boldsymbol{p}$ to the boundary $B$ (Fig. \\ref{fig:Feature} right). The distance is positive if $\\boldsymbol{p}$ lies inside the boundary and negative if it lies outside. The SM feature is essentially the signed distance transform of a myocardial boundary generated by the shape model. Each SM feature is defined by the shape parameters $\\boldsymbol{b}$. During training, an SM feature is created by random uniform sampling of each $b_{i}$ in the range of $\\pm s_{feature}\\sqrt { \\lambda _{ i } }$ where $s_{feature}$ is set to 1 in all our experiments. The binary SM feature test, parameterized by $\\boldsymbol{b}$ and a threshold $\\tau$, is written as:\n\\begin{equation}\n{ t }_{ \\textrm{SM} }^{ \\boldsymbol{b},\\tau  }(\\boldsymbol{p})=\\begin{cases} 1,\\qquad \\textrm{if}\\quad D(\\boldsymbol{p},B(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb}))>\\tau  \\\\ 0,\\qquad \\textrm{otherwise}. \\end{cases}\n\\end{equation}\n\\noindent where $D(.)$ is the function that computes $d$. Depending on the binary test outcome, pixel $\\boldsymbol{p}$ will go to the left (1) or right (0) child node of the current split node. During training, the RF learns the values of $\\boldsymbol{b}$ and $\\tau$ that best split the training pixels at a node. The SM features explicitly impose a global shape constraint in the RF framework. The random sampling of $\\boldsymbol{b}$ also allows the RF to learn plausible shape variations of the myocardium.\n\\subsubsection{\\textit{Shape Model Fitting:}}\nThe RF output is a probability map which cannot be used directly in subsequent analysis and application. Simple post-processing on the probability map such as thresholding and edge detection can produce segmentations with inaccurate and incoherent boundaries due to the nature of the pixel-based RF classifier. Our SMRF fits the shape model to the RF probability map to extract a final myocardial boundary that is smooth and which preserves the integrity of the myocardial shape. The segmentation accuracy is also improved as the shape constraint imposed by the shape model can correct some of the misclassifications made by the RF.\n\nLet ${ T }_{ \\boldsymbol{\\theta}}$ be a pose transformation defined by the pose parameter $\\boldsymbol{\\theta}$ which includes translation, rotation and scaling. The shape model fitting is then an optimization problem where we want to find the optimal values of $(\\boldsymbol{b},\\boldsymbol{\\theta})$ such that the model best matches the RF probability map under some shape constraints. We minimize the following objective function:\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\boldsymbol{b},\\boldsymbol{\\theta}}{\\text{min}}\n& & { { \\left\\| { \\boldsymbol{I} }_{ \\textrm{RF} }-{ \\boldsymbol{I} }_{ \\textrm{M} }({ T }_{ \\boldsymbol{\\theta}  }(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb})) \\right\\|  }^{ 2 }+\\alpha \\frac { 1 }{ K } \\sum _{ i=1 }^{ K }{ \\frac { \\left| { b }_{ i } \\right|  }{ \\sqrt { { \\lambda  }_{ i } }  }  }  } \\\\\n& \\text{subject to}\n& & -{ s }_{ fit }\\sqrt { { \\lambda  }_{ i } } <{ b }_{ i }<{ s }_{ fit }\\sqrt { { \\lambda  }_{ i } }, \\; i = 1, \\ldots, K.\n\\end{aligned}\n\\end{equation}\nThe first term of the objective function compares how well the match is between the model and the RF probability map $\\boldsymbol{I}_{\\textrm{RF}}$. $\\boldsymbol{I}_{\\textrm{M}}(.)$ is a function that converts the landmarks generated by the shape model into a binary mask of the myocardial shape. This allows us to evaluate a dissimilarity measure between the RF segmentation and the model by computing the sum of squared difference between the RF probability map and the model binary mask. The second term of the objective function is a regularizer which imposes a shape constraint. It is related to the probability of a given shape \\cite{DBLP:journals\/pr\/CristinacceC08} and ensures that it does not deviate too much away from its mean shape. $\\alpha$ is the weighting given to the regularization term and its value is determined empirically. Finally, an additional shape constraint is imposed on the objective function by limiting the upper and lower bounds of $b_i$ to allow for only plausible shapes. $s_{fit}$ is set to 2 in all our experiments. The optimization is carried out using direct search which is a derivative-free solver from the MATLAB global optimization toolbox. At the start of the optimization, each $b_i$ is initialized to zero. Pose parameters are initialized such that the model shape is positioned in the image center with no rotation and scaling.\n\\section{Experiments}\n\\subsubsection{\\textit{Datasets:}}\n2D+t MCE sequences were acquired from 15 individuals using a Philips iE33 ultrasound machine and SonoVue as the contrast agent. Each sequence is taken in the apical 4-chamber view under the triggered mode which shows the left ventricle at end-systole. One 2D image was chosen from each sequence and the myocardium manually segmented by two experts to give inter-observer variability. This forms a dataset of 15 2D MCE images for evaluation. Since the appearance features of the RF are not intensity invariant, all the images are pre-processed with histogram equalization to reduce intensity variations between different images. The image size is approximately 351$\\times$303 pixels.\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.55\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{Fig3}\n  \\caption{}\n  \\label{fig:ResultVisual}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{Fig4}\n  \\caption{}\n  \\label{fig:ResultQuant}\n\\end{subfigure}\n\\caption{(a) Visual segmentation results with one MCE example on each row. First three columns: Probability maps from classic RF, position feature RF and SMRF respectively. Last column: Ground truth boundary (\\textit{red}) and the SMRF boundary (\\textit{blue}) obtained from fitting the shape model to the SMRF probability map in the third column. \\textit{Black arrows} indicate papillary muscle. (b) Segmentation accuracy of different RF classifiers at different tree depths.}\n\\label{fig:Result}\n\\end{figure}\n\\subsubsection{\\textit{Validation Methodology:}}\nWe compared our SMRF segmentation results to the classic RF that uses appearance features \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}, as well as RFs that use other contextual features such as entanglement \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} and position features \\cite{lempitsky_random_2009}. We also compared our results to repeated manual segmentations and the Active Shape Model (ASM) method \\cite{DBLP:journals\/tmi\/GinnekenFSRV02}. Segmentation accuracy is assessed quantitatively using pixel classification accuracy, Dice and Jaccard indices, Mean Absolute Distance (MAD) and Hausdorff Distance (HD). To compute the distance error metrics (MAD and HD), a myocardial boundary is extracted from the RF probability map using the Canny edge detector. This is not required for the SMRF in which the shape model fitting step directly outputs a myocardial boundary.\n\\\\\n\\indent\nWe performed leave-one-out cross-validation on our dataset of 15 images. The RF parameters are determined experimentally and then fixed for all experiments. 20 trees are trained with maximum tree depth of 24. 10\\% of the pixels from the training images are randomly selected for training. The RF and the shape model fitting were implemented in C\\# and MATLAB respectively. Given an unseen test image, RF segmentation took 1.5min with 20 trees and shape model fitting took 8s on a machine with 4 cores and 32GB RAM. RF training took 38mins.\n\\section{Results}\nFig. \\ref{fig:ResultVisual} qualitatively shows that our SMRF probability map (column 3) has smoother boundary and more coherent shape than the classic RF (column 1) and position feature RF (column 2). Fitting the shape model to the SMRF probability map produces the myocardial boundary (\\textit{blue}) in column 4. The fitting guides the RF segmentation especially in areas where the probability map has a low confidence prediction. In the example on the second row, our SMRF predicts a boundary that correctly excludes the papillary muscle (\\textit{black arrows}). This is often incorrectly included by the other RFs due to its similar appearance to the myocardium.\n\\begin{table}[]\n\\centering\n\\caption{Quantitative comparison of segmentation results between the proposed SMRF and other methods. Results presented as (Mean $\\pm$ Standard Deviation).}\n\\label{table:Result}\n\\begin{tabular}{|M{10.5em}|c|c|c|c|r|}\n\\hline\n & \\multicolumn{1}{c|}{Accuracy} & \\multicolumn{1}{c|}{Dice} & \\multicolumn{1}{c|}{Jaccard} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}MAD\\\\   (mm)\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}HD\\\\   (mm)\\end{tabular}} \\\\ \\hline\nIntra-observer & 0.96$\\pm$0.01 & 0.89$\\pm$0.02 & 0.80$\\pm$0.03 & 1.02$\\pm$0.26 & 3.75$\\pm$0.93 \\\\\nInter-observer & 0.94$\\pm$0.02 & 0.84$\\pm$0.05 & 0.72$\\pm$0.07 & 1.59$\\pm$0.57 & 6.90$\\pm$3.24 \\\\\nASM \\cite{DBLP:journals\/tmi\/GinnekenFSRV02} & 0.92$\\pm$0.03 & 0.77$\\pm$0.08 & 0.64$\\pm$0.11 & 2.23$\\pm$0.81 & 11.44$\\pm$5.23 \\\\ \\hline\nClassic RF & 0.91$\\pm$0.04 & 0.74$\\pm$0.12 & 0.60$\\pm$0.14 & 2.46$\\pm$1.36 & 15.69$\\pm$7.34 \\\\\nEntangled RF \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} & 0.91$\\pm$0.05 & 0.75$\\pm$0.13 & 0.62$\\pm$0.15 & 2.43$\\pm$1.62 & 15.06$\\pm$7.92 \\\\\nPosition Feature RF \\cite{lempitsky_random_2009} & \\textbf{0.93$\\pm$0.03} & \\textbf{0.81$\\pm$0.10} & 0.69$\\pm$0.13 & 1.81$\\pm$0.84 & 9.51$\\pm$3.80 \\\\\nSMRF & \\textbf{0.93$\\pm$0.03} & \\textbf{0.81$\\pm$0.10} & \\textbf{0.70$\\pm$0.12} & \\textbf{1.68$\\pm$0.72} & \\textbf{6.53$\\pm$2.61} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nTable. \\ref{table:Result} compares the quantitative segmentation results of our SMRF to other methods. Both SM features and position features encode useful structural information that produces more accurate RF probability maps than the classic RF and entangled RF. This is reflected by the higher Dice and Jaccard indices. For MAD and HD metrics, SMRF outperforms all other RF methods because the shape model fitting step in SMRF produces more accurate myocardial boundaries than those extracted using the Canny edge detector. In addition, SMRF also outperforms ASM \\cite{DBLP:journals\/tmi\/GinnekenFSRV02} and comes close to the inter-observer variations.\n\nFig. \\ref{fig:ResultQuant} compares the segmentation accuracy of the probability maps of different RF classifiers. Our SMRF obtained higher Jaccard indices than the classic and entangled RFs at all tree depths. At lower tree depths, SMRF shows notable improvement over the position feature RF. The SM features have more discriminative power than the position features as it captures the explicit geometry of the myocardium using the shape model. The SM feature binary test partitions the image space using more complex and meaningful myocardial shapes as opposed to position feature which simply partitions the image space using straight lines. This provides a stronger global shape constraint than the position feature and allows a decision tree to converge faster to the correct segmentation at lower tree depths. This gives the advantage of using trees with smaller depths which speeds up both training and testing.\n\\section{Conclusion}\nWe presented a new method SMRF for myocardial segmentation in MCE images. We showed how our SMRF utilizes a statistical shape model to guide the RF segmentation. This is particular useful for MCE data whose image intensities are affected by many variables and therefore prior knowledge of myocardial shape becomes important in guiding the segmentation. Our SMRF introduces a new SM feature which captures the global myocardial structure. This feature outperforms other contextual features to allow the RF to produce a more accurate probability map. Our SMRF then fits the shape model to the RF probability map to produce a smooth and coherent final myocardial boundary that can be used in subsequent perfusion analysis. In future work, we plan to validate our SMRF on a larger, more challenging dataset which includes different cardiac phases and chamber views.\n\\subsubsection*{Acknowledgments.} The authors would like to thank Prof. Daniel Rueckert, Liang Chen and other members from the BioMedIA group for their help and advice. This work was supported by the Imperial College PhD Scholarship.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\\subsection{Background}\nThe present paper studies phase transitions from an ergodic theory and dynamical system viewpoint. It investigates the relations between renormalization, substitutions and phase transition initiated in \\cite{baraviera-leplaideur-lopes} and continued in \\cite{BL}.\n\nPhase transitions are an important topic in statistical mechanics and also in probability theory (see {\\em e.g.\\ } \\cite{georgii,grimmett}). \nThe viewpoint presented here is different for several reasons. One of them is that, here, the geometry of the lattice is not relevant\\footnote{and we only consider a one-dimensional lattice.}, whereas in statistical mechanics, \nthe geometry of the lattice is the most relevant part. \n\nA {\\em phase transition} is characterized by a lack of analyticity of the \npressure function. \nThis definition of phase transition is inspired by statistical mechanics \nand is now standard for dynamical systems, see \\cite{bowen, ruelle, sinai}. \nGiven a dynamical systems, say $(X,T)$, and a potential $\\varphi:X\\to{\\mathbb R}$, the pressure function is given by \n$$\n\\CP(\\beta):=\\sup\\left\\{h_{\\mu}(T)+\\beta \\int\\varphi\\,d\\mu\\right\\},\n$$\nwhere the supremum is taken over the invariant probability measures $\\mu$, $h_{\\mu}(T)$ is the Kolmogorov entropy and $\\beta$ is a real parameter. \n\nFor a uniformly hyperbolic dynamical system $(X,T)$ and \na H\\\"older continuous potential $\\varphi$, the pressure function $\\beta\\mapsto \\CP(\\beta)$ is analytic (see {\\em e.g.\\ } \\cite{bowen, ruelle, keller}). Even if analyticity is usually considered as a very rigid property and thus quite rare, it turns out that proving non-analyticity for the pressure function is not so easy. \nCurrently, this has become an important challenge in smooth ergodic theory to produce and study phase transitions, see {\\em e.g.\\ } \\cite{Makarov-Smirnov, coronel-Rivera-Letelier, BL, iommi-todd}. \n\nTo observe phase transitions, one has to weaken hyperbolicity of the system\nor of regularity of the potential; it is the latter one that we continue to investigate here. Our dynamical system is the full shift, which is uniformly hyperbolic. \nThe first main question we want to investigate is thus what potentials $\\varphi$ will produce phase transitions. More precisely, we are looking for a machinery to produce potentials with phase transitions. \n\nThe main purpose of \\cite{baraviera-leplaideur-lopes} was to investigate possible relation between {\\em renormalization} and phase transition. In the shift\nspace  $(\\{0,1\\}^{{\\mathbb N}}, \\sigma)$, a renormalization is a function $H$ satisfying an equality of the form \n\\begin{equation}\\label{equ1-renorm}\n\\sigma^{k}\\circ H=H\\circ \\sigma.\n\\end{equation}\n The link with potentials was made in \\cite{baraviera-leplaideur-lopes} by introducing a renormalization operator $\\CR$ acting on potentials and related to a solution $H$ for \\eqref{equ1-renorm}.\n\nIt is easy to check that constant length $k$ substitutions are \nsolutions to the renormalization equation.\nIn \\cite{BL}, we studied  the Thue-Morse case substitution,\nwhich has constant length $2$. \nHere we investigate the Fibonacci substitution, which is not of constant length. Several reasons led us to study the Fibonacci case:\n\n$\\bullet$ Together with the Thue-Morse substitution, the Fibonacci substitutions is the most ``famous'' substitution and it has been well-studied. In particular, the dynamical properties of their respective attracting sets are well-known and this will be\nused extensively in this paper for the Fibonacci shift. As a result, we were \nable to describe the relevant fixed point of renormalization exactly. \nInformation of the left and right-special words in these attractors is \na key ingredient to prove existence of a phase transition; it \nis a crucial issue in the relations between \nsubstitutions and phase transitions.  \n\n$\\bullet$ The type of phase transition we establish is a {\\em freezing phase transition}. This means that beyond the phase transition ({\\em i.e.,\\ } for large $\\beta$), \nthe pressure function is affine and equal to its asymptote, and the equilibrium state ({\\em i.e.,\\ } ground state) is the unique shift-invariant measure\nsupported on an aperiodic subshift space, sometimes called {\\em quasi-crystal}.\nOne open question in statistical mechanics (see \\cite{vanEnter?}) is whether freezing phase transitions can happen and whether {\\em quasi-crystal ground state}\ncan be reach at {\\em positive temperature}. An affirmative answer was given for the Thue-Morse quasi-crystal in \\cite{BL}; we show here this also holds for the Fibonacci quasi-crystal. \n\n$\\bullet$ \nWe think that Fibonacci shift opens the door to study more cases. One natural question is whether any quasi-crystal can be reached as a ground state at positive temperature. In this context we emphasize that the Fibonacci substitution also has a Sturmian shift, that is, it is related to the irrational rotation with angle the golden mean $\\gamma:=\\frac{1+\\sqrt5}2$. \nWe expect that the machinery developed here for the Fibonacci substitution can\nbe extended to the Sturmian shift associated to general irrational rotation\nnumbers \n(although those with bounded entries in the continued fraction expansion\nwill be the easiest), possibly to rotations on higher-dimensional tori, and also to more general substitutions. \n\n\n \n\n\\subsection{Results}\n\nLet $\\S = \\{0,1\\}^{{\\mathbb N}}$ be the full shift space; points in $\\S$ \nare sequences $x:=(x_{n})_{n\\ge 0}$ or equivalently infinite {\\em words} $x_{0}x_{1}\\ldots$. \nThroughout we let $\\overline x_j = 1-x_j$ denote the opposite symbol.\nThe dynamics is the left-shift \n$$\n\\sigma: x=x_{0}x_{1}x_{2}\\ldots\\mapsto x_{1}x_{2}\\ldots.\n$$\nGiven a word $w=w_{0}\\ldots w_{n-1}$ of {\\em length} $|w|=n$, the corresponding\n{\\em cylinder} (or {\\em $n$-cylinder}) is the set of infinite words starting as $w_{0}\\ldots w_{n-1}$. We use the notation\n$C_n(x)=[x_0\\dots x_{n-1}]$\nfor the $n$-cylinder containing $x=x_{0}x_{1}\\ldots$ \nIf $w=w_{0}\\ldots w_{n-1}$ is a word with length $n$ and $w'=w'_{0}\\ldots$ a word of any length, the {\\em concatenation} $ww'$ is the word $w_{0}\\ldots w_{n-1}w'_{0}\\ldots$. \n\nThe Fibonacci substitution on $\\S$ is defined by: \n$$\nH: \\begin{cases}\n0 \\to 01\\\\\n1 \\to 0.\n\\end{cases}\n$$\nand extended for words by the concatenation rule $H(ww')=H(w)H(w')$.\nIt is convenient for us to count the Fibonacci numbers starting with index $-2$:\n\\begin{equation}\\label{eq:Fibo}\nF_{-2} = 1,\\ F_{-1} = 0,\\ F_0=1, \\ F_1 = 1,\\ F_2 = 2,\\ F_{n+2} = F_{n+1} + F_n,\n\\end{equation}\nWe have\n\\begin{equation}\\label{eq:Fiboa}\nF_{n}^{a}:=|H^{n}(a)| = \\begin{cases}\nF_{n+1} & \\text{ if } a = 0,\\\\\nF_{n} & \\text{ if } a = 1.\n\\end{cases}\n\\end{equation}\nThe Fibonacci substitution has a unique fixed point\n$$\n\\rho = 0\\ 1\\ 0\\ 01\\ 010\\ 01001\\ 01001010\\ 0100101001001\\dots\n$$\nWe define the orbit closure ${\\mathbb K} = \\overline{\\cup_n \\sigma^n(\\rho)}$; it forms a subshift\nof $(\\S, \\sigma)$ associated to $\\rho$. More properties on ${\\mathbb K}$ are given in Section~\\ref{sec-H-K-R}.\n\n\\bigskip\nWe define the renormalization operator \nacting on potentials $V:\\S\\to {\\mathbb R}$ by \n$$\n(\\CR V)(x)= \\begin{cases}\n V\\circ \\sigma\\circ H(x)+V\\circ H(x) & \\text{ if }x\\in[0],\\\\\n V\\circ H(x) & \\text{ if }x\\in[1].\n\\end{cases}\n$$\nWe are interested in finding fixed points for $\\CR$ and, where possible, studying their stable leaves,\n{\\em i.e.,\\ } potentials converging to the fixed point under iterations of $\\CR$. \nContrary to the Thue-Morse substitution, \nthe Fibonacci substitution is not of constant length. This is the source of several complications, in particular for the correct expression for $\\CR^{n}$. \n\nFor  $\\alpha>0$, let $\\CX_{\\alpha}$ be the set of functions $V: \\S \\to {\\mathbb R}$ \nsuch that $}%{\\displaystyle V(x) \\sim n^{-\\alpha}$ if $d(x,{\\mathbb K})=2^{-n}$. \nMore precisely, $\\CX_{\\alpha}$ is the set of functions $V$ such that:\n\\begin{enumerate}\n\\item $V$ is continuous and non-negative. \n\\item There exist two continuous functions $g,h:\\S \\to {\\mathbb R}$, satisfying $}%{\\displaystyle h_{|{\\mathbb K}}\\equiv 0$ and $g>0$, such that \n$$\nV(x) = \\frac{g(x)}{n^\\alpha} + \\frac{h(x)}{n^\\alpha} \\quad \\text{ when } \\quad \nd(x,{\\mathbb K}) = 2^{-n}.\n$$\n\\end{enumerate}\nWe call $g$ the \\textit{$\\alpha$-density}, or just the \\textit{density} \nof $V\\in \\CX_{\\alpha}$. Continuity and the assumption $h_{|{\\mathbb K}}\\equiv 0$ imply \nthat $h(x)\/n^{\\alpha} = o(n^{-\\alpha})$.\n\nOur first theorem achieves the existence of a fixed point for $\\CR$ and shows that the germ of $V$ close to ${\\mathbb K}$, {\\em i.e.,\\ } its $\\alpha$-density, allows us to \ndetermine the stable leaf of that fixed point. \n\nGiven a finite word $w$, let\n$\\kappa_a(w)$ denote the number of symbols $a\\in\\{0,1\\}$ in $w$.\nIf $x \\in \\S \\setminus {\\mathbb K}$, we denote by $\\widetilde\\kappa_{a}(x)$ the number of symbols $a$ in the finite word $x_{0}\\ldots x_{n-1}$ where $d(x,{\\mathbb K})=2^{-n}$. \n\n\\begin{theorem} \\label{theo-fixedpoint}\nIf $V \\in \\CX_\\alpha$, with $\\alpha$-density function $g$, then\n$$\n\\lim_{k\\to\\infty} \\CR^{k}V(x) =\n\\begin{cases}\n\\quad \\infty & \\text{ for all  } x\\in \\S\\setminus {\\mathbb K} \\text{ if } \\alpha < 1; \\\\[2mm]\n\\quad 0 & \\text{ for all  } x\\in \\S \\text{ if } \\alpha > 1; \\\\[2mm]\n\\quad }%{\\displaystyle\\int g \\ d\\mu_{\\mathbb K} \\cdot \\widetilde V(x)\n& \\text{ for all  } x\\in \\S \\text{ if } \\alpha = 1,\n\\end{cases}\n$$\nwhere  $\\widetilde V\\in\\CX_{1}$ is a fixed point for $\\CR$, given by \n\\begin{equation}\\label{eq:tildeV}\n\\widetilde V(x) = \n\\begin{cases}\n \\log\\left(}%{\\displaystyle\\frac{\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma}{}%{\\displaystyle\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma-1}\\right) & \\text{ if } x \\in [0];\\\\[4mm]\n \\log\\left(}%{\\displaystyle\\frac{\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}}{}%{\\displaystyle\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}-1}\\right) & \\text{ if } x \\in [1].\n\\end{cases}\n\\end{equation}\n\\end{theorem}\nThis precise expression of $\\widetilde V$ corresponds to a $\\alpha$-density\n$\\tilde g(x) = \\gamma^2\/(2\\gamma-1)$ if $x \\in [0] \\cap K$ and $\\tilde g(x) =\n\\gamma\/(2\\gamma-1)$ if $x \\in [1] \\cap {\\mathbb K}$,\nand $\\int \\widetilde V(x) d\\mu_{\\mathbb K} = 1$.\n\nOur second theorem suggests that renormalization for potentials is a machinery to produce potentials with phase transition. \nWe recall that a {\\em  freezing phase transition} is characterized\nby the fact that the pressure is of the form \n$$\n\\CP(\\beta)=a\\beta+b \\quad \\text{ for } \\beta\\ge\\beta_{c}\n$$\nand that the equilibrium state is fixed for $\\beta\\ge\\beta_c$.\nThe word ``freezing'' comes from the fact that in statistical mechanics \n$\\beta$ is the inverse of the temperature (so the temperature \ngoes to $0$ as $\\beta\\to+\\infty$) and that a {\\em ground-state} \nis reached at positive temperature $1\/\\beta_c$, see \\cite{CLT, Dyson}. \n\n\\begin{theorem}\\label{theo-pt}\nAny potential $\\varphi:=-V$ with $V\\in \\CX_{1}$ admits a freezing phase transition at finite $\\beta$: there exists $\\beta_{c}>0$ such that \n\\begin{enumerate}\n\\item for $0\\le \\beta<\\beta_{c}$ the map $\\CP(\\beta)$ is analytic, there exists a unique equilibrium state for $\\beta \\varphi$ and this measure has full support;\n\\item for $\\beta>\\beta_{c}$, $\\CP(\\beta)=0$ and  $\\mu_{{\\mathbb K}}$ is the unique \nequilibrium state for $\\beta \\varphi$.  \n\\end{enumerate}\n\\end{theorem}\n\nThese two theorems explain a link between substitution, renormalization and phase transition on quasi-crystals: a substitution generates a quasi-crystal but also allows to define a renormalization operator acting on the potentials. This operator has some fixed point, and the stable leaf of that fixed point furnishes a family of potentials with freezing phase transition. \n\n\\subsection{Outline of the paper}\nIn Section~\\ref{sec-H-K-R} we recall and\/or prove various\nproperties of the Fibonacci subshift and its special words.\nWe establish the form of $H^n$ and $\\CR^nV$ for arbitrary $n$ and relate this\nto (special words of) the Fibonacci shift.\nIn Section~\\ref{sec-prooftheofix}, after  clarifying the role of accidents on\nthe computation of $\\CR^nV$, we prove Theorem~\\ref{theo-fixedpoint}.\nSection~\\ref{sec-proffthpt} deals with the thermodynamic formalism.\nFollowing the strategy of \\cite{leplaideur-synth} we specify and estimate the required (quite involved) quantities that are the core of the proof of Theorem~\\ref{theo-pt}.\n\n\\section{Some more properties of $H$, ${\\mathbb K}$ and $\\CR$}\\label{sec-H-K-R}\n\\subsection{The set \\boldmath ${\\mathbb K}$ as Sturmian subshift  \\unboldmath}\nIn addition to being a substitution subshift, $({\\mathbb K},\\sigma)$ is the Sturmian subshift associated to the golden mean rotation, $T_{\\gamma}:x \\mapsto x+\\gamma \\pmod 1$. The golden mean is\n$\\gamma=\\frac{1+\\sqrt5}2$  and it satisfies $\\gamma^{2}=\\gamma+1$. \n\nFixing an orientation on the circle, \nlet $\\arc{ab}$ denote the arc of points between $a$ and $b$ in the circle \nin that orientation. \nIf we consider the itinerary of $2\\gamma$ under the action of $T_{\\gamma}$ with the code $0$ if the point belongs to $\\arc{0\\gamma}$ and $1$ if the point belongs to $\\arc{\\ga0}$ (see Figure~\\ref{fig-codi-fibo}), we get $\\rho$, the fixed point of the substitution.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{fibocode3.pdf}\n\\caption{Coding for Fibonacci Sturmian subshift.}\n\\label{fig-codi-fibo}\n\\end{center}\n\\end{figure}\n\nThere is an almost ({\\em i.e.,\\ } up to a countable set) one-to-one correspondence between points in ${\\mathbb K}$ and codes of orbits of $(\\SS, T_\\gamma)$,\n expressed by the commutative diagram\n$$\n\\begin{array}{ccc}\\SS & \\stackrel{}%{\\displaystyle T_\\gamma}{\\longrightarrow} & \\SS \\\\\\pi\\downarrow & \\circlearrowleft & \\downarrow\\pi \\\\{\\mathbb K} & \\stackrel{}%{\\displaystyle\\sigma}{\\longrightarrow} & {\\mathbb K}\\end{array}\n$$\nand $\\pi$ is a bijection, except at points $T_\\gamma^{-n}(\\gamma) \\in \\SS$, $n \\ge 0$.\nSince Lebesgue measure is the unique $T_\\gamma$-invariant probability measure,\n$\\mu_{\\mathbb K} := \\text{Leb} \\circ \\pi^{-1}$ is the unique\ninvariant probability measure of $({\\mathbb K},\\sigma)$.  \n\nWe will use the same terminology for both ${\\mathbb K}$ and $\\SS$. \nFor instance, a cylinder $C_{n}(x)$ for $x \\in \\SS$ is an interval\\footnote{Some work has to be done to check that it actually is an interval.}, \nwith the convention that $C_{n}(x)=\\pi^{-1}(C_{n}(\\pi(x)))$, and we may confuse \na point $x \\in \\SS$ and its image $\\pi(x) \\in {\\mathbb K}$. \n\n\\begin{definition}\\label{def-words}\nLet $\\CA_{{\\mathbb K}}$ denote the set of finite words that appear in $\\rho$.\n A word $\\omega:=\\omega_{0}\\ldots \\omega_{n-1}\\in \\CA_{{\\mathbb K}}$ is said to be {\\em left-special} if $0w$ and $1w$ both appear in $\\CA_{{\\mathbb K}}$. It is {\\em right-special} if $w0$ and $w1$ both appear in $\\CA_{{\\mathbb K}}$.\nA left and right-special word is called {\\em bi-special}. A {\\em special} word is either left-special or right-special. \n\\end{definition}\n \nSince $\\rho$ has $n+1$ words of length $n$ \n(a characterization of Sturmian words), there is exactly one left-special \nand one right-special word of length $n$.\nThey are of the form $\\rho_{0}\\ldots \\rho_{n-1}$ and $\\rho_{n-1}\\ldots \\rho_{0}$\nrespectively, which can be seen from the forward itinerary\nof $x \\approx \\gamma$ and backward itinerary of $x \\approx 0$ in the circle. \nSometimes the left and right-special word merge into a single bi-special \nword $\\omega$, but only\none of the two words $0\\omega0$, and $1\\omega1$ appears in $\\CA_{{\\mathbb K}}$, see \\cite[Section 1]{arnoux-rauzy}, the construction of $\\Gamma_{n+1}$ from $\\Gamma_{n}$.\n\n\\begin{proposition}\\label{prop-bispecialfibo}\nBi-special words in $\\CA_{{\\mathbb K}}$ are of the form $\\rho_{0}\\ldots \\rho_{F_m-3}$\nand for each $m \\ge 3$, $\\rho_{0}\\ldots \\rho_{F_m-3}$ is bi-special.\n\\end{proposition}\n\nWe prove this proposition at the end of Section~\\ref{subsec:Hn}\n\n\\subsection{Results for \\boldmath $H^{n}$ \\unboldmath}\\label{subsec:Hn}\nWe recall that $\\kappa_{a}(w)$ is the number of symbol $a$ in the finite word $w$. \n\n\\begin{lemma}\\label{lem-lengthHn}\nFor any finite word $w$, the following recursive relations hold:\n\\begin{eqnarray*}\n\\kappa_0(H^n(w)) &=& F_n \\kappa_0(w) + F_{n-1} \\kappa_1(w);\\\\\n\\kappa_1(H^n(w)) &=& F_{n-1} \\kappa_0(w) + F_{n-2} \\kappa_1(w);\\\\\n|H^n(w)| &=& F_{n+1} \\kappa_0(w) + F_n \\kappa_1(w) = |H^{n-1}(w)|+|H^{n-2}(w)|, \n\\end{eqnarray*}\nwhere $|H^{0}(w)|=|w|,\\ |H^{1}(w)|=|H(w)|$.\n\\end{lemma}\n\nSince we have defined $F_{-2} = 1$ and $F_{-1} = 0$, see \\eqref{eq:Fibo},\nthese formulas hold for $n = 0$ and $n=1$ as well.\n\n\\begin{proof}\nSince $H^n(0)$ contains $F_{n+1}$ zeroes and $F_{n-1}$ ones,\nwhile  $H^n(0)$ contains $F_{n-1}$ zeroes and $F_{n-2}$ ones,\nthe first two lines follow from concatenation.\nThe third line is the sum of the first two, and naturally\nthe recursive relation follows from the same recursive relation\nfor Fibonacci numbers.\n\\end{proof}\n\nSince $({\\mathbb K}, \\sigma, \\mu_{\\mathbb K})$ is uniquely ergodic, and isomorphic to \n$(\\SS, T_\\gamma, \\text{Leb})$, we immediately get that\n\\begin{equation}\\label{eq:symfreq}\n\\lim_{n \\to +\\infty} \\frac{\\kappa_a(H^n(w))}{ |H^n(w)| } =\n\\begin{cases} \n|\\arc{0\\gamma}| = \\frac1\\gamma & \\text{ if } a = 0, \\\\\n|\\arc{\\ga0}| = 1-\\frac1\\gamma & \\text{ if } a = 1. \n\\end{cases}\n\\end{equation}\n\n\\iffalse\n\\begin{definition}\\label{def-stopblock}\nWe say that  $x, y \\in \\S$ coincide up to a {\\em stopping block} if there exists a word $w$ such that \n$$x=w01\\text{ and }y=w10.$$\nWe shall also write that $(x,y)$ has a stopping block, and\/or that $x$ and $y$ have a stopping block. \n\\end{definition}\n\nThe Fibonacci substitution is not of constant length, and this \ncomplicates the calculation how long two points $H^{n}(x)$ and $H^{n}(y)$ coincide if we know how long $x$ and $y$ coincide. \nThe main interest of stopping-block is that it propagates itself under the iterations of $H^{n}$. \n\\fi\n\n\\begin{lemma}\\label{lem-prop-stopbloc}\nAssume that $x$ and $y$ have a maximal common prefix $w$.\nThen $H^{n}(x)$ and $H^{n}(y)$ coincide for $T_{n}(w)+F_{n+2}-2$ digits, where $T_{n}(w)$ is defined by \n\\begin{equation}\\label{eq:Tnw}\nT_{0}= |w|,\\ T_{1}=|H(w)|,\\ T_{n+2}(w) = T_{n+1}(w)+T_{n}(w).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor $x = w0\\dots$ and $y = w1\\dots$, we find\n\\begin{align*}\n\\begin{array}{c} w \\\\ w \\\\\\end{array}\n& \\begin{array}{|c|}\\hline 0  \\ {\\hbox{{\\it 1}} \\!\\! I}   \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow}\n\\begin{array}{cc} H(w) & 0\\ \\\\ H(w) & 0\\ \\\\\\end{array}\n\\begin{array}{|c|}\\hline 1 \\\\0 \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow}\\begin{array}{cccc} H^2(w) & 0 & 1 & 0\\ \\\\ H^2(w) & 0 & 1 & 0\\ \\end{array}\n\\begin{array}{|c|}\\hline 0 \\\\ 1 \\\\\\hline \\end{array} \\\\[3mm]\n& \\stackrel{H}{\\longrightarrow}\n\\begin{array}{ccccccc} H^3(w) & 0 & 1 & 0 & 0 & 1 & 0\\ \\\\\nH^3(w) & 0 & 1 & 0 & 0 & 1 & 0\\ \\end{array}\n\\begin{array}{|c|}\\hline 1 \\\\0  \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow} \\ \\cdots\n\\end{align*}\nwhere we used that $H(a)$ starts with $0$ for both $a=0$ and $a=1$.\nWe set $T_n(w) = |H^n(w)|$, then the recursive formula \\eqref{eq:Tnw}\nfollows as in Lemma~\\ref{lem-lengthHn}.\n\nIterating $H$ on the words $01$ and $10$, we find:\n\\begin{equation}\\label{eq:Hn0110}\n\\begin{array}{|cc|}\\hline 0 & 1 \\ {\\hbox{{\\it 1}} \\!\\! I}  & 0 \\\\\\hline \\end{array}\\stackrel{H}{\\longrightarrow}\\begin{array}{c} 0 \\\\0 \\\\\\end{array}\\begin{array}{|cc|}\\hline 1 & 0 \\\\0 & 1 \\\\\\hline \\end{array}\\stackrel{H}{\\longrightarrow}\\begin{array}{ccc}0 & 1 & 0 \\\\0 & 1 & 0\\end{array}\\begin{array}{|cc|}\\hline 0 & 1 \\ {\\hbox{{\\it 1}} \\!\\! I}  & 0 \\\\\\hline \\end{array}\n\\stackrel{H}{\\longrightarrow}\\begin{array}{cccccc}0 & 1 & 0 & 0 & 1 & 0 \\\\0 & 1 & 0 & 0 & 1 & 0\\end{array}\n\\begin{array}{|cc|}\\hline 1 & 0 \\\\0 & 1 \\\\\\hline \\end{array}\\ .\n\\end{equation}\nThus $|H^n(10)| = |H^n(01| = F_{n+2}$ and the common prefix\nof $H^n(10)$ and $H^n(01)$ has length $F_{n+2}-2$ is precisely the same as\nthe common block of $H^n(w0)$ and $H^n(w0)$ between $H^n(w)$ and\nthe first difference.\n\nTherefore, $H^n(x)$ and $H^n(y)$\ncoincide for $T_n(w) + F_{n+2}-2$ digits.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-coinc-hn-rho}\nFor $x \\in {\\mathbb K}$ and $n\\in{\\mathbb N}$, $H^{n}(x)$ and $\\rho$ coincide for at least $F_{n+3}-2$ digits if $x\\in [0]$ and for at least $F_{n+2}-2$ digits if $x\\in[1]$. \n\\end{corollary}\n\n\\begin{proof}\nIf $x \\in [0]$, then, by Lemma~\\ref{lem-prop-stopbloc}, \n$H^n(x)$ coincides with $H^n(\\rho) = \\rho$\nfor at least $T_n(0) + F_{n+2}-2$ digits.\nBut $T_n(0) = |H^n(0)| = F_{n+1}$, so $T_n(0) + F_{n+2}-2 = F_{n+3}-2$.\n\nIf $x \\in [1]$, then $H(x) \\in [0]$ and the previous argument gives that \n$H^n(x)$ coincides with $H^n(\\rho) = \\rho$\nfor at least $F_{n+2}-2$ digits.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop-bispecialfibo}]\nWe iterate the blocks $0\\cdot01$, $0\\cdot10$ and $1\\cdot01$ under $H$:\\\\\n$$\n\\begin{array}{|cccc|}\\hline 0 & \\cdot & 0 & 1 \\\\ 0 & \\cdot & 1 & 0 \\\\ 1 & \\cdot & 0 & 1 \\\\ \\hline \\end{array}\n\\stackrel{H}{\\longrightarrow}\n\\begin{array}{|cc|}\\hline 0 & 1 \\\\ 0 & 1 \\\\ & 0  \\\\ \\hline \\end{array}\\\n\\begin{array}{c} 0 \\\\0 \\\\ 0 \\end{array}\\\n\\begin{array}{|cc|}\\hline 1 & 0 \\\\ 0 & 1 \\\\  1 & 0 \\\\ \\hline \n\\end{array}\\stackrel{H}{\\longrightarrow}\n\\begin{array}{|ccc|}\\hline \\dots\\!\\!\\! & 1 & 0 \\\\ \\dots\\!\\!\\! & 1 & 0 \\\\ & 0 & 1 \\\\ \\hline \\end{array}\\\n\\begin{array}{ccc} 0 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 0 \\end{array}\\\n\\begin{array}{|cc|}\n\\hline 1 & 0 \\\\ 0 & 1 \\\\  1 & 0 \\\\ \\hline  \n\\end{array}\n\\stackrel{H}{\\longrightarrow} \\cdots\\ , \\\\\n$$\nso the common central block here is bi-special, and it is the same \nas the common block $v$ of $H^n(01)$ and $H^n(10)$ \nof length $F_{n+2}-2$ in the proof of Lemma~\\ref{lem-prop-stopbloc}.\nThus we have found the bi-special word of length $F_{n+2}-2$, and every\nprefix and suffix of $v$ is left and right-special respectively.\nThe fact that these are the only bi-special words can be derived from \nthe Rauzy graph for this Sturmian shift, see\n{\\em e.g.\\ } \\cite[Sec. 1]{arnoux-rauzy}.\nIn their notation, there is a bi-special word of length $k$\nif the two special nodes in the Rauzy graph coincide: $D_k = G_k$.\nThe lengths of the two ``buckles'' of non-special nodes between $D_k = G_k$\nare two consecutive Fibonacci numbers minus one, as follows from the \ncontinued fraction expansion \n$$\n\\gamma=1+\\frac1{1+\\frac1{1+\\ddots}}.\n$$\nTherefore, the complexity satisfies\n$$\nk+1 = p(k) = \\#\\{ \\text{nodes of Rauzy graph of order } k\\}\n= F_n-1 + F_{n-1}-1 + 1,\n$$\nso indeed only the numbers $k = F_{n+1}-2$ can be the lengths of bi-special \nwords.\n\\end{proof}\n\n\n\\subsection{Iterations of the renormalization operator}\n\nThe renormalization operator  for potentials can be rewritten under as \n(recall the definition of $F_{n}^{a}$, $a=0,1$, from \\eqref{eq:Fiboa})\n\\begin{equation}\\label{equ-def-cr}\n\\CR V|_{[a]} = \\sum_{j=0}^{F^{a}_{1}-1} V \\circ \\sigma^j \\circ H|_{[a]}.\n\\end{equation}\nThis general formula may be extended to other substitutions and leads to an expression for $\\CR^{n}V$.  The main result here is Lemma~\\ref{lem:RkV}, where we show that\n\\begin{equation}\\label{eq:RnV}\n(\\CR^n V)(x) = \\sum_{j=0}^{F_{n^{*}}-1} V \\circ \\sigma^j \\circ H^n(x),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:n*}\nn^* = \\begin{cases}\nn+1 & \\text{ if } x \\in [0],\\\\\nn & \\text{ if } x \\in [1].\n\\end{cases}\n\\end{equation}\nThe substitution $H$ solves a renormalization equation of the form \\eqref{equ1-renorm}. If $x=0x_{1}\\ldots$, then $H(x)=01H(x_{1})\\ldots$ and $\\sigma^{2}\\circ H(x)=H\\circ \\sigma(x)$. If $x=1x_{1}\\ldots$ then we simply have \n$\\sigma\\circ H(x)=H\\circ \\sigma(x)$. The renormalization equation is thus more complicated than for the constant length case. We need an expression for iterations of $H$ and $\\sigma$. \n\\begin{lemma}\\label{lem:commute_shift_H}\nGiven $k \\ge 0$ and $a=0,1$, let \n$w = w_1w_2\\dots w_{F^{a}_{k}} = H^k(a)$.\nThen for every $0 \\le i < F^{a}_{k}$ we have\n$$\nH \\circ \\sigma^i \\circ H^k|_{[a]} = \\sigma^{| H(w_1\\dots w_i)|} \\circ H^{k+1}|_{[a]}.\n$$\n\\end{lemma}\n\\begin{proof}\nFor $k = 0$ this is true by default and for $k= 1$, this is precisely\nwhat is done in the paragraph before the lemma.\nLet us continue by induction, assuming that the statement is true for $k$.\nThen $\\sigma^i$ removes the first $i$ symbols of $w = H^k(a)$,\nwhich otherwise, under $H$, would be extended to a word of\nlength $|H(w_1\\dots w_i)|$. We need this number of shifts\nto remove $H(w_1\\dots w_i)$ from $H([w]) = H^{k+1}([a])$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:RkV}\nFor every $k \\ge 0$ and $a=0,1$, we have\n$$\n\\CR^kV|_{[a]} = S_{F^{a}_{k}}V \\circ H^k|_{[a]},\n$$\nwhere $S_nV = \\sum_{i=0}^{n-1} V \\circ \\sigma^i$ denotes the $n$-th ergodic sum.\n\\end{lemma}\n\n\\begin{proof}\nFor $k = 0$ this is true by default. For $k = 1$, this follows by the definition of the renormalization operator $\\CR$.\nLet us continue by induction, assuming that the statement is true for $k$.\nWrite $w = H^k(a)$ and $t_i = |H(w_i)| = F_{w_i}$.\nThen\n\\begin{eqnarray*}\n\\CR^{k+1}V|_{[a]} &=& (\\CR V) \\circ S_{F^a_k}V \\circ H^k|_{[a]} \\hskip 1cm \\text{\\small (Induction assumption)}\\\\\n &=& \\sum_{i=0}^{F^{a}_{k}-1}  \\left( \\sum_{j=0}^{t_i-1} V \\circ \\sigma^j \\circ H \\right)  \\sigma^i\\circ H^k|_{[a]}\\hskip 1cm \\text{\\small (by formula \\eqref{equ-def-cr})}\\\\\n &=& \\sum_{i=0}^{F^{a}_{k}-1} \\left( \\sum_{j=0}^{t_i-1} V \\circ \\sigma^{j+|H(w_1\\dots w_i)|} \\circ H \\right) \\circ H^k|_{[a]}  \\hskip 1cm\\text{\\small (by Lemma~\\ref{lem:commute_shift_H})}\\\\\n &=& \\sum_{l=0}^{F^{a}_{k+1}-1} V \\circ \\sigma^l \\circ H^{k+1}|_{[a]}, \n\\end{eqnarray*}\nas required.\n\\end{proof}\n\n\n\n\n\n\\subsection{Special words are sources of accidents}\nOverlaps of  $\\rho$ with itself are strongly related to bi-special words. They are of prime importance to determine the fixed points of $\\CR$ and their \nstable leaves, see {\\em e.g.\\ } formula \\eqref{equ-crkV} below. Dynamically, they correspond to what we call {\\em accident} in the time-evolution of the distance between the orbit and ${\\mathbb K}$.  \n\n\n\n\nFor most $x$ close to ${\\mathbb K}$, $d(\\sigma(x),{\\mathbb K}) = 2d(x,{\\mathbb K})$, but\nthe variation of $d(\\sigma^{j}(x),{\\mathbb K})$ is not always monotone with respect to $j$. When it decreases, it generates an accident:\n\\begin{definition}\\label{def-accident}\nLet $x\\in\\S$ and $d(x,{\\mathbb K})=2^{-n}$. If $d(\\sigma(x),{\\mathbb K})\\le 2^{-n}$, we say that we have an {\\em accident} at $\\sigma(x)$. \nIf there is an accident at $\\sigma^{j}(x)$, then we shall simply say we have an accident at $j$. \n\\end{definition}\n\nThe next lemma allows us to detect accidents. \n\\begin{lemma}\\label{lem-accident-bispecial}\nLet $x=x_{0}x_{1}\\ldots$ coincide with some $y\\in{\\mathbb K}$ for $d$ digits. Assume that the first accident occurs at $b$. Then $x_{b}\\ldots x_{d-1}$ is a bi-special word in $\\CA_{{\\mathbb K}}$. Moreover, the word $x_{0}\\ldots x_{d-1}$ is not right-special. \n\\end{lemma}\n\\begin{proof}\nBy definition of accident, there exists $y$ and $y'$ in ${\\mathbb K}$ such that $d(x,{\\mathbb K})=d(x,y)$ and $d(\\sigma^{b}(x),{\\mathbb K})=d(\\sigma^{b}(x),y')$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.7]{accident.pdf}\n\\caption{Accident and bi-special words}\n\\label{fig-accident}\n\\end{center}\n\\end{figure}\nFigure~\\ref{fig-accident} shows that the word $x_{b}\\ldots x_{d-1}$ is bi-special because its two extensions $y$ and $y'$ in ${\\mathbb K}$ \nhave different suffix and prefix for this word. \n\nIt remains to prove that $x_{0}\\ldots x_{d-1}$ is not right-special. If it was, then $x_{0}\\ldots x_{d-1}x_{d}=y_{0}\\ldots y_{d-1}\\overline{y_{d}}$ \nwould a ${\\mathbb K}$-admissible word, thus $d(x,{\\mathbb K})\\le 2^{-(d+1)}\\neq 2^{-d}$. \n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{theo-fixedpoint}}\\label{sec-prooftheofix}\n\\subsection{Control of the accidents under iterations of $\\CR$}\nNext we compute $\\CR^{n}V$ and show that accidents do not crucially\nperturb the Birkhoff sum involved. This will follow from Corollaries~\\ref{coro-dHnK} and \\ref{cor-coinc-hn-rho}.\n\nNote that Lemma~\\ref{lem-prop-stopbloc} shows that $H$ is one-to-one. The next proposition explains the relation between the attractor ${\\mathbb K}$ and its image by $H$. \n\n\\begin{proposition}\\label{prop-K-H}\nThe subshift ${\\mathbb K}$ is contained in $H({\\mathbb K}) \\cup \\sigma\\circ H({\\mathbb K})$. More precisely, if $[0]\\cap {\\mathbb K} \\subset H({\\mathbb K})$ and $x\\in[1]\\cap {\\mathbb K} \\subset \\sigma\\circ H({\\mathbb K})$.  \n\\end{proposition}\n\n\\begin{proof}\nFirst note that Lemma~\\ref{lem-prop-stopbloc} shows that $H$ is one-to-one. We also recall that the word $11$ is forbidden in ${\\mathbb K}$. Hence, each digit $1$\nin $x=x_{0}x_{1}x_{2}\\ldots\\in {\\mathbb K}$ is followed and preceded by a digit $0$ (unless the $1$ is in first position). \n\nAssuming $x_{0}=0$, we can unique split $x$ into blocks of the form \n$0$ and $01$.\nIn this splitting, we replace \neach single $0$ by $1$ and each pair $01$ by $0$. \nThis produces a new word, say $y$, and by construction, $H(y)=x$. \nThis operation is denoted by $H^{-1}$. \nIt can be used on finite words too, provided that the last digit is $1$. \nIf $x_{0}=1$, we repeat the above construction with $0x$, and $x=\\sigma\\circ H(y)$. \n\nIt remains to prove that $y\\in{\\mathbb K}$. \nFor every $x\\in{\\mathbb K}$, there is a sequence $k_{n}\\to\\infty$ \nsuch that $\\sigma^{k_{n}}(\\rho)\\to x$. Assume again that $x_0 = 0$.\nThen we can find a sequence $l_n \\sim k_n\/\\gamma$ such that \n$H \\circ \\sigma^{l_n}(\\rho)=\\sigma^{k_{n}}(\\rho)$.\nTherefore $\\lim_n \\sigma^{l_n}(\\rho) \\in {\\mathbb K}$, and this limit is indeed the sequence\n$y$ that satisfies $H(y) = x$.\nFinally, for $x_0 = 1$, we repeat the argument with $0x$.\n\\end{proof}\n\n\n\\begin{corollary}\\label{coro-dHnK}\nIf $d(x,{\\mathbb K})=d(x,y)$ with $y\\in {\\mathbb K}$, then $d(H^{n}(x),{\\mathbb K})=d(H^{n}(x),H^{n}(y))$ for $n\\ge 0$.\n\\end{corollary}\n\\begin{proof}\nWrite $x=wa$ and $y=w\\overline a$ where $a$ is an unknown digit and $\\overline a$ its opposite. Note that $H^{n}(x)$ starts with $0$ for any $n\\ge 1$.\nAssume that there is some $z\\in {\\mathbb K}$ such that $d(H(x),z)<d(H(x),H(y))$.\n\n\\paragraph{{\\it Case 1:}} $x=w0\\ldots$ and $y=w1\\ldots$. Necessarily, $y=w10$. Therefore $H(x)=H(w)01\\ldots$ and $H(y)=H(w)001\\ldots$. By assumption, $z$ coincide with $H(x)$ longer than $H(y)$, which shows that $z$ starts as \n$z=H(w)01\\ldots$\nConsequently $H^{-1}(z)=w0\\dots$ and this contradicts that $d(x,{\\mathbb K})=d(x,y)$. \n\n\\paragraph{{\\it Case 2:}} $x=w1b\\ldots$ and $y=w0\\ldots$. Then $H(x)=H(w)00\\ldots$  (since $H(b)$ starts with $0$ regardless what $b$ is)\nand $H(y)=H(w)01\\ldots$ Again $z$ coincides with $H(x)$ longer than $H(y)$ and thus $z$ starts as $H(w)00$. The $0$ before last position is necessarily a single zero for the $H^{-1}$-procedure and thus $H^{-1}(z)$ coincide with  $x$ for longer than $y$. This is a contradiction.\n\nConsequently for both cases we have shown $d(H(x),{\\mathbb K})=d(H(x),H(y))$. \nThe result follows by induction. \n\\end{proof}\n\nWe recall that by \\eqref{eq:RnV}, $\\CR^{n}V$ is given by a Birkhoff sum, of\n$F_{n^*}$ where $n^* = n+1$ or $n$ as in \\eqref{eq:n*}.\nTo compute $(\\CR^{n}V)(x)$, we need an estimate for $d(\\sigma^{j}(H^{n}(x)),{\\mathbb K})$, for $0 \\le j\\le F_{n+1}-1$ or $0 \\le j\\le F_{n}-1$. The key point is that no accident can occur for these $j$. This follows from the next lemma. \n\n\\begin{lemma}\\label{lem-no-accident}\nThe sequence $H^n(x)$ has no accident in the first $F_{n^*}$ entries.\n\\end{lemma}\n\n\\begin{proof}\nWe give the proof for $x \\in [1]$, so $n^* = n$. The proof for $x \\in [0]$ is analogous.\nBy Corollary~\\ref{cor-coinc-hn-rho}, $H^n(x)$ coincides for at least $F_{n+2}-2$\ndigits with $\\rho$.\nIf an accident happens in the first $F_n$ digits, say at entry $0 \\le j < F_n$,\nthen by Lemma~\\ref{lem-accident-bispecial}, a bi-special word starts\nat $j$, which by Proposition~\\ref{prop-bispecialfibo} is a suffix\nof $\\rho$ of length $F_m-2$ for some $m$.\nSince we have an accident, $j+F_m-2 \\ge F_{n+2}-1$, so $m > n+1$.\n\n\n\n\nHence $\\rho_{0}\\ldots\\rho_{F_{n+2}-1}$ can be written as $BBB'$ where $B$ is the suffix of $\\rho$ of length $j$ and $B'$ is a suffix of $\\rho$ of length $\\ge |B|\/\\gamma$.\nClearly $B$ starts with $0$. We can split it uniquely into blocks $0$ and $01$, and $B$ fits an integer number of such blocks, because if\nthe final block would overlap with the second appearance of $B$, then $B$ \nwould start with $1$, which it does not.\n\nTherefore we can perform an inverse substitution $H^{-1}$, for  each block $B$ and also for $B'$ because we cal globally do $H^{-1}$ for $\\rho_{0}\\ldots\\rho_{F_{n+2}-1}$. We  find $H^{-1}(BBB') = CCC'$ which  has the same characteristics.\nRepeating this inverse iteration, we find that $\\rho$ starts with $0101$, or with $00$, \na contradiction.\n\\end{proof}\n\nLet $N(x,n)$ be the integer such that $2^{-N(x,n)} = d(H^{n}(x),{\\mathbb K})$.\nBy the previous lemma $d(\\sigma^{j}(H^{n}(x)){\\mathbb K})=2^{-(N(x,n)-j)}$\nfor every $j<F_{n^{*}}$. For the largest value $j = F_{N^*}$,\nwe have  $d(\\sigma^{j}(H^{n}(x)){\\mathbb K}) = 2^{-(T_{n}+F_{n+2}-2-F_{n^{*}})}$.\nTherefore, if $g$ is the $\\alpha$-density function for $V$, then we obtain\n\\begin{equation}\\label{equ-crkV}\n(\\CR^{n}V)(x)=\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{(N(x,n)-j)^{\\alpha}} +o\\left(\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{(N(x,n)-j)^{\\alpha}}\\right).\n\\end{equation}\n\n\\subsection{Proof of Theorem~\\ref{theo-fixedpoint}}\n\\subsubsection{$\\widetilde V$ is a fixed point}\nA simple computation shows that $\\CR$ fixes $\\widetilde V$ from\n\\eqref{eq:tildeV}. Assume $x\\notin{\\mathbb K}$ is such that $\\widetilde\\kappa_{0}(x)=n$ and $\\widetilde\\kappa_{1}(x)=m$ (see the definition of $\\widetilde\\kappa_{a}$ above  the statement of Theorem~\\ref{theo-fixedpoint}).\nThen, by Lemmas~\\ref{lem-lengthHn} and \\ref{lem-prop-stopbloc}, \nand the fact that $H(x)$ starts with $0$, we get\n\\begin{eqnarray*}\n\\widetilde\\kappa_{0}(H(x)) = n+m+1\\ & \\qquad &\\widetilde\\kappa_{0}(\\sigma \\circ H(x))  =  n+m,\\\\\n\\widetilde\\kappa_{1}(H(x))  =  n \\qquad \\qquad & \\qquad & \\widetilde\\kappa_{1}(\\sigma \\circ H(x)) =  n.\n\\end{eqnarray*}\n \\begin{itemize}\n\\item If $x$ starts with $0$, then $H(x)$ starts with $01$ and \n\\begin{eqnarray*}\n(\\CR\\widetilde V)(x)&=& \\widetilde V(H(x))+\\widetilde V\\circ \\sigma(H(x))\\\\\n&=& \\log\\left(\\frac{(n+m+1)+\\frac1\\gamma n+\\gamma}{(n+m+1)+\\frac1\\gamma n+\\gamma-1}\\right)+\\log\\left(\\frac{\\gamma(n+m)+ n+\\gamma^{2}}{\\gamma(n+m)+ n+\\gamma^{2}-1}\\right)\\\\\n&=& \\log\\left(\\frac{(n+m+1)+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma}\\right)+\\log\\left(\\frac{n+m+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma-\\frac1\\gamma}\\right)\\\\\n&=& \\log\\left(\\frac{n+m+1+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma-\\frac1\\gamma}\\right)\\\\\n&=& \\log\\left(\\frac{n(1+\\frac1\\gamma)+m+\\gamma+1}{n(1+\\frac1\\gamma)+m+\\gamma-\\frac1\\gamma}\\right)\\qquad\\qquad  \\text{\\small since } \\gamma^{2}=\\gamma+1\\\\\n&=& \\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma(\\gamma-1)}\\right)=\\log\\left(\\frac{n+\\frac1\\gamma m+\\gamma}{n+\\frac1\\gamma m+\\gamma-1}\\right)=\\widetilde V(x).\n\\end{eqnarray*}\n \n\\item If $x$ starts with $1$, then $H(x)$ starts with $0$ and \n\\begin{eqnarray*}\n(\\CR\\widetilde V)(x)&=& \\widetilde V(H(x))=\\log\\left(\\frac{(n+m+1)+ \\frac1\\gamma n+\\gamma}{(n+m+1)+\\frac1\\gamma n+\\gamma-1}\\right)\\\\\n&=&\n\\log\\left(\\frac{\\gamma(n+m+1)+ n+\\gamma^{2}}{\\gamma(n+m+1)+ n+\\gamma^{2}-\\gamma}\\right)\\\\\n&=&\n\\log\\left(\\frac{n(\\gamma+1)+\\gamma m+\\gamma+\\gamma^{2}}{n(\\gamma+1)+\\gamma m+\\gamma^{2}}\\right)\\\\\n&=& \\log\\left(\\frac{\\gamma^{2}n+\\gamma m+\\gamma^{3}}{\\gamma^{2}n+\\gamma m+\\gamma^{2}}\\right)\\\\\n&=&\n\\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma}\\right)\n=\\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma^{2}-1}\\right) =\\widetilde V(x).\n\\end{eqnarray*}\n\\end{itemize}\n\n\n\\subsubsection{A Toeplitz summation}\nNext, we consider $V\\in\\CX_{1}$. We shall see in the proof that the convergence for these $V$ automatically implies that for $V\\in\\CX_{\\alpha}$ with $\\alpha\\ne1$, the same computations work but with either too light or too heavy denominators. \n\nIn the same spirit, we shall see that $}%{\\displaystyle \\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{N(x,n)-j}$ actually converges, which immediately yields that  $}%{\\displaystyle o\\left(\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{N(x,n)-j}\\right)$  converges to $0$. We can thus forget any term of that kind. \n\nRecall that by Lemmas~\\ref{lem-lengthHn} and \\ref{lem-prop-stopbloc}\nand Corollary~\\ref{coro-dHnK} we have\n$$\nN(x,n) := \\log_2 d(H^n(x), {\\mathbb K}) = T_n+F_{n+2}-2 \\quad \\text{ for } \\\nT_{n}:=F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x).\n$$\nWe thus have to compute the limit of \n$$\n\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)+F_{n+2}-(j+2)}\n$$\nas $n^* \\to +\\infty$, where $n^{*} = n+1$ if $x \\in [0]$ and $n^* = n$ if $x\\in[1]$. Moreover $g$ is a non-negative continuous function, hence uniformly continuous. Furthermore, for $y \\in {\\mathbb K}$ closest to $x$,\nthe point $\\sigma^{F_{n^{*}}} \\circ H^{n}(x)$ coincides with $\\sigma^{F_{n^{*}}} \\circ H^{n}(y)$ for at least $F_{n}-2$ digits.\nThere exists a sequence  $\\varepsilon_{n}\\downarrow 0$ such that \n$$\\left|g\\circ \\sigma^{j}(H^{n}(x))-g\\circ \\sigma^{j}(H^{n}(y))\\right|\\le \\varepsilon_{n},$$\nfor every $j\\le F_{n^{*}}-1$.\n\nFinally, Binet's formula $F_{n+1}-\\gamma F_n = \\sqrt{5} \\gamma^{-(n-1)}$ shows that \n\\begin{eqnarray*}\nF_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)\\!&+&\\!F_{n+2}-(j+2)\\\\\n&=& F_{n+1}(\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma-\\frac{j}{F_{n+1}})(1+\\varepsilon'_{n}) \\\\\n&=& F_{n}(\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}-\\frac{j}{F_{n}})(1+\\varepsilon''_{n}),\n\\end{eqnarray*}\nwhere $\\varepsilon'_{n}$ and $\\varepsilon''_{n}$ tend to $0$ as $n \\to +\\infty$. \n\nCombining $\\varepsilon$, $\\varepsilon'$ and $\\varepsilon''$ in a single $o(1)$, \nwe can rewrite the above equalities to \n\\begin{equation}\\label{equ1-toeplitz}\n \\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)+F_{n+2}-(j+2)}=\n\\frac{1+o(1)}{F_{n^{*}}} \\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ\\sigma^{j}\\circ H^{n}(y)}{X_{n}-\\frac{j}{F_{n^{*}}}},\n\\end{equation}\nwhere $X_{n}=\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma$ if $x \\in [0]$ \nand $X_{n}=\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}$ if $x\\in[1]$. \n\n\\subsubsection{Convergence of the weighted sum in \\eqref{equ1-toeplitz}}\nThe reader can verify that we are here considering a Toeplitz summation method, with a regular matrix (see \\cite[ Definition 7.5]{kuipers-niederreiter}  and \\cite{weber-crass}), up to a renormalization factor, which is the limit of \n$$\n\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac1{X_{n}-\\frac{j}{F_{n^{*}}}}.\n$$\nThis expression is a Riemann sum, and converges to $\\widetilde V(x)$\nas $n^* \\to \\infty$. \n\nFrom \\cite{weber-crass} we have that the M\\\"uller (see \\cite{muller}) criterion applies, and we get that for $\\mu_{{\\mathbb K}}$-almost every $z\\in{\\mathbb K}$, \n\\begin{equation}\n\\label{equ-cv-toepli}\n\\lim_{n\\to+\\infty}\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(z)}{X_{n}-\\frac{j}{F_{n^{*}}}}=\\widetilde V(x)\\int g\\,d\\mu_{{\\mathbb K}}.\n\\end{equation}\n\nNevertheless, our expression in \\eqref{equ1-toeplitz} is different, because the point $z$ we are considering is $H^{n}(y)$ and thus depends on $n$.\nA priori, this may generates fluctuations in the convergence, but\nwe prove here that this is not the case.\n\nThe main argument is that $({\\mathbb K},\\sigma)$ is uniquely ergodic. We claim that this implies that the convergence in \\eqref{equ-cv-toepli} is uniform in $z$. \n\nIndeed, if it is not uniform, we can find $\\varepsilon > 0$ and \na sequence of $z_{n}$ such that for every $n$, \n$}%{\\displaystyle |\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(z_{n})}{X_{n}-\\frac{j}{F_{n^{*}}}} - \\widetilde V(x)\\int g\\,d\\mu_{{\\mathbb K}}| > \\varepsilon$ \nfor every $n$. \nThen any accumulation point $\\mu_{\\infty}$ of the family of measures \n$$\n\\mu_{n}:=\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{1}{X_{n}-\\frac{j}{F_{n^{*}}}}\\delta_{\\sigma^{j}(z_{n})}\n$$\nis $\\sigma$-invariant (because $F_{n^{*}}\\to+\\infty$), supported on ${\\mathbb K}$, and \n$\\int g\\,d\\mu_{\\infty}\\neq \\int g\\,d\\mu_{{\\mathbb K}}$. This would contradict the unique ergodicity for $({\\mathbb K}, \\sigma)$. \n\nTherefore, the convergence in \\eqref{equ-cv-toepli} is uniform in $z$ and this shows that \n$$\n\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(H^{n}(y))}{X_{n}-\\frac{j}{F_{n^{*}}}} \\to \\widetilde V(x) \\cdot \\int g\\,d\\mu_{{\\mathbb K}}.\n$$ \nThis finishes the proof of Theorem~\\ref{theo-fixedpoint}.\n\n\n\\section{Proof of Theorem~\\ref{theo-pt}}\\label{sec-proffthpt}\n\n\\subsection{The case \\boldmath $-\\log \\frac{n+1}{n}$ \\unboldmath}\n\\label{subsec-logcase}\nWe first consider the potential $\\varphi(x) = -\\log \\frac{n+1}{n}$ if $d(x, {\\mathbb K}) = 2^{-n}$, leaving the general potential in $\\CX_1$ for later.\n\n\\subsubsection{Strategy, local equilibria}\nFix some cylinder $J$ such that the associated word, say $\\omega_{J}$, does not appear in $\\rho$ (as {\\em e.g.\\ } 11). We follow the induction method presented in \\cite{leplaideur-synth}. Let $\\tau$ be the first return time into $J$ (possibly $\\tau(x)=+\\infty$), and consider the family of transfer operators \n\\begin{eqnarray*}\n\\CL_{Z,\\beta}:\\psi&\\mapsto& \\CL_{Z,\\beta}(\\psi)\\\\\nx&\\mapsto& \\CL_{Z,\\beta}(\\psi)(x):=\\sum_{n=1}^{+\\infty}\\sum_{\\stackrel{y\\in J\\ \\tau(y)=n}{ \\sigma^{n}(y)=x}}e^{\\beta \\cdot (S_{n}\\varphi)(y)-nZ}\\psi(y),\n\\end{eqnarray*}\nwhich acts on the set of continuous functions $\\psi:J\\to{\\mathbb R}$.  \nFollowing \\cite[Proposition 1]{leplaideur-synth}, for each $\\beta$ there exists $Z_{c}(\\beta)$ such that $\\CL_{Z,\\beta}$ is well defined for every $Z>Z_{c}(\\beta)$. \nBy \\cite[Theorem 1]{leplaideur-synth}, $Z_{c}(\\beta)\\ge 0$ because the pressure of the dotted system \n(which in the terminology of \\cite{leplaideur-synth} is the system restricted to the trajectories that avoid $J$)\nis larger (or equal) than the pressure of ${\\mathbb K}$ which is zero. \n\nWe shall prove\n\\begin{proposition}\n\\label{prop-spectral&zc}\nThere exists $\\beta_{0}$ such that $\\CL_{0,\\beta}({1\\!\\!1}_{J})(x)<1$ for every $\\beta>\\beta_{0}$ and $x\\in J$.\n\\end{proposition}\nWe claim that if Proposition~\\ref{prop-spectral&zc} holds, then\n\\cite[Theorem 4]{leplaideur-synth} proves that $\\CP(\\beta)=0$ for every $\\beta>\\beta_{0}$, and $\\mu_{{\\mathbb K}}$ is the unique equilibrium state for $\\beta \\varphi$. \n\nTo summarize \\cite{leplaideur-synth} (and adapt it to our context), the \npressure function satisfies (see Figure~\\ref{fig-graphes-press}\\footnote{We will see that $\\CL_{0,\\beta}({1\\!\\!1}_{J})(x)$ is a constant function on $J$.}),\n$$\nZ_{c}(\\beta) \\le \\CP(\\beta) \\le \\max(\\log(\\CL_{0,\\beta}({1\\!\\!1}_{J})),0).\n$$\n\nAs long as $\\CP(\\beta)>0$, there is a unique equilibrium state and it has full support. In particular this shows that the construction does not depend on the choice of $J$. If Proposition~\\ref{prop-spectral&zc} holds, then \n$$Z_{c}(\\beta)=\\CP(\\beta)=\\max(\\log(\\CL_{0,\\beta}({1\\!\\!1}_{J})),0)=0,\n\\text{ for } \\beta>\\beta_{c}$$\nand $\\mu_{{\\mathbb K}}$ is the unique equilibrium state because $\\CL_{0,\\beta}({1\\!\\!1}_{J})<1$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=.5]{graphe1na.pdf}\n\\caption{The Pressure between $Z_{c}(\\beta)$ and $\\log\\lambda_{0,\\beta}:=\\CL_{0,\\beta}({1\\!\\!1}_{J})$}\n\\label{fig-graphes-press}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 1}\\label{subsubsec-upper=series}\nWe reduce the problem to the computation of a series depending on $\\beta$. \nNote that $\\varphi(x)$ only depends on the distance from $x$ to ${\\mathbb K}$. This shows that if $x, x' \\in J$ and $y, y' \\in J$ are such that \n$$y=\\omega x\\ , \\quad \u00a0y'=\\omega x',$$\nwith $\\omega\\in \\{0,1\\}^{n}$, $\\tau(y)=\\tau(y')=n$, then \n$$(S_{n}\\varphi)(y)=(S_{n}\\varphi)(y').$$\nIn other words, $\\CL_{Z,\\beta}({1\\!\\!1}_{J})$ is a constant function, and then equal to the spectral radius $\\lambda_{Z,\\beta}$ of $\\CL_{Z,\\beta}$.  \n\nConsequently, to compute $\\lambda_{Z,\\beta}$, it suffices to compute the sum of all \n$}%{\\displaystyle e^{\\beta \\cdot (S_{n}\\varphi)(\\omega)-nZ}$, where $\\omega$ is a word of length $n+|\\omega_{J}|$, starting and finishing as $\\omega_{J}$. Such a word $\\omega$ can also be seen as a path of length $n$ starting from $J$ and returning (for the first time) to $J$ at time $n$. \n\nWe split such a path in several sub-paths. We fix an integer $N$ and say that the path is {\\em free} at time  $k$ if $\\omega_{k}\\ldots \\omega_{n-1}\\omega_{J}$ is at distance  larger than $2^{-N}$ to ${\\mathbb K}$. Otherwise, we say that we have an {\\em excursion}. The path is thus split into intervals of free moments and excursions. \nWe assume that $N$ is chosen so large that $0$ is a free moment. This also shows that for every $k\\le n$, $d(\\sigma^{k}(\\omega\\omega_{J}),{\\mathbb K})$ is determined by $\\omega_{k}\\ldots \\omega_{n-1}$. \n\nIf $k$ is a free time, $}%{\\displaystyle \\varphi(\\sigma^{k}(\\omega\\omega_{J}))\\le A_{N}:=-\\log\\left(1+\\frac 1N\\right)$. Denote by $k_{0}$ the maximal integer such that \n$k$ is a free time for every $k\\le k_{0}$. Then $S_{k_{0}+1}\\varphi\\le (k_{0}+1)A_{N}$ and there are fewer than $2^{k_{0}+1}$ such prefixes of length $k_0+1$. \n\nNow, assume that every $j$  for $k_{0}+1\\le j\\le k_{0}+k_{1}$ is an excursion time, and assume that $k_{1}$ is the maximal integer with this property. To the contribution $(S_{k_{0}+1}\\varphi)(\\omega\\omega_{J})$ we must add the contribution $(S_{k_{1}}\\varphi)(\\sigma^{k_{0}+1}(\\omega\\omega_{J}))$ of the excursion. \nThen we have a new interval of free times, and so on. \nThis means that we can compute $\\CL_{0,\\beta}({1\\!\\!1}_{J})$ by gluing together paths with the same decompositions  of free times and excursion times. If we denote by $C_{E}$ the total contribution of all paths with exactly one excursion (and only starting at the beginning of the excursion), then we have \n\\begin{equation}\n\\label{equ1-upperboundCL0}\n\\lambda_{0,\\beta}=\\CL_{0,\\beta}({1\\!\\!1}_{J}) \\le \\sum_{k=1}^{+\\infty}\\left(\\sum_{k_{0}=0}^{+\\infty}e^{(k_{0}+1)(\\beta A_{N}+\\log2)}\\right)^{k+1}C_{E}^k.\n\\end{equation}\nThe sum in $k$ accounts for $k+1$ intervals of free moment with $k$ intervals of excursions times between them. The sum in $k_{0}$ accounts for the possible length $k_{0}+1$ for an interval of free times. \nThese events are maybe not independent but the sum in \\eqref{equ1-upperboundCL0} includes all paths, possible or not, and therefore yields an upper bound. \n\nThe integer $N$ is fixed, and we can take $\\beta$ so large that $\\beta A_{N}<-\\log2$. This shows that the sum in $k_{0}$ in \\eqref{equ1-upperboundCL0}  converges and is as close to $0$ as we \nwant if $\\beta$ is taken sufficiently large. \n\nTo prove Proposition~\\ref{prop-spectral&zc}, it is thus sufficient to prove that $C_{E}$ can be made as small as we want if $\\beta$ increases. \n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 2}\\label{subsubsec-splitCE}\nWe split excursions according to their number of accidents, see Definition~\\ref{def-accident}. \nLet $x$ be a point at a beginning of an excursion. \n \nLet $B_{0}:=0=b_{0}$, $B_1 := b_1>b_{0},\\ B_2 := b_1+b_2>b_{1},\\ \nB_3 := b_1+b_2+b_3, \\dots ,\nB_M := b_1+b_2+ \\dots + b_M$\nbe the times of accidents in the excursion.\nThere is $y_0 \\in {\\mathbb K}$ such that $x$ shadows $y_0$ at the beginning\nof the excursion, say for $d_0$ iterates.\nLet $y_i \\in {\\mathbb K}$, $i = 1, \\dots, M$, be the points that $x$ starts to shadow \nat the $i$-th accident, for $d_i$ iterates.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{excursion.pdf}\n\\caption{Accidents during an excursion.}\n\\label{fig-excursion}\n\\end{center}\n\\end{figure}\n\nThen by Lemma~\\ref{lem-accident-bispecial}, $x_{b_{i+1}}\\ldots x_{d_{i}}$ is bi-special and by Proposition~\\ref{prop-bispecialfibo}, $d_i-b_{i+1} = F_{n_{i+1}}-2$ for some $n_{i+1}$. \n\n\\begin{remark}\\label{rem-yispe}\nWe emphasize that the first $d_i$ entries of $y_i$\ndo not form a special word. Indeed, it is neither right-special (due to\nLemma~\\ref{lem-accident-bispecial}) nor left-special, because otherwise there would be an accident earlier. \n\\hfill $\\blacksquare$\\end{remark}\n\nIf there are $M+1$ accidents (counting the first as 0), the ergodic sums  for $\\varphi$ are \n\\begin{eqnarray*}\n(S_{b_{i+1}}\\varphi)(\\sigma^{B_{i}} (x) ) &=& \n \\sum_{k=0}^{b_{i+1}-1}\\varphi \\circ \\sigma^{B_{i}+k} (x)\\\\\n &=& \n \\sum_{k=0}^{b_{i+1}-1} -\\log \\frac{d_{i}+1-k}{d_{i}-k} \\\\\n&=& -\\log \\frac{d_{i}+1}{d_{i}+1-b_{i+1}} = -\\log(1+  \\frac{b_{i+1}}{d_i+1-b_{i+1}}),\n\\end{eqnarray*}\nfor $0\\le i\\le M-1$, while the ergodic sum of the tail of the excursion is\n\n\\begin{equation}\n\\label{equ-estiEM+1}\n(S_{d_M}\\varphi)(\\sigma^{B_M} (x)) =  \n\\sum_{k=0}^{d_M-1} \\varphi \\circ \\sigma^{B_M+k}( x) =\n-\\log \\frac{d_M+1}{N+1}.\n\\end{equation}\n\nWe set $}%{\\displaystyle \\mathbf{ e}_{i} := e^{\\beta \\cdot (S_{b_{i}}\\varphi)(\\sigma^{B_{i-1}} (x) )}$ for $i=1\\ldots M$ and $}%{\\displaystyle \\mathbf{ e}_{M+1} := e^{\\beta \\cdot (S_{d_M}\\varphi)(\\sigma^{B_M} (x))}$. \nComputing $C_{E}$, we can order excursions according to their number of accidents ($M+1$) and then according to the contribution of each accident. Let $E_{i}$ stand for the total contribution of all possible $\\mathbf{ e}_{i}$'s \nbetween accidents $i-1$ and $i$. \nThen\n\\begin{equation}\\label{equ-defCE}\nC_{E}=\\sum_{M=0}^{+\\infty}\\prod_{i=1}^{M+1}E_{i}.\n\\end{equation}\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 3}\\label{subsubsec-computCE}\nLet us now find an upper bound for $E_{i}$. \nBy definition, $E_{i}$ is the sum over the possible $d_{i-1}$ and $b_{i}$ of $\\mathbf{ e}_{i}$. \n\nRecall $d_{i-1}-b_{i}=F_{n_{i}}-2$, so $b_{i}$ and $F_{n_{i}}$\ndetermine $d_{i-1}$. \nThe key idea is that $F_{n_{i}}$ and $F_{n_{i+1}}$ determine the possible values of $b_{i}$. \nThis implies that $E_{i}$ can be written as an expression over the $F_{n_{i}}$ and $F_{n_{i+1}}$. \n\n\\medskip\n$\\bullet$ For $2\\le i\\le M$ each $\\mathbf{ e}_{i}$ depends on $F_{n_{i}}$ and $b_{i}$. \nLet us show that  for $2\\le i\\le M$, $b_{i}$ depends on $n_{i}$ and $n_{i-1}$. \nIndeed, the sequence $y_{i} \\in {\\mathbb K}$ coincides for\n$F_{n_{i}}-2$ initial symbols with $\\rho$, and from entry $b_{i+1}$ has another\n$d_{i} - b_{i+1} = F_{n_{i+1}}-2$ symbols in common with the head of $\\rho$,\nbut differs from $x_{B_{i}+ d_i}$ at entry $d_i$, see Figure~\\ref{fig-excursion}.\nThus we need to find all the values of $d_i > F_{n_i}-2$ such that\n$\\rho_0 \\dots \\rho_{d_i-1}$ ends the bi-special word $\\rho_0 \\dots \\rho_{F_{n_{i+1}-3}}$ but is itself not bi-special.\nThe possible starting positions of this appearance\nof $\\rho_0 \\dots \\rho_{F_{n_{i+1}-3}}$ are the required numbers $b_{i+1}$.\n\n\\begin{lemma}\\label{lem:bij}\nLet us denote by $b_{i+1}(j)$, $j \\ge 1$, the $j$-th value that\n $b_{i+1}$ can assume. Then \n\\begin{equation}\\label{equ-estibj}\nb_{i+1}(j)\\ge\\max(F_{n_{i}}-F_{n_{i+1}},F_{n_{i}-1})+j F_{n_{i+1}-2}.\n\\end{equation}\n\\end{lemma}\n\nThis will allow us to find an upper bound for $E_{i}$ for $1\\le i\\le M-1$\nlater in this section. \n\n\\begin{proof}\nWe abbreviate the bi-special words $L_k = \\rho_0 \\dots \\rho_{F_k-3}$\nfor $k \\ge 4$. \nFor the smallest value $d_i \\ge F_{n_i}-2$ so that\n$\\rho_0 \\dots \\rho_{d_i-1}$ ends in (but is not identical to) a block $L_{n_{i+1}}$,\nthis block starts at entry:\n$$\nb_{i+1}(0) =\n\\begin{cases}\nF_{n_i}-F_{n_{i+1}} &\\text{ if } n_{i+1} < n_i \\text{ and } n_i-n_{i+1} \\text{ is even,}\\\\\nF_{n_i}-F_{n_{i+1}-1} &\\text{ if } n_{i+1} < n_i  \\text{ and } n_i-n_{i+1} \\text{ is odd,}\\\\\nF_{n_{i+1}-1} &\\text{ if } n_{i+1} \\ge n_i.\n\\end{cases}\n$$\nHowever, if $n_{i+1} < n_i$ then $d_i = F_{n_i}-2$ and if \n $n_{i+1} \\ge n_i$ then $d_i = F_{n_{i+1}+1}-2$ in this case, and thus\n$\\rho_0 \\dots \\rho_{d_i-1}$ is right-special, contradicting\nLemma~\\ref{lem-accident-bispecial}.\nTherefore we need to wait for the next appearance of $L_{n_{i+1}}$.\nFor the Rauzy graph of the Fibonacci shift, the bi-special word $L_k$\nis the single node connecting loops of length $F_{k-1}$ and $F_{k-2}$,\nsee \\cite[Section 1]{arnoux-rauzy}. Therefore the gap between two \nappearances of $L_k$ is always $F_{k-2}$ or $F_{k-1}$.\nThis gives $b_{i+1}(j+1) \\ge b_{i+1}(j) + F_{n_{i+1}-2}$ for all $j \\ge 0$\nand \\eqref{equ-estibj} follows.\n\\end{proof}\n\n$\\bullet$ For $i=1$, formula \\eqref{equ-estibj} can be applied, if we introduce the quantity $n_{0}$, coinciding with the overlap of the end of the previous \n``fictitious'' word, say $y_{-1}$. The point is that $y_{0}$ is the ``beginning'' of the excursion, thus the first accident. Then, $F_{n_{0}}\\le N$ and $F_{n_{1}}>N$, which yields $n_{0}<n_{1}$. Therefore $b_{1}=F_{n_{0}}-2+}%{\\displaystyle \\frac{j}\\gamma(F_{n_{1}}-2)$ with $j\\ge 0$. \n\n$\\bullet$ The estimation \n$$\nE_{M+1} = \\sum_{d \\ge 1} e^{-\\beta \\log(\\frac{F_{N_m}+d}{N+1})}\n= \\sum_{d \\ge 1} \\left( \\frac{F_{N_m}+d}{N+1} \\right)^{-\\beta}\n\\le \\frac{N+1}{\\beta-1} \\left( \\frac{F_{N_m}}{N+1} \\right)^{1-\\beta}\n$$ \nfollows from by \\eqref{equ-estiEM+1}, with $d_M=F_{n_M}+d$ and $d\\ge 1$. \n\n\\medskip\nRecall that within excursions, all $F_{n_{j}} \\ge N+1$\nfor all $j$, where $N$ can be chosen as large as we want.\nBy Binet's formula $F_n = \\frac{1}{\\sqrt{5}}(\\gamma^{n+1}-(-1\/\\gamma)^{n+1})$, \nwe can replace $F_{n}$ by $}%{\\displaystyle\\frac{\\gamma^{n+1}}{\\sqrt 5}$, and  \nwe can also treat the quantities $-1$ as negligible compared to $\\gamma^{n_i}$.  \nTherefore, assuming that $\\beta > 1$, we obtain\n\\begin{eqnarray*}\nE_{i} &=& \\sum_{j \\ge 1} \ne^{}%{\\displaystyle-\\beta  \\log\\left(1 +  \\frac{ \\max(F_{n_{i}}-F_{n_{i+1}},F_{n_{i}-1}) + j F_{n_{i+1}-2}}{F_{n_{i+1}}-1}\\right)} \\\\\n&\\simeq& \\sum_{j \\ge 1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )+ j\/\\gamma^2\\right)^{-\\beta}\\\\\n&\\le& \\frac{\\gamma^2}{\\beta-1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}.\n\\end{eqnarray*} \nfor $2\\le i\\le M$\n\nLet $P\\approx }%{\\displaystyle \\frac{\\log\\frac{N}{\\sqrt5}}{\\log\\gamma}$ be the largest integer $n$ such that $F_{n}\\le N$. Then \\eqref{equ-defCE} yields \n\\begin{align}\\label{eq:est0}\n\\nonumber C_{E}\\le & \\sum_{M=0}^{+\\infty}\\left(\\frac{\\gamma^2}{\\beta-1}\\right)^{M}\\frac{(N+1)}{\\beta-1} \\ \\cdot \\\\\n& \\sum_{\\stackrel{n_{1},\\ldots n_{M}> P}{n_{0}\\le P}} E_{1}\\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\gamma^{(P-n_{M})(\\beta-1)}.\n\\end{align}\n\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 4}\\label{subsubsec-CEsmall}\nWe show that $C_{E}\\to0$ as $\\beta\\to+\\infty$. \n\n\n\\begin{proposition}\n\\label{prop-Cezero}\nThere exists $A=A(\\beta)\\in (0,1)$ with $\\lim_{\\beta\\to+\\infty} A = 0$ such that \n$$\nC_{E}\\le 2P\\, \\frac{N+1}{\\beta-1}\\sum_{n=1}^{+\\infty}\\gamma^{-n(\\beta-1)}\\sum_{M=0}^{+\\infty}A^{M}\\sum_{i=0}^{M}\\frac{n^{i}}{i!}.\n$$\n\\end{proposition}\n\nBefore proving this proposition, we show that is finishes the proof of Proposition~\\ref{prop-spectral&zc}. \nThe series has only positive terms. Clearly, $}%{\\displaystyle \\sum_{M=0}^{+\\infty}A^{M}\\sum_{i=0}^{M}\\frac{n^{i}}{i!}\\le \\frac1{1-A}e^{n}$, so the main sum converges if\n$\\gamma^{\\beta-1}>e$. Thus Proposition~\\ref{prop-Cezero} implies that $C_{E}\\to 0$ as $\\beta\\to+\\infty$. \n\nTherefore, inequality \\eqref{equ1-upperboundCL0} shows that if $\\beta\\to+\\infty$, \nthen $\\lambda_{0,\\beta}\\to0$ too, and hence Proposition~\\ref{prop-spectral&zc} is proved. \n\nThe rest of this subsection is then devoted to the proof of Proposition~\\ref{prop-Cezero}. \n\\begin{lemma}\n\\label{lem-gamma-n}\nLet $\\eta$ and $y$ be positive real numbers. Then for every $n$, \n$$\\int_{y}^{\\infty}x^{n}e^{-\\eta (x-y)}dx=\\sum_{j=0}^{n}\\frac{n!}{j!}\\frac{y^{j}}{\\eta^{n+1-j}}.$$\n\\end{lemma}\n\\begin{proof}\nSet $u_{n}:=}%{\\displaystyle \\int_{y}^{\\infty}x^{n}e^{-\\eta (x-y)}dx$. \nThen\n\\begin{eqnarray*}\nu_{n}&=& \\int_{0}^{\\infty}(x+y)^{n}e^{-\\eta x}dx\\\\\n&=&\\left[\\frac{-1}{\\eta}(x+y)^{n}e^{-\\eta x}\\right]_{0}^{\\infty}+\\frac{n}\\eta\\int_{0}^{\\infty}(x+y)^{n-1}e^{-\\eta x}\\\\\n&=& \\frac{y^{n}}\\eta+\\frac{n}\\eta u_{n-1}.\n\\end{eqnarray*}\nThe formula follows by induction.\n\\end{proof}\n\n\nLet $n$ be some positive integer and $\\xi$ and $\\zeta$ two positive real numbers. We consider a matrix $D_{n} = (d_{n,i,j})_{i=1, j=1}^{n+1, n}$ with $n+1$ rows and $n$ columns defined by \n$$\nd_{n,i,j}:= \\begin{cases}\n \\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1} & \\text{ if } i \\le j,\\\\[1mm]\n \\frac\\xi{j} & \\text{ if }i=j+1,\\\\[1mm]\n 0 &\\text{ if }i>j+1.\n\\end{cases}\n$$\nor in other words:\n$$\nD_n=\\left(\\begin{array}{cc cc ccc}\n0!\\zeta & 1!\\zeta^2 & 2!\\zeta^3 & \\ldots & (j-1)!\\zeta^j & \\ldots &   (n-1)!\\zeta^{n} \\\\\n\\xi & \\zeta & \\ldots &  &  &  & (n-2)!\\zeta^{n-1} \\\\\n0 & \\frac\\xi2 & \\zeta &  &  &  & \\vdots \\\\\n0 & 0 & \\frac\\xi3 & \\ddots & \\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1} & &  \\vdots\\\\\n\\vdots &  & 0 & \\ddots & \\ddots &  & \\vdots\\\\\n\\vdots &  & & 0 & \\frac{\\xi}{j} & \\zeta & \\zeta^2 \\\\\n0 &  &  &  & 0 & \\frac{\\xi}{n-1} & \\zeta \\\\\n0 & 0 & \\ldots & \\ldots & 0 & 0 & \\frac{\\xi}{n} \n\\end{array}\\right).\n$$\nWe call $\\mathbf{ w}$ non-negative (and write $\\mathbf{ w}\\succeq 0$) if all a entries of $\\mathbf{ w}$ are non-negative. This defines a partial ordering on vectors by \n$$\\mathbf{ w}' \\succeq \\mathbf{ w} \\iff \\mathbf{ w}'-\\mathbf{ w}\\succeq 0.$$  \n\n\\begin{lemma}\n\\label{lem-matrix}\nAssume $0<\\zeta<1$ and \nset $K:=\\frac1{1-\\zeta}$. Then, for every $n$,\n$$\nD_n \\cdot \\left(\\begin{array}{c}}%{\\displaystyle\\frac{K^{n-1}}{0!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{1!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{2!} \\\\[1mm]\n\\vdots \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{(n-1)!}\\end{array}\\right)\n\\preceq\n\\left(\\begin{array}{c}}%{\\displaystyle\\frac{K^{n}}{0!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{1!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{2!} \\\\[1mm]\n\\vdots \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{n!}\\end{array}\\right).$$\n\\end{lemma}\n\n\\begin{proof}\nThis is just a computation. For the first row we get \n$$\n\\sum_{j=1}^{n}(j-1)!\\zeta^{j}.\\frac{K^{n-1}}{(j-1)!}\\le K^{n-1}.\\frac\\zeta{1-\\zeta}\\le K^{n}.$$\nFor row $i>1$ we get \n$$\\frac1{(i-1)}\\frac{K^{n-1}}{(i-2)!}+\\sum_{j=i}^{n}\\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1}\\frac{K^{n-1}}{(j-1)!}=\\frac{K^{n-1}}{(i-1)!}\\left(1+\\zeta+\\zeta^{2}\\ldots\\right)\\le \\frac{K^{n}}{(i-1)!}.$$\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{prop-calcul-matrix-majo}\nSet $\\zeta:=}%{\\displaystyle\\frac1{(\\beta-1)\\log\\gamma}$ and $K=\\frac{1}{1-\\zeta}$. \nConsider $M$ integers $n_{1},\\ldots n_{M}$, with $n_{M}> P$. Then, \nfor every $M\\ge 2$, \n$$\n\\sum_{n_1, \\dots, n_{M-1} > P} \\prod_{i=1}^{M-1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} ) \\right)^{1-\\beta}\\le K^{M-1}\\sum_{i=0}^{M-1}\\frac{(n_{M}-P)^{i}}{i!}$$\n\\end{proposition}\n\\begin{proof}\nNote that \n\\begin{align*}\n\\sum_{n_1, \\dots, n_{M-1} > P} & \\prod_{i=1}^M \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\\\\n= & \\sum_{n_{M-1}=1}^{\\infty}\\left(\\ldots\\left(\\sum_{n_{2}=1}^{\\infty}\\left(\\sum_{n_{1}=1}^{\\infty}\\right.\\right.\\right.\n\\left(1+\\max(\\gamma^{n_{1} - n_{2}}-1,\\gamma^{n_{1}-n_{2}-1} ))^{1-\\beta}\\right)\\cdot\\\\\n& \\left.\\left((1+\\max(\\gamma^{n_{2} - n_{3}}-1,\\gamma^{n_{2}-n_{3}-1} ))^{1-\\beta}\\right)\\ldots\\right) \\\\\n& \\left(1+\\max(\\gamma^{n_{M-1} - n_{M}}-1,\\gamma^{n_{M-1}-n_{M}-1} \\right)^{1-\\beta}.\n\\end{align*}\nThis means that we can proceed by induction. Now \n\\begin{align*}\n\\sum_{n_{1}=P+1}^{\\infty} & (1+\\max(\\gamma^{n_{1} - n_{2}}-1,\\gamma^{n_{1}-n_{2}-1} ))^{1-\\beta}\\\\\n&\\le\\int_{P}^{n_{2}} (1+\\gamma^{x-n_{2}-1} ))^{1-\\beta}dx\n+\\int_{n_{2}}^{\\infty} (\\gamma^{x - n_{2}})^{1-\\beta}dx\\\\\n&\\le n_{2}-P+\\int_{n_{2}}^{\\infty}e^{-(\\beta-1)\\,(x-n_{2})\\,\\log\\gamma}\\,dx\\\\\n&= n_{2}-P+\\int_{n_{2}}^{\\infty}e^{-\\frac{x-n_{2}}{\\zeta}}\\,dx,\n\\end{align*}\nbecause $\\zeta=\\frac1{(\\beta-1)\\log\\gamma}$. This shows that the result holds for $M=2$.\n\nAssuming that the sum for $M=p$ is of the form \n$\\sum_{j=0}^{p-1}a_{j}(n_{p}-P)^{j}$, we compute the sum for $M=p+1$.\n\\begin{align*}\n\\sum_{n_{p}=P+1}^{\\infty} & \\sum_{j=0}^{p-1}  a_{j}\\frac{(n_{p}-P)^{j}}{(1+\\max(\\gamma^{n_{p}-n_{p+1}}-1,\\gamma^{n_{p}-n_{p+1}-1} ))^{\\beta-1}} \\\\ \n&\\le\\ \\sum_{j}a_{j}\\int_{P}^{n_{p+1}}\\frac{(x-P)^{j}}{(1+\\gamma^{x-n_{p+1}-1})^{\\beta-1}}\\,dx+\n\\sum_{j}a_{j}\\int_{n_{p+1}}^{\\infty}\\frac{(x-P)^{j}}{(\\gamma^{x-n_{p+1}})^{\\beta-1}}\\,dx\\\\\n&\\le \\sum_{j}\\frac{a_{j}(n_{p+1}-P)^{j+1}}{(j+1)}+\\int_{n_{p+1}}^{\\infty}(x-P)^{j}e^{-\\frac{x-n_{p+1}}{\\zeta}}\\,dx.\n\\end{align*}\nSet $}%{\\displaystyle \\mathbf{ w}\\cdot\\mathbf{ w}'=\\sum w_{i}w'_{i},$\nfor vectors $\\mathbf{ w}=(w_{1},\\ldots, w_{p+1})$ and $\\mathbf{ w}'=(w'_{1},\\ldots, w'_{p+1})$. \nLemma~\\ref{lem-gamma-n} yields \n \\begin{align*}\n\\sum_{n_{p}=P+1}^{\\infty} & \\sum_{j=0}^{p-1} a_{j}\\frac{(n_{p}-P)^{j}}{(1+\\max(\\gamma^{n_{p}-n_{p+1}}-1,\\gamma^{n_{p}-n_{p+1}-1}))^{\\beta-1}}\\\\\n&\\le \\sum_{j}\\frac{a_{j}}{(j+1)}(n_{p+1}-P)^{j+1} + \n\\sum_{i=0}^{j}\\frac{j!}{i!}\\zeta^{j-i+1}(n_{p+1}-P)^{i}\\\\\n&\\le D_{p}\\left(\\begin{array}{c}a_0 \\\\a_1 \\\\ \\vdots \\\\ a_{p-1}\n\\end{array}\\right) \\cdot \n\\left(\\begin{array}{c}1 \\\\n_{p+1} \\\\\\vdots \\\\n_{p+1}^{p}\\end{array}\\right).\n\\end{align*}\nLemma~\\ref{lem-matrix} concludes the proof of the induction. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop-Cezero}]\nWe have just proven that \n\\begin{align*}\n\\sum_{n_{1},\\ldots n_{M}>P} & \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\gamma^{(P-n_{M})(\\beta-1)} \\\\\n& \\le K^{M-1}\\sum_{n_{M}=P+1}^{+\\infty}\\sum_{j=0}^{M-1}\\frac{(n_{M}-P)^{j}}{j!}\\gamma^{(n_{M}-P)(\\beta-1)}.\n\\end{align*}\n\nIt remains to sum over $n_{0}$. Note that in that case, there are only $P$ terms of the form \n$}%{\\displaystyle \\sum_{j=0}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}+\\frac{j}{\\gamma}\\right)^{\\beta}}$ because $n_{0}\\le P<n_{1}$ for each possible $n_{0}$, \n\\begin{eqnarray*}\n}%{\\displaystyle \\sum_{j=0}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-1}+\\frac{j}{\\gamma^2}\\right)^{\\beta}}&=&\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}\\right)^{\\beta}}+}%{\\displaystyle \\sum_{j=1}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}+\\frac{j}{\\gamma^2}\\right)^{\\beta}}\\\\\n&\\le & 1+\\frac{\\gamma^2}{\\beta-1}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}\\right)^{\\beta-1}}\\\\\n&\\le & 1+\\frac{\\beta-1}2,\n\\end{eqnarray*}\nfor $\\beta \\ge \\sqrt{2} \\gamma$.\n Finally, inequality \\eqref{eq:est0} yields \n$$C_{E}\\leq 2P\\, \\frac{N+1}{\\beta-1}\\sum_{M=0}^{+\\infty}A^{M}\\sum_{n=1}^{+\\infty}\\gamma^{n(1-\\beta)}\\sum_{j=0}^{M-1}\\frac{n^{j}}{j!},$$\nwith $}%{\\displaystyle A:=\\frac{\\gamma}{\\beta-1}K=\\frac{\\gamma}{\\beta-1-\\frac1{\\log\\gamma}}$. This tends to $0$ as $\\beta \\to +\\infty$. \n\\end{proof}\n\n\n\\subsection{End of the proof of Theorem~\\ref{theo-pt}}\n\n\\subsubsection{End of the case $-\\log\\frac{n+1}n$}\nProposition~\\ref{prop-spectral&zc} shows that there exists some minimal $\\beta_{0}$ such that $\\lambda_{0,\\beta}<1$ for every $\\beta>\\beta_{0}$. \nThis also shows that $\\CP(\\beta)=0$ for $\\beta>\\beta_{0}$. \nSince $\\CP(\\beta)$ is a continuous and convex function, it is constant for \n$\\beta>\\beta_{0}$. As $\\CP(0)=\\log2$, there exists \na minimal $\\beta_{c} > 0$ such that $\\CP(\\beta)>0$ for every $0\\le \\beta<\\beta_{c}$. \nClearly, $\\beta_{c}\\le \\beta_{0}$. \n\nWe claim that for $\\beta<\\beta_{c}$, there exists a unique equilibrium state and that it has full support. Indeed, there exists at least one equilibrium state, say $\\mu_{\\beta}$, and at least one cylinder, say $J$, has positive $\\mu_{\\beta}$-measure. \nTherefore, we can induce on this cylinder, and the form of potential (see \\cite[Theorem 4]{leplaideur-synth}) shows that there exists a unique local equilibrium state. It is a local Gibbs measure and therefore $\\mu_{\\beta}$ is uniquely determined on each cylinder, and unique  and with full support (due to the mixing property). \n\nWe claim that the pressure function $\\CP(\\beta)$ is analytic on $[0,\\beta_{c}]$. \nIndeed, each cylinder $J$ has positive $\\mu_{\\beta}$-measure and the associated $Z_{c}(\\beta)$ is the pressure of the dotted system (that is: restricted to the trajectories that avoid $J$). This set of trajectories has a pressure strictly smaller than $\\CP(\\beta)$ because otherwise, several equilibrium states would coexist. \nTherefore $\\CP(\\beta)$ is determined by the implicit equation $\\lambda_{\\CP(\\beta),\\beta}=1$ and $\\CP(\\beta)>Z_{c}(\\beta)$ for $\\beta \\in [0,\\beta_{c}]$. \nThe Implicit Function Theorem shows that $\\CP(\\beta)$ is analytic. \n \nFor $\\beta\\ge \\beta_{c}$, the pressure $\\CP(\\beta)=0$ and for cylinders $J$ as above, we have $Z_{c}(\\beta)\\ge 0$. This shows that $Z_{c}(\\beta)=0$ for every $\\beta\\ge \\beta_{c}$. Due to the form of the potential, $\\lambda_{0,\\beta}$ is continuous and decreasing  in $\\beta$. \n\nWe claim that $\\beta_{c}=\\beta_{0}$. \nIndeed, assume by contradiction $\\beta<\\beta_{c}$. Then $\\lambda_{0,\\beta_{c}}>1$, since  otherwise (because $\\lambda_{0,\\beta}$ being strictly decreasing in $\\beta$), \n$\\lambda_{0,\\beta_{c}}\\le 1$ would yield that $\\lambda_{0,\\beta}<1$ for every $\\beta>\\beta_{c}$.\nThis would imply $\\beta_{c} \\ge \\beta_{0}$ (recall that $\\beta_{0}$ is minimal with this property). \nNow, for fixed $\\beta$, $Z\\mapsto \\lambda_{Z,\\beta}$ is continuous and strictly decreasing and goes to $0$ at $Z\\to+\\infty$. Therefore, if $\\lambda_{0,\\beta_{c}}>1$ then there exists $Z>0$ such that $\\lambda_{Z,\\beta_{c}}=1$. The local equilibrium state for this $Z$ generates\n some $\\sigma$-invariant probability measure\\footnote{Since $Z_{c}(\\beta_{c})=\\CP(\\beta_{c})=0<Z$, the expectation of the return time is comparable to $\\left|\\frac{\\partial\\CL_{Z,\\beta_{c}}({1\\!\\!1}_{J})}{\\partial Z}\\right|$, which converges.} with pressure for $\\beta \\varphi$ equal to $Z$, thus positive, and this is in contradiction with $\\CP(\\beta_{c})=0$. \nThis proves that $\\beta_{c}=\\beta_{0}$ and finishes the proof of Theorem~\\ref{theo-pt} in the case that $V(x) = -\\log \\frac{n+1}{n}$. \n\n\\subsection{The general case \\boldmath $V\\in \\CX_{1}$ \\unboldmath}\nFor $V \\in \\CX_1$, there exists $\\kappa>0$ such that \n$$-V \\le \\kappa\\varphi.$$\n This shows that the pressure function is constant equal to zero for $\\beta\\ge \\beta_{0}\/\\kappa$. Again, the pressure is convex, thus non-increasing and continuous. We can define $\\beta'_{c}$ such that $\\CP(\\beta)>0$ for$0\\le\\beta\\le \\beta'_{c}$ and $\\CP(\\beta)=0$ for $\\beta\\ge \\beta'_{c}$. \n \n The rest of the argument is relatively similar to the previous discussion. We deduce that for $\\beta<\\beta'_{c}$, there exists a unique equilibrium state, it has full support and $\\CP(\\beta)$ is analytic on this interval. For $\\beta\\ge \\beta'_{c}$, it is not clear that $\\lambda_{0,\\beta}$ decreases in $\\beta$. However,\nwe do not really need this argument, because if $\\lambda_{0,\\beta}>1$, then the decrease of $Z\\mapsto \\lambda_{Z,\\beta}$ (which follows from convexity argument and $}%{\\displaystyle\\lim_{Z\\to+\\infty}\\lambda_{Z,\\beta}=0)$, is sufficient to produce a contradiction. \n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\n\nThe main paradigm for adapting pretrained models for classification~\\cite{GPT, UniLM, BERT} is fine-tuning via an explicit classifier head. However, an alternative approach has arisen: adapting the pretrained language model directly as a predictor through autoregressive text generation~\\cite{GPT2} or completion of a cloze task~\\cite{ASimpleMethod}. This method is notably used in T5 fine-tuning~\\cite{T5} leading to state-of-the-art results on the SuperGLUE benchmark~\\cite{SuperGLUE}.\n\n\\blfootnote{Code available at \\url{https:\/\/github.com\/TevenLeScao\/pet}}\n\n\nOne argument made for classification by direct language generation is that it allows us to pick custom \\textit{prompts} for each task~\\cite{Decathlon}. \nWhile this approach can be used for zero-shot classification~\\cite{Zeroshot} or priming~\\cite{GPT3}, it can also be used in fine-tuning to provide extra task information to the classifier, especially in the low-data regime~\\cite{PET, PET2}.\n\nIf this argument is indeed true, it is natural to ask how it impacts the sample efficiency of the model, or more directly, \\textit{how many data points is a prompt worth?} As with many low-data and pretraining-based problems, this \nquestion is complicated by the fine-tuning setup, training procedure, and prompts themselves. We attempt to isolate these variables through diverse prompts, multiple runs, and best practices in low-training data fine-tuning. We introduce a metric, the \\textit{average data advantage}, for quantifying the impact of a prompt in practice.\n\n\nOur experiments find that the impact of task-targeted prompting can nicely be quantified in terms of direct training data, and that it varies over the nature of different tasks. On MNLI~\\cite{MNLI}, we find that using a prompt contributes approximately 3500 data points. On SuperGLUE, it adds approximately 280 data points on RTE~\\cite{RTE} and up to 750 on BoolQ~\\cite{BoolQ}. In low- to medium-data settings, this advantage can be a real contribution to training a model. \n\n\n\\section{Related Work}\n\n\nPrompting has been used both for zero-shot and fine-tuning based methods. Zero-shot  approaches attempt to answer a task with a prompt without fine-tuning through generation~\\cite{GPT2}. GPT3~\\cite{GPT3} extends this approach to a supervised priming method by taking in training data as priming at inference time, so it can attend to them while answering. T5~\\cite{T5} and other sequence-to-sequence pretrained models use standard word-based fine-tuning with a marker prompt to answer classification tasks with strong empirical success. Our setting differs in that we are interested in using task-based prompts and fine-tuning, in-between the T5 and GPT2 setting. \n\nOur setting most closely resembles PET~\\cite{PET, PET2}, which claims that task-specific prompting helps transfer learning, especially in the low-data regime. However, in order to reach the best possible results on SuperGLUE, PET introduces several other extensions: semi-supervision via additional pseudo-labeled data, ensembling models trained with several different prompts, and finally distilling the ensemble into a linear classifier rather than a language model. Our aim is to isolate the specific contributions of prompting within supervised fine-tuning. \n\nFinally, recent papers have experimented with discovering prompts through automated processes tailored to the language model~\\cite{HowCanWeKnow, AutomaticVerbalizer}. We limit ourselves to human-written prompts, as we are interested into \nwhether prompting itself specifically adds information to the supervised task. It is an interesting question as to whether automatic prompts can have this same impact (relative to the training data they require). \n\n\\section{Comparison: Heads vs Prompts}\n\n\n\nConsider two transfer learning settings for text classification: \\textit{head-based}, where a generic head \nlayer takes in pretrained representations to predict an output class; \\textit{prompt-based}, where a \ntask-specific pattern string is designed to coax the model into producing a textual output corresponding to \na given class. Both can be utilized for fine-tuning with supervised training data, but prompts further allow\nthe user to customize patterns to help the model. \n\nFor the \\textit{prompt} model we follow the notation from PET~\\cite{PET} and decompose a prompt into a \\textit{pattern} and a \\textit{verbalizer}. The \\textit{pattern} turns the input text into a cloze task, i.e. a sequence with a masked token or tokens that need to be filled. Let us use as example an excerpt from SuperGLUE task BoolQ~\\cite{BoolQ}, in which the model must answer yes-or-no binary questions. In order to let a language model answer the question in \\textit{italics}, our pattern is in \\textbf{bold}~\\cite{PET2}:\n\n\\begin{quote}\n\\small\n \"Posthumous marriage -- Posthumous marriage (or necrogamy) is a marriage in which one of the participating members is deceased. It is legal in France and similar forms are practiced in Sudan and China. Since World War I, France has had hundreds of requests each year, of which many have been accepted.\n\\textbf{Based on the previous passage, \\textit{can u marry a dead person in france ?} <MASK>}\"\n\\end{quote}\n\n\n\nThe masked word prediction is mapped to a \\textit{verbalizer} which produces a class. (here \"Yes\": True. \"No\": False\\footnote{The correct answer here is, of course, \\textit{yes}. Originated in 1803 as Napoleon rose to power, this practice was mainly to the benefit of war widows.}). \nSeveral \\textit{pattern-verbalizer pairs} (\\textit{PVPs}) could be used for a single task, differing either through the pattern, the verbalizer, or both. Fine-tuning is done by training the model to produce the correct verbalization. The loss is the cross-entropy loss between the correct answer and the distribution of probabilities amongst the tokens in the verbalizer. We re-use pattern choices from~\\citet{PET2}; examples are available in Appendix~\\ref{prompts}.\n\n\n\n\n\n\n\n\\section{Experimental Setting}\n\n\nWe run all experiments with the same pretrained checkpoint, \\textit{roberta-large} (355M parameters) from RoBERTa~\\cite{Roberta}, which we load from the \\textit{transformers}~\\cite{Transformers} library.\\footnote{After experimenting with RoBERTa, AlBERT~\\cite{Albert} and BERT~\\cite{BERT}, we found \\textit{roberta-large} to have the most consistent performance.}\nIn line with previous observations~\\cite{Feather,Finetuning,Mixout}, head-based fine-tuning performance varies considerably. We follow recommendations of~\\citet{Stability} and~\\citet{Revisiting} to train at a low learning rate ($10^{-5}$) for a large number of steps (always at least $250$, possibly for over 100 epochs).\n\n\n\n\n\n\n\\begin{figure*}[h!]\n\\centering\n\\hspace*{-1cm}\\includegraphics[width=1.1\\textwidth]{Graphs\/Master_figure.png}\n\\caption{Prompting vs head (classifier) performance across data scales, up to the full dataset, for six SuperGLUE tasks. Compares the best prompt and head performance at each level of training data across 4 runs. Highlighted region shows the accuracy difference of the models. Cross-hatch region highlights the lowest- and highest- accuracy matched region in the curves. The highlighted area in this region is used to estimate the data advantage. }\n\\label{main_figure}\n\\end{figure*}\n\nWe perform our evaluation on SuperGLUE and MNLI~\\cite{MNLI}. These datasets comprise a variety of tasks, all in English, including entailment (MNLI, RTE~\\cite{RTE}, CB~\\cite{CB}), multiple choice question answering (BoolQ~\\cite{BoolQ}, MultiRC~\\cite{MultiRC}), and common-sense reasoning (WSC~\\cite{WSC}, COPA~\\cite{COPA}, WiC~\\cite{WiC}). We do not include ReCoRD~\\cite{ReCORD} in our comparisons as there is no head model to compare with, since it is already a cloze task. Data sizes range from $250$ data points for CB to $392,702$ for MNLI. As test data is not publicly available for SuperGLUE tasks, we set aside part of training (from $50$ for CB, COPA and MultiRC to $500$ for BoolQ) to use for development, and evaluate on their original validation sets. For MNLI, we use the available matched validation and test sets. \n\nWe compare models across a scale of available data, starting with $10$ data points and increasing exponentially (as high-data performance tends to saturate) to the full dataset. For example, for MultiRC, which has 969 data points initially, we start by reserving 50 data points for development. This leaves us with 919 training points, and we train models with 10, 15, 20, 32, 50, 70, 100, 150, 200, 320, 500, 750, and 919 training points. We run every experiment 4 times in order to reduce variance, for a total of 1892 training runs across all tasks. At every point, we report the best performance that has been achieved at that amount of data or lower. Full graphs are presented in Appendix~\\ref{reduction}.\n\n\n\n\\section{Results}\n\n\n\n\n\nFigure~\\ref{main_figure} shows the main results comparing head- and prompt-based fine-tuning with the best-performing pattern on that task. \nPrompting enjoys a substantial advantage on every task, except for WiC as is reported in previous results~\\cite{PET2}.\nBoth approaches improve with more training data, but prompting remains better by a varying amount. Many tasks in SuperGLUE have relatively few data points, but we also see an advantage in large datasets like BoolQ and MNLI.\n\nTo quantify how many data points the prompt is worth, we first isolate the $y$-axis band of the lowest- and highest- accuracy where the two curves match in accuracy.\\footnote{We assume asymptotically the two curves would match, but are limited by data.} The horizontal line at these points represents the advantage of prompting. We then take the integral in this region, i.e. area between the linearly-interpolated curves\\footnote{In areas where the head model is better, if any, get subtracted from the total.}, divided by the height of the band. The area has the dimension of a quantity of data points times the metric unit, so dividing by the performance range yields a \\# of data points advantage. \nAs low data training is sensitive to noise, in addition to following best training practices we \nrun several different experiments for each $x$-point. We use a bootstrapping approach to estimate confidence over these runs. Specifically, we hold out one of the 4 head runs and 4 prompt runs (16 combinations total), and compute the standard deviation of those outcomes.\n\n\nWe report these quantities for every task in Table~\\ref{main_table} as \\textit{Average advantage}. For almost all the tasks, we see that prompting gives a substantial advantage in terms of data efficiency, adding the equivalent of hundreds of data points on average.\n\n\n\n\n\n\n\n\\begin{table*\n\n\\hspace*{-0.7cm}\\begin{tabular}{@{}l rrrrrrrr@{}}\n\\toprule \n& \\multicolumn{8}{c}{Average Advantage (\\# Training Points)} \\\\\n& MNLI & BoolQ & CB & COPA & MultiRC* & RTE & WiC & WSC\\\\\\midrule \\multicolumn{1}{l}{\\textit{P vs H}} & $3506\\pm536$ & $752\\pm46$ & $90\\pm2$ & $288\\pm242$ & $384\\pm378$  & $282\\pm34$ & $-424\\pm74$ & $281\\pm137$ \\\\\n\\hline\n\\multicolumn{1}{l}{\\textit{P vs N}} & $150\\pm252$ & $299\\pm81$ & $78\\pm2$ & -& $74\\pm56\\phantom{0}$ & $404\\pm68$ & $-354\\pm166$ & -\\\\\n\\multicolumn{1}{l}{\\textit{N vs H}} & $3355\\pm612$ & $453\\pm90$ & $12\\pm1$ & -& $309\\pm320$ & $-122\\pm62$ & $-70\\pm160$ & -\\\\\n\\bottomrule\n\\end{tabular}\n\n\\caption{Average prompting advantage in number of data points for MNLI \\& SuperGLUE tasks. \\textit{P} denotes the prompt model, \\textit{H} the head model. On average across performance levels, an MNLI prompt model yields the results of an MNLI head model trained with 3500 additional data points. Confidence levels are based on a multiple random runs (see text). \\textit{N} indicates a null-verbalizer prompting task that replaces the verbalizer with a non-sensical mapping.  *The comparison band of MultiRC is too small as the head baseline fails to learn beyond majority class; we use the full region for a lower-bound result.}\n\\label{main_table}\n\\end{table*}\n\n\n\n\n\n\\section{Analysis}\n\n\\paragraph{Impact of Pattern vs Verbalizer}\n\nThe intuition of prompts is that they introduce a task description in natural language,\neven with few training points. \nTo better understand the zero-shot versus adaptive nature of prompts,\nwe consider a \\textit{null verbalizer}, a control with a verbalizer that cannot yield semantic information without training. For every task that requires filling in one word (which excludes the more free-form COPA and WSC), we replace the verbalizers, for example, \"yes\", \"no\", \"maybe\", \"right\" or \"wrong\",  with random first names.\n\nTable~\\ref{main_table} shows the advantage of the standard prompt over the null verbalizer to estimate this control.\nWe see that for small data tasks such as CB, the null verbalizer removes much of the benefits of prompting. However, with more training data, the model seems to adapt the verbalizer while still gaining the inductive bias benefits of the pattern. Figure~\\ref{neutral_run_figure} showcases this dynamic on MNLI. This result further shows that prompting yields data efficiency even if it is not directly analogous to the generation process of training. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Graphs\/MNLI_partial_neutral_run.png}\n\\caption{Comparison of full prompt and null verbalizer advantage on MNLI at lower data scales.}\n\\label{neutral_run_figure}\n\\end{figure}\n\n\\paragraph{Impact of Different Prompts}\n\nIf the prompt acts as a description of the task, one would expect different valid descriptions to vary in their benefits. In order to compare the different prompts we used on each task, we chart the median performance for each of them under different runs. In nearly every experiment, we find that the confidence intervals of those curves largely overlap, implying that prompt choice is not a dominant hyperparameter, i.e. the variance across random seeds usually outweighs the possible benefits of prompt choice. One exception is the low-data regime of BoolQ, where one of the prompts enjoys a significant few-shot advantage over the others. We plot this curve for MultiRC in Figure~\\ref{median_comparison} and the rest in Appendix~\\ref{all_results}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Graphs\/median_f1_multirc.png}\n\\caption{Median performance on MultiRC across runs for three prompts. Differences are inconsistent and eclipsed by the variance within one prompt's runs.}\n\\label{median_comparison}\n\\end{figure}\n\n\\paragraph{Metric sensitivity}\n\n\nWe treat each metric linearly in calculating advantage; alternatively, we could re-parameterize the $y$ axis for each task. This choice does not have a consistent effect for or against prompting. For example, emphasizing gains close to convergence increases prompting advantage on CB and MNLI but decreases it on COPA or BoolQ. \n\n\n\n\n\n\\section{Conclusion}\n\nWe investigate prompting through a systematic study of its data advantage. Across tasks, prompting consistently yields a varying improvement throughout the training process. Analysis shows that prompting is mostly robust to pattern choice, and can even learn without an informative verbalizer. On large datasets, prompting is similarly helpful in terms of data points, although they are less beneficial in performance. In future work, we hope to study the mechanism and training dynamics of the prompting benefits.\n\n\\section{Impact statement}\n\nSignificant compute resources were used to run this paper's experiments. A single experiment (defined as one model run, at one data level, on one task) was quite light-weight, taking usually a little under an hour on a single Nvidia V100. However, as we computed a little under two thousand runs, this adds up to about 1800 GPU hours, to which one must add around 400 GPU hours of prototyping and hyper-parameter searching. Those 2200 GPU hours would usually have necessitated the release of about 400kg of CO2, about 40\\% of a transatlantic flight for a single passenger, in the country where we ran the experiments, although we used a carbon-neutral cloud compute provider.\n\nThe main benefit of prompting, rather than compute efficiency, is data efficiency. Although we ran all of our experiments on English, we hope that this property will be especially helpful in low-resource language applications. In a sense, a practitioner could then remedy the lack of task-specific data in their language by introducing information through a prompt. However, this comes with the inherent risk of introducing human biases into the model. Prompt completion also suffers from biases already present within the language model~\\cite{Babysitter}. This could cause a prompted model to repeat those biases in classification, especially in the few-shot setting where prompting mostly relies on the pretrained model.\n\n\\section{Acknowledgments}\n\nWe thank Steven Cao and Joe Davison for the discussions about prompting that initially spurred this paper. We further thank Timo Schick for making the code for PET available and for discussions about performance replication. We lastly thank Canwen Xu, Yacine Jernite, Victor Sanh, Dimitri Lozeve and Antoine Ogier for their help with the figures and writing of this draft.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe propagation and acceleration\nof cosmic rays (CRs) are governed by their interactions\nwith magnetic fields. Astrophysical magnetic fields are turbulent and, \ntherefore, the resonant and non-resonant (e.g. transient time damping, or TTD)\ninteraction of cosmic rays with MHD turbulence is the accepted\n principal mechanism to scatter and isotropize\ncosmic rays \\citep[see][]{Schlickeiser02}. In addition, efficient scattering is essential for the acceleration of cosmic rays. \nFor instance, scattering of cosmic rays back into the shock is a\nvital component of the first order Fermi acceleration \\citep[see][]{Longairbook}. At the same time, stochastic acceleration by turbulence is \nentirely based on scattering. The dynamics of cosmic rays in MHD turbulence holds the key to all high energy astrophysics and related problems. \n\nWe live in an exciting era when we are starting to test fundamental processes taking place at the beginning of the Universe, at the event horizon of black holes, when the nature of dark matter and dark energy is being probed etc. Using computers many researchers make sophisticated complex models to confront the observations in unprecedented details. In the mean time, with the launching of the new facilities, we have much more observational data available than ever before. For instance, CHANDRA observations of supernova\nremnants provide a strong constraint to diffusion coefficients and\/or magnetic fields near the shock \\citep[see, e.g.][]{Bamba05, PYL05}; \nthe diffuse\ngamma-ray measurements from Fermi from the Galactic disc have been successfully used to\nphenomenologically constrain numerical modeling of cosmic rays, e.g., with GALPROP \\citep{Ackermann12}; observations of solar\nenergetic particles (SEP) have been also fruitful over the past decades and lead to better understanding of transport in the solar\nwind \\citep[see a review by][and references therein]{Horbury05_SEP}. These developments make it urgent that we understand the key physical processes underlying astrophysical phenomena, can parameterize them and, if necessary, use as a subgrid input in our computer models.\n\n\nAt present, the propagation of the CRs is an advanced theory, which makes\nuse both of analytical studies and numerical simulations. However,\nthese advances have been done within the turbulence paradigm which\nis being changed by the current research in the field.\nInstead of the empirical 2D+slab model of turbulence, numerical\nsimulations suggest anisotropic Alfv\\'enic modes following \\cite[GS95]{GS95} scalings (an analog of 2D, but not an\nexact one, as the anisotropy changes with the scale involved) + fast modes \\citep{CL02_PRL}. These progresses resulted in important revisions on the theory of cosmic ray transport (see review by \\citealt{LBYO} and references therein). The GS95 turbulence injected on large scales and its extensions to compressible medium is less efficient in scattering of CRs compared to the estimates made assuming that magnetic turbulence consists of plane waves moving parallel to magnetic field \\citep{Chandran00, YL02}. Fast compressible modes, on the other hand, are demonstrated as the dominant scattering agent in spite of various damping processes they are subjected to \\citep{YL02, YL04, YL08}\n\nAt the same time, one should not disregard the possibilities of generation of additional perturbations on small scales by CR themselves. For instance,  the slab Alfv\\'enic perturbation can be created, e.g., via streaming instability \\citep[see][]{Wentzel74, Cesarsky80}. Instabilities induced by anisotropic distribution of CRs were also suggested as a possibility to scatter CRs \\citep[]{Lerche, Melrose74}. Particularly at shock front, studies of instabilities have been one of the major efforts since the acceleration efficiency is essentially determined by the confinement at the shock front and magnetic field amplifications. Examples of the new developments in the field include, current driven instability \\citep{Bell2004}, vorticity generation at curved shock \\citep{Giac_Jok2007}, through Baroclinic effect \\citep{Inoue09}, through precursor \\citep{BJL09}, etc. This field is rich in its own and we shall not dwell upon it in this chapter.\n\nIn fact, the small scale instabilities and large scale turbulence are not independent of each other. {\\em First} of all, the instability generated waves can be damped through nonlinear interaction with the large scale turbulence \\citep[henceforth YL02, YL04]{YL02, YL04}. In the case of anisotropic GS95 turbulence, the efficiency is reduced \\citep{FG04}. Nonetheless, owing to the non-linear damping, the instabilities can only grow in a limited range, e.g., $\\sim< 100$GeV in interstellar medium for the streaming instability \\citep{FG04, YL04}. \n{\\em Secondly}, the large scale compressible turbulence also generate small scale waves through firehose, gyroresonance instability, etc \\citep{Schek06, LB06, YL11, Santos-Lima}. \n\nPropagation of CRs perpendicular to mean magnetic field\nis another important problem for which one needs to take into account both large and small scale interactions in tested models of turbulence. Indeed, if one takes only the diffusion along the magnetic field line and field line random walk \\citep[FLRW][]{Jokipii1966, Jokipii_Parker1969, Forman1974}, compound (or subdiffusion) would arise. Whether the subdiffusion is realistic in fact depends on the models of turbulence chosen \\citep{YL08, Yan:2011valencia}. In this chapter we review current understandings to this question within the domain of numerically tested models of MHD turbulence.\n\nIn what follows, we introduce the basic mechanisms for the interactions between particles and turbulence in \\S2. We discuss the cosmic ray transport in large scale turbulence, including both analytical and numerical studies in \\S3. Applications to cosmic ray propagation is presented in \\S4. In \\S5, we consider the perpendicular transport of cosmic rays on both large and small scales. We shall also discuss the issue of super-diffusion and the applicability of sub-diffusion. In \\S6, we concentrate on the issue of self-confinement in the presence of preexisting turbulence and dwell on, in particular, the streaming instability at supernova remnant shocks and its implication for CR acceleration. \\S7, we address the issue of gyroresonance instability of CRs and its feedback on large scale compressible turbulence. Summary is provided in \\S8.\n\n\\section{Interactions between turbulence and particles}\n\\label{basics}\nBasically there are\ntwo types of resonant interactions: gyroresonance acceleration\nand transit acceleration (henceforth TTD). The resonant condition is $\\omega-k_{\\parallel}v\\mu=n\\Omega$ ($n=0, \\pm1,2...$),\nwhere $\\omega$ is the wave frequency, $\\Omega=\\Omega_{0}\/\\gamma$\nis the gyrofrequency of relativistic particle, $\\mu=\\cos\\xi$,\nwhere $\\xi$ is the pitch angle of particles. TTD formally corresponds to $n=0$ and it requires compressible perturbations.  \n\nThe Fokker-Planck equation is generally used to describe\nthe evolvement of the gyrophase-averaged distribution function $f$,\n\n\\[\n\\frac{\\partial f}{\\partial t}=\\frac{\\partial}{\\partial\\mu}\\left(D_{\\mu\\mu}\\frac{\\partial f}{\\partial\\mu}+D_{\\mu p}\\frac{\\partial f}{\\partial p}\\right)+\\frac{1}{p^{2}}\\frac{\\partial}{\\partial p}\\left[p^{2}\\left(D_{\\mu p}\\frac{\\partial f}{\\partial\\mu}+D_{pp}\\frac{\\partial f}{\\partial p}\\right)\\right],\\]\n where $p$ is the particle momentum. The Fokker-Planck coefficients\n$D_{\\mu\\mu},D_{\\mu p},D_{pp}$ are the fundamental physical parameters\nfor measuring the stochastic interactions, which are determined by\nthe electromagnetic fluctuations \\citep[see][]{SchlickeiserMiller}:\n\nGyroresonance happens when the Doppler shifted wave frequency matches the Larmor frequency of a particle. In quasi-linear theory (QLT), the Fokker-Planck\ncoefficients are given by  \\citep[see][]{SchlickeiserMiller, YL04}\n\n\\begin{eqnarray}\n\\left(\\begin{array}{c}\nD_{\\mu\\mu}\\\\\nD_{pp}\\end{array}\\right)  =  {\\frac{\\pi\\Omega^{2}(1-\\mu^{2})}{2}}\\int_{\\bf k_{min}}^{\\bf k_c}dk^3\\delta(k_{\\parallel}v_{\\parallel}-\\omega \\pm \\Omega)\n\\left[\\begin{array}{c}\n\\left(1+\\frac{\\mu V_{ph}}{v\\zeta}\\right)^{2}\\\\\nm^{2}V_{A}^{2}\\end{array}\\right]\\times\\nonumber\\\\\n\\times\\left\\{ \\left[J_{2}^{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)+J_{0}^{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)\\right]\n\\left[\\begin{array}{c}\nM_{{\\mathcal{RR}}}({\\mathbf{k}})+M_{{\\mathcal{LL}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{RR}}}({\\mathbf{k}})+K_{{\\mathcal{LL}}}({\\mathbf{k}})\\end{array}\\right]\\right.\\nonumber\\\\\n\\left.-2J_{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)J_{0}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)\n\\left[e^{i2\\phi}\\left[\\begin{array}{c}\nM_{{\\mathcal{RL}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{RL}}}({\\mathbf{k}})\\end{array}\\right]+e^{-i2\\phi}\\left[\\begin{array}{c}\nM_{{\\mathcal{LR}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{LR}}}({\\mathbf{k}})\\end{array}\\right]\\right]\\right\\} ,\\label{gyro}\n\\end{eqnarray}\nwhere $\\zeta=1$ for Alfv\\'{e}n modes and $\\zeta=k_{\\parallel}\/k$\nfor fast modes, $k_{min}=L^{-1}$, $k_c=\\Omega_{0}\/v_{th}$\ncorresponds to the dissipation scale, $m=\\gamma m_{H}$ is the relativistic\nmass of the proton, $v_{\\perp}$ is the particle's velocity component\nperpendicular to $\\mathbf{B}_{0}$, $\\phi=\\arctan(k_{y}\/k_{x}),$\n${\\mathcal{L}},{\\mathcal{R}}=(x\\pm iy)\/\\sqrt{2}$ represent left and\nright hand polarization. $M_{ij}$ and $K_{ij}$ are the correlation tensors of magnetic and velocity fluctuations.  \n\nFrom the resonance condition, we know that the most important interaction\noccurs at $k_{\\parallel}=k_{\\parallel,res}=\\Omega\/v_{\\parallel}$.\nThis is generally true except for small $\\mu$ (or scattering near\n$90^{\\circ}$). \n\nTTD happens due to the resonant interaction with parallel magnetic mirror force. Particles can be accelerated by when they are in phase with the waves either by interacting with oscillating parallel electric field (Landau damping), or by moving magnetic mirrors (TTD). When particles are trapped by moving in the same speed with waves, an appreciable amount of interactions can occur between waves and particles. Since head-on collisions are more frequent than that trailing collisions, particles gain energies. Different from gyroresonance, the resonance function of TTD is broadened even for CRs with small pitch angles. The formal resonance peak $k_{\\parallel}\/k=V_{ph}\/v_{\\parallel}$ favors quasi-perpendicular modes. However, these quasi-perpendicular modes cannot form an effective mirror to confine CRs because the gradient of magnetic perturbations along the mean field direction $\\nabla_{\\parallel}\\mathbf{B}$ is small. As we will show later in \\S\\ref{NLT_sec}, the resonance is broadened in nonlinear theory \\citep[see][]{YL08}.    \n\n\n\n\\section{Scattering of cosmic rays}\n\\label{scattering}\n\n\\subsection{Scattering by Alfv\\'{e}nic turbulence}\n\\label{Alf_scatter}\nAs we discussed in $\\S$2, Alfv\\'{e}n modes are anisotropic, eddies\nare elongated along the magnetic field, i.e., $k_{\\perp}>k_{\\parallel}$.\nThe scattering of CRs by Alfv\\'{e}n modes is suppressed first because\nmost turbulent energy goes to $k_{\\perp}$ due to the anisotropy of\nthe Alfv\\'{e}nic turbulence so that there is much less energy left\nin the resonance point $k_{\\parallel,res}=\\Omega\/v_{\\parallel}\\sim r_{L}^{-1}$.\nFurthermore, $k_{\\perp}\\gg k_{\\parallel}$ means $k_{\\perp}\\gg r_{L}^{-1}$\nso that cosmic ray particles have to be interacting with lots of eddies\nin one gyro period. This random walk substantially decreases the scattering\nefficiency. The scattering by Alfv\\'en modes was studied in YL02. In case that the pitch angle $\\xi$ not close to 0, the analytical result is  \\begin{equation}\n\\left[\\begin{array}{c}\nD_{\\mu\\mu}\\\\\nD_{pp}\\end{array}\\right]=\\frac{v^{2.5}\\mu^{5.5}}{\\Omega^{1.5}L^{2.5}(1-\\mu^2)^0.5}\\Gamma[6.5,k_c^{-\\frac{2}{3}}k_{\\parallel,res}L^{\\frac{1}{3}}]\\left[\\begin{array}{c}\n1\\\\\nm^{2}V_{A}^{2}\\end{array}\\right],\\label{ana}\\end{equation}\nwhere $\\Gamma[a,z]$ is the incomplete gamma function. The presence\nof this gamma function in our solution makes our results orders of\nmagnitude larger than those%\n\\footnote{The comparison was done with the resonant term in Chandran (2000) as the nonresonant term is spurious %\n} in \\cite{Chandran00}, who employed\nGS95 ideas of anisotropy, but lacked the quantitative\ndescription of the eddies. However,\nthe scattering frequency,\n\n\\begin{equation}\n\\nu=2D_{\\mu\\mu}\/(1-\\mu^{2}),\\label{nu}\n\\label{nu}\n\\end{equation}\nare nearly $10^{10}$ times lower than the estimates for isotropic and slab model (see Fig.~\\ref{impl} {\\em left}). {\\em It is clear that for most interstellar circumstances, the scattering by Alfv\\'enic turbulence is suppressed.} As the anisotropy of the Alfv\\'{e}n modes is increasing with the\ndecrease of scales, the interaction with Alfv\\'{e}n modes becomes\nmore efficient for higher energy cosmic rays. When the Larmor radius\nof the particle becomes comparable to the injection scale, which is\nlikely to be true in the shock region as well as for very high energy cosmic\nrays in diffuse ISM, Alfv\\'{e}n modes get important.\n\n\\subsection{Cosmic ray scattering by compressible MHD turbulence}\n\nAs we mentioned earlier, numerical simulations of MHD turbulence supported the GS95 model of turbulence,\nwhich does not have the \"slab\" Alfv\\'enic modes that produced most of the scattering in the earlier models\nof CR propagation. Can the turbulence that does not appeal to CRs back-reaction (see \\S 4) produce \nefficient scattering? \n\nIn the models of ISM turbulence \\citep[]{Armstrong95, Mckee_Ostriker2007}, where the injection happens at large scale, \nfast modes were identified as a scattering agent for cosmic rays in interstellar medium \\cite[]{YL02,YL04}.\nThese works made use of the quantitative description of turbulence\nobtained in \\cite{CL02_PRL}  to calculate\nthe scattering rate of cosmic rays. \n\nDifferent from Alfv\\'en and slow modes, fast modes are isotropic \\citep{CL02_PRL}. Indeed they are subject to both collisional and collisionless damping. The studies in \\cite{YL02, YL04} demonstrated, nevertheless, that the scattering by fast modes dominates in most cases in spite of the damping\\footnote{On the basis of weak turbulence theory, \\cite{Chandran2005} has argued that high-frequency \nfast waves, which move mostly parallel to magnetic field, generate Alfv\\'en waves also moving mostly parallel to magnetic field. We expect\nthat the scattering by thus generated Alfv\\'en modes to be similar to the scattering by the fast modes created by them. Therefore\nwe expect that the simplified approach adopted in \\cite{YL04} and the papers that followed to hold.} (see Fig.\\ref{impl} {\\em right}).\n\\begin{figure*} [h!t] \n{\\includegraphics[width=0.45\\textwidth]{YL_fig1a.eps} \n\\includegraphics[width=0.45\\textwidth]{comp.eps}\n} \n\\caption{\\small {\\em Left:} rate of CR scattering by\nAlfv\\'en waves versus CR energy.  The lines at the top of the figure are\nthe accepted estimates obtained for Kolmogorov turbulence. The dotted\ncurve is from \\cite{Chandran00}. The analytical calculations are given\nby the solid line with our numerical calculations given by\ncrosses;  {\\em Right:} the scattering by fast modes,  dashed line represents the case without damping for fast modes included, the solid and dash-dot line are the results taking into account collisionless damping.}\n\\label{impl}\n\\end{figure*}\nMore recent studies of cosmic ray propagation and acceleration that explicitly appeal to the effect of\nthe fast modes include \\citet{Cassano_Brunetti, Brunetti_Laz, YL08, YLP08}.\nIncidentally, fast modes have been also identified as primary agents for the acceleration of charged dust particles \\cite{YL03,YLD04}.\n\n\n\\subsection{Nonlinear theory of diffusion}\n\\label{NLT_sec}\n\nWhile QLT allows easily to treat the CR dynamics in a local magnetic\nfield system of reference, a key assumption in QLT, that the particle's orbit is unperturbed, makes one wonder about the limitations of the approximation. Indeed, while QLT provides simple physical insights into scattering, it is known to have problems. For instance, it fails in treating $90^\\circ$ scattering  \\citep[see][]{Volk:1973, Volk:1975, Jones:1973, Jones:1978, Owens:1974, Goldstein:1976, Felice90degree} and perpendicular transport \\citep[see][]{Kota_Jok2000, Matthaeus:2003}. \n\nIndeed, many attempts have been made to improve the QLT and various non-linear\n theories have been attempted (see \\citealt{Dupree:1966}, V\\\"olk 1973, 1975, \nJones, Kaiser \\& Birmingham 1973, Goldstein 1976). Currently we observe a surge\nof interest in finding way to go beyond QLT. Examples include the nonlinear guiding center theory \\citep[see][]{Matthaeus:2003}, second-order \nquasilinear theory \\citep{Shalchi_SQT, Qin_NLT, LeRoux:2007}, etc. Most of the analysis were limited to traditional 2D+slab models of MHD turbulence. An important step was taken in Yan \\& Lazarian (2008), where non-linear effect was accounted for in treating CR scattering in the type of MHD turbulence that are supported by numerical simulations. The results have been applied to both solar flares (Yan, Lazarian \\& Petrosian 2008) and grain acceleration \\citep{HLS12}. Below, we introduce the nonlinear theory and their applications to both particle transport and acceleration in incompressible and compressible turbulence based on the results from Yan \\& Lazarian (2008).\n\nThe basic assumption of the quasi-linear theory is that particles follow unperturbed orbits. In reality, particle's pitch angle varies gradually with the variation of the magnetic field due to conservation of adiabatic invariant $v_\\bot^2\/B$, where $B$ is the total strength of the magnetic field \\citep[see][]{Landau:1975}. Since B is varying in turbulent field, so are the projections of the particle speed $v_\\bot$ and $v_\\|$.\n This results in broadening of the resonance. The variation of the velocity is mainly caused by the magnetic perturbation $\\delta B_\\|$ in the parallel direction. This is true even for the incompressible turbulence we discussion in this section. For the incompressible turbulence, the parallel perturbation arises from the pseudo-Alfv\\'en modes. The perpendicular perturbation $\\delta B_\\bot$ is higher order effect, which we shall neglect here.\n\nThe propagation of a CR can be described as a combination of a motion of its guiding center and CR's motion about its guiding center. \nBecause of the dispersion of the pitch angle $\\Delta\\mu$ and therefore of the parallel speed $\\Delta v_\\|$, the guiding center is perturbed about the mean position $<z>=v\\mu t$ as they move along the field lines. As a result, the perturbation $\\delta B({\\bf x},t)$  that the CRs view when moving along the field gets a different time dependence. The characteristic phase function $e^{ik_\\|z(t)}$ of the perturbation $\\delta B({\\bf x},t)$ deviates from that for plane waves. Assuming the guiding center has a Gaussian distribution along the field line, \\begin{equation}\nf(z)=\\frac{1}{\\sqrt{2\\pi}\\sigma_z}e^{-\\frac{(z-<z>)^2}{2\\sigma_z^2}},\n\\label{gauss}\n\\end{equation}\none gets by integrating over z, \\begin{equation}\n\\int_{-\\infty}^{\\infty} dze^{ik_\\| z}f(z)= e^{ik_\\|<z>}e^{-k_\\|^2\\sigma_z^2\/2}. \n\\label{phase}\n\\end{equation}\nThe first adiabatic invariant gives us \n\\begin{equation}\n\\sigma_z^2=<\\Delta v_\\|^2>t^2=\\frac{v^4}{v_\\|^2}\\left(\\frac{<\\delta B_\\parallel^2>}{B_0^2}\\right)t^2.\n\\end{equation}\n\n\nInsert the Eq.(\\ref{phase}) into the expression of $D_{\\mu\\mu}$ (see V\\\"olk 1975, \\citealt{YL04}), we obtain\n\n\\begin{eqnarray}\nD_{\\mu\\mu}&=&\\frac{\\Omega^2(1-\\mu^2)}{B_0^2}\\int d^3k\\sum_{n=0}^{\\infty}R_n(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)\\nonumber\\\\\n&&\\left[I^A({\\bf k})\\frac{n^2J_n^2(w)}{w^2}+\\frac{k_\\|^2}{k^2}J^{'2}_n(w)I^M({\\bf k})\\right],\n\\label{general}\n\\end{eqnarray} \nFollowing are the definitions of the parameters in the above equation. $\\Omega, \\mu$ are the Larmor frequency and pitch angle cosine of the CRs. $J_n$ represents Bessel function, and $w=k_\\bot v_\\bot\/\\Omega=k_\\bot LR\\sqrt{1-\\mu^2}$, where $R=v\/(\\Omega l)$ is the dimensionless rigidity of the CRs, $L$ is the injection scale of the turbulence. $k_\\bot, k_\\|$ are the components of the wave vector ${\\bf k}$ perpendicular and parallel to the mean magnetic field, $\\omega$ is the wave frequency. $I^A({\\bf k})$ is the energy spectrum of the Alfv\\'en modes and $I^M({\\bf k})$ represents the energy spectrum of magnetosonic modes. In QLT, the resonance function $R_n=\\pi\\delta(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)$. Now due to the perturbation of the orbit, it should be   \n\\begin{eqnarray}\n&&R_n(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)\\nonumber\\\\\n&=&\\Re\\int_0^\\infty dt e^{i(k_\\|v_\\|+n\\Omega-\\omega) t-\\frac{1}{2}k_\\|^2<\\Delta v_\\|^2>t^2}\\nonumber\\\\\n&=&\\frac{\\sqrt{\\pi}}{|k_\\|\\Delta v_\\||}\\exp\\left[-\\frac{(k_\\|v \\mu-\\omega+n\\Omega)^2}{k_\\|^2\\Delta v_\\|^2}\\right]\\nonumber\\\\\n&\\simeq&\\frac{\\sqrt{\\pi}}{|k_\\||v_\\bot \\sqrt{M_A}}\\exp\\left[-\\frac{(k_\\|v \\mu-\\omega+n\\Omega)^2}{k_\\|^2v_\\bot^2M_A}\\right]\n\\label{resfunc}\n\\end{eqnarray}\nwhere $M_A\\equiv \\delta V\/v_A=\\delta B\/B_0$ is the Alfv\\'enic Mach number and $v_A$ is the Alfv\\'en speed. We stress that Eqs.~(\\ref{general},\\ref{resfunc}) are generic, and applicable to both incompressible and compressible medium. \n\nFor gyroresonance ($n=\\pm 1,2,...$), the result is similar to that from QLT for $\\mu\\gg \\Delta \\mu=\\Delta v_\\|\/v$. In this limit, Eq.(\\ref{general}) represents a sharp resonance and becomes equivalent to a $\\delta$-function when put into Eq.(\\ref{general}).  \nIn general, the result is different from that of QLT, especially at $\\alpha\\rightarrow 90^\\circ$, the resonance peak happens at $k_{\\|,res}\\sim \\Omega\/\\Delta v$ in contrast to the QLT result \n$k_{\\|,res}\\sim\\Omega\/v_\\|\\rightarrow \\infty$. We shall\nshow below, that due to the anisotropy, the scattering coefficient $D_{\\mu\\mu}$ is still very small if the Alfv\\'en and the pseudo-Alfv\\'en modes are concerned. \n\nOn the other hand, the dispersion of the $v_\\parallel$ means that CRs with a much wider range of pitch angle can be scattered by the compressible modes through TTD \n($n=0$), which is marginally affected by the anisotropy and much more efficient than the gyroresonance. In QLT, the projected particle speed should be comparable to phase speed of the magnetic field compression according to the $\\delta$ function for the TTD resonance.. This means that only particles with a specific pitch angle \ncan be scattered. For the rest of the pitch angles, the interaction is still dominated by gyroresonance, which efficiency is negligibly small for the Alfv\\'enic anisotropic turbulence (see \\S\\ref{Alf_scatter}). With the resonance broadening, however, wider range of pitch angle can be scattered through TTD, including $90^\\circ$. \n\n\n\\subsection{Results from test particle simulations}\n\nWe live in an era when we can test various processes in astrophysics and numerical studies have become an important part of theoretical efforts. Test particle simulation has been used to study CR scattering and\ntransport \\cite{Giacalone_Jok1999, Mace2000}. The aforementioned studies, however, used synthetic\ndata for turbulent fields, which have several disadvantages.\nCreating synthetic turbulence data which has scale-dependent\nanisotropy with respect to the local magnetic field (as observed\nin \\citealt{CV00} and \\citealt{MG01}) is difficult\nand has not been realized yet.  Also,\nsynthetic data normally uses Gaussian statistics and\ndelta-correlated fields, which is hardly appropriate for\ndescription of strong turbulence. \n\nUsing the results of direct numerical MHD simulations as the input data, \\cite{BYL2011} and \\cite{Xu_Yan} performed test particle simulations. Their results show good correspondence with the analytical predictions. We briefly summarize the results here. As shown in Fig.\\ref{xx_yy}, particles' motion is diffusive both along the magnetic field (x direction) and across the field (y direction). Moreover, the scattering coefficient shows the same pitch angle dependence as that predicted in \\cite{YL08}, namely the scattering is most efficient for large pitch angles due to the TTD mirror interaction (see Fig. \\ref{xx_yy} {\\em left}).  \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{duu.eps}\n\\includegraphics[width=0.45\\textwidth]{ratio.jpg}\n\\caption{{\\em Left}: dimensionless CR scattering coefficient $D_{\\mu\n    \\mu}\/\\Omega$ vs the pitch angle $\\mu$. It is dominated by TTD resonant mirror interaction with compressible modes; {\\em right:} Diffusive behavior of the particles displayed in the tracing\n  simulations.  Both the parallel and perpendicular transport are normal diffusion, and the ratio of their diffusion coefficients is $\\sim M_A^4$, consistent with the analytical prediction in \\cite{YL08} \\citep*[from][]{Xu_Yan}.}\n\\label{xx_yy}\n\\end{figure}\n\n\n\\section{Cosmic ray propagation in Galaxy}\n\\label{results}\nThe scattering by fast modes is influenced by the medium properties as the fast modes are subject to linear damping, e.g., Landau damping.\n Using the approach above we revisit the problem of the CR propagation in the selected phases of the ISM (see Table~\\ref{ch1t1} for a list of fiducial parameters appropriate for the idealized phases\\footnote{The parameters of idealized interstellar phases are a subject of debate. Recently, even the entire concept of the phase being stable\n entities has been challenged \\citep[see][and ref. therein]{Gazol:2007}. Indeed different parts\n of interstellar medium can exhibit variations of these parameters \\citep[see][and ref. therein]{Wolfire:2003}}) assuming that turbulence is injected on large scales.\n\\begin{table*}\n{\\footnotesize \\begin{tabular}{ccccccc}\n\\hline\n\\hline \n ISM&\nhalo&\n HIM&\n WIM&\n WNM&\n CNM&\n DC\\tabularnewline\n\\hline\nT(K)&\n $2\\times 10^6$&\n $1\\times10^{6}$&\n 8000&\n 6000&\n 100&\n 15\\tabularnewline\n$c_S$(km\/s)&\n130&\n91&\n8.1&\n7&\n0.91&\n0.35\\tabularnewline\nn(cm$^{-3}$)&\n $10^{-3}$&\n $4\\times10^{-3}$&\n 0.1&\n 0.4&\n 30&\n 200\\tabularnewline\n$l_{mfp}$(cm)&\n$4\\times 10^{19}$&\n$2\\times10^{18}$&\n$6\\times10^{12}$&\n$8\\times10^{11}$&\n$3\\times10^{6}$&\n$10^{4}$\\tabularnewline\nL(pc)&\n 100&\n 100&\n 50&\n 50&\n 50&\n 50\\tabularnewline\nB($\\mu$G)&\n5&\n2&\n5&\n5&\n5&\n15\\tabularnewline\n$\\beta$&\n0.28&\n3.5&\n0.11&\n0.33&\n0.42&\n0.046\\tabularnewline\ndamping&\n collisionless&\n collisional&\n collisional&\n neutral-ion&\n neutral-ion&\n neutral-ion\\tabularnewline\n\\hline\n\\hline\n\\end{tabular}\n\\caption{The parameters of idealized ISM phases and relevant damping. The\ndominant damping mechanism for turbulence is given in the last line. HIM=hot ionized medium, CNM=cold neutral medium, WNM=warm neutral\nmedium, WIM=warm ionized medium, DC=dark cloud.}}\n\\label{ch1t1}\n\\end{table*}\n\n\\subsection{Halo}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{f5.eps}\n\\includegraphics[width=0.45\\textwidth]{f9.eps}\n\\caption{\\small {\\em Left}: The turbulence truncation scales in Galactic halo and warm ionized medium (WIM). The damping curves flattens around $90^\\circ$ due to field line wandering (dotted lines, see \\citealt*{YL04, LVC04}); For WIM, both viscous and collisionless damping are applicable; {\\em right}: The mean free paths in two different phases of ISM: halo (solid line) and WIM (dashed line). At lower energies ($\\sim<100$GeV), the different dependence in WIM is owing to the viscous damping \\citep[from][]{YL08}.}\n\\label{mfp}\n\\end{figure}\n\nIn Galactic halo (see Table~\\ref{ch1t1}), the Coulomb collisional mean free path is $\\sim 10$ pc, the plasma is thus in a collisionless regime. The cascading rate \nof the fast modes is \\citep{CL02_PRL}\n\\begin{equation}\n\\tau_k^{-1}=(k\/L)^{1\/2}\\delta V^2\/V_{ph}.\n\\label{tcasfast}\n\\end{equation}\n\nBy equating it with the collisionless damping rate \n\\begin{equation}\n\\Gamma_{c} = \\frac{\\sqrt{\\pi\\beta}\\sin^{2}\\theta}{2\\cos\\theta}kv_A\\times \\left[\\sqrt{\\frac{m_e}{m_i}}\\exp\\left(-\\frac{m_e}{\\beta m_i\\cos^2\\theta}\\right)+5\\exp\\left(-\\frac{1}{\\beta\\cos^{2}\\theta}\\right)\\right],\n\\label{Ginz}\n\\end{equation}\nwe obtain the turbulence truncation scale $k_c$:\n\\begin{equation}\nk_c L\\simeq \\frac{4M_A^4m_i\\cos^2\\theta}{\\pi m_e\\beta\\sin^4\\theta}\\exp\\left(\\frac{2m_e}{\\beta m_i\\cos^2\\theta}\\right).\n\\label{landauk}\n\\end{equation}\nwhere $\\beta=P_{gas}\/P_{mag}$.\n\nThe scale $k_c$ depends on the {\\it wave pitch angle} $\\theta$, which makes\nthe damping anisotropic. As the turbulence undergoes cascade and the waves propagate in a turbulent medium, the angle $\\theta$ is changing.\nAs discussed in YL04 the field wandering defines the spread of angles. During one cascading time, the fast modes propagate a distance \n$v\\tau_{cas} $ and see an angular deviation $\\tan \\delta \\theta \\simeq \\sqrt{\\tan^2\\delta \\theta_\\parallel+\\tan^2 \\delta\\theta_\\perp}$, which is\n\\begin{equation}\n\\tan \\delta\\theta \\simeq  \\sqrt{\\frac{M_A^2\\cos\\theta}{27(kL)^{1\/2}}+\\left(\\frac{M_A^2\\sin^2\\theta}{kL}\\right)^{1\/3}}\n\\label{dthetaB}\n\\end{equation}\nAs evident, the damping scale given by Eq.(\\ref{landauk}) varies considerably especially when $\\theta\\rightarrow 0$ and $\\theta\\rightarrow 90^\\circ$. For the quasi-parallel modes, the randomization ($\\propto (kL)^{-1\/4}$) is negligible since the turbulence cascade continues to very small scales. On small scales, most energy of the fast modes is contained in these quasi-parallel modes \\citep*{YL04, Petrosian:2006}.\n\nFor the quasi-perpendicular modes, the damping rate (Eq.\\ref{Ginz}) should be averaged over the range $90^\\circ-\\delta\\theta$ to $90^\\circ$. Equating Eq.(\\ref{tcasfast}) and Eq.(\\ref{Ginz}) averaged over $\\delta\\theta$, we get the averaged damping wave number (see Fig.\\ref{mfp} {\\em left}). The field line wandering has a marginal effect on the gyroresonance, whose interaction with the quasi-perpendicular modes is negligible (YL04). However, TTD scattering rates of moderate energy CRs ($<10$TeV) will be decreased owing to the increase of the damping around the $90^\\circ$ (see Fig.\\ref{mfp} {\\em left}). For higher energy CRs, the influence of damping is marginal and so is that of field line wandering. \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{f7.eps}\n\\includegraphics[width=0.45\\textwidth]{f8.eps}\n\\caption{Pitch angle diffusion coefficients in halo and WIM. Upper lines in the plots represent the contribution from TTD and lower lines are for gyroresonance \\citep[from][]{YL08}.}\n\\label{fastcompr}\n\\end{figure}\n\nThe QLT result on gyroresonance in the range $\\mu>\\Delta \\mu$ provides a good approximation to the non-linear results \\citep{YL08}. For CRs with sufficiently small rigidities, the resonant fast modes ($k_{res}\\approx 1\/(R\\mu)$) are on small scales with a quasi-slab structure (see Fig.\\ref{mfp} {\\em left}). For the scattering by these quasi-parallel modes, the analytical result that follows from QLT approximation \\citep[see][]{YL04} for the gyroresonance is\\footnote{It can be shown that the QLT result follows from our more general results (see Eqs.\\ref{general}, \\ref{resfunc}) if we put $\\Delta \\mu \\rightarrow 0$. This justifies our use of the analytical approximation.} \n\n\\begin{equation}\n\\left[\\begin{array}{c}\nD^{G}_{\\mu\\mu}\\\\\nD^{G}_{pp}\\end{array}\\right]=\\frac{\\pi v \\mu^{0.5}(1-\\mu^{2})}{4LR^{0.5}}\\left[\\begin{array}{c}\n\\frac{1}{7}[1+(R\\mu)^2]^{-\\frac{7}{4}}-(\\tan^{2}\\theta_c+1)^{-\\frac{7}{4}}\\\\\n\\frac{m^{2}V_{A}^{2}}{3}\\left\\{[1+(R\\mu)^2]^{-\\frac{3}{4}}-(\\tan^{2}\\theta_c+1)^{-\\frac{3}{4}}\\right\\}\\end{array}\\right]\n\\label{lbgyro}\\end{equation}\nwhere $\\tan\\theta_c={k_{\\perp,c}}\/{k_{\\parallel,res}}$.\n\nOnce we know the functional form of the $D_{\\mu\\mu}$, we can obtain the corresponding mean free path \\citep{Earl:1974}:\n\\begin{equation}\n\\lambda_\\|\/L=\\frac{3}{4}\\int^1_0 d\\mu \\frac{v(1-\\mu^2)^2}{(D^T_{\\mu\\mu}+D^G_{\\mu\\mu})L},\n\\end{equation}\nwhere $D^T_{\\mu\\mu}$ is the contribution from TTD interaction and can be obtained using the nonlinear theory (see \\citealt{YL08}, and also \\S\\ref{NLT_sec}) with the inertial range of fast modes determined for the local medium (see, e.g. \\ref{landauk} in the case of collisionless damping). \n\nThe mean free path is sensitive to the scattering by gyroresonance at small pitch angles, due to the influence of damping on the fast modes on small scales. Fig.\\ref{fastcompr} shows the pitch angle diffusion of CRs with different energies due to the TTD and gyroresonance.\n \nThe weak dependence of the mean free path (see Fig.\\ref{mfp} {\\em right}) of the moderate energy (e.g$<1$TeV) CRs in halo results from the fact that gyroresonance changes marginally with the CR energy (see Fig.\\ref{fastcompr}). This is associated with the damping in collisionless medium. We expect that similar flat dependence can happen in any collisionless medium. This can be a natural explanation of the puzzling ``Palmer Concensus\" \\citep{Palmer:1982}, the same trend observed in solar wind.\n\n\\subsection{Warm Ionized Medium}\n\nIn warm ionized medium, the Coulomb collisional mean free path is $l_{mfp}=6\\times 10^{12}$ cm and the plasma $\\beta\\simeq0.11$. Suppose that the turbulence energy is injected from large scale, then the compressible turbulence is subjected to the viscous damping besides the collisionless damping. \nBy equating the viscous damping rate with the cascading rate (Eq.\\ref{tcasfast}), we obtain the following truncation scale, \n\\begin{eqnarray}\nk_{c}L=x_c\\left\\{\\begin{array}{rl}(1-\\xi^2)^{-\\frac{2}{3}} & \\beta\\ll 1\\\\\n(1-3\\xi^2)^{-\\frac{4}{3}} & \\beta\\gg 1\\end{array}\\right.\n\\end{eqnarray}\nwhere $x_c=\\left[\\frac{6\\rho\\delta V^2L}{\\eta_0V_A}\\right]^{\\frac{2}{3}}$,\n $\\eta_0$ is the longitudinal viscosity. In the low $\\beta$ regime, the motions are primarily perpendicular to the magnetic field so that $\\partial v_{x}\/\\partial x=\\dot{n}\/n\\sim\\dot{B}\/B$. The longitudinal viscosity enters here as the result of distortion of the Maxiwellian distribution \\citep[see][]{Braginskii:1965}. The transverse energy of the ions increases during compression because of the conservation of adiabatic invariant $v_{\\perp}^{2}\/B$. If the rate of compression is faster than that of collisions, the ion distribution in the momentum space is bound to be distorted from the Maxiwellian isotropic sphere to an oblate spheroid with the long axis perpendicular to the magnetic field. As a result, the transverse pressure gets greater than the longitudinal pressure, resulting in a stress $\\sim\\eta_{0}\\partial v_{x}\/\\partial x$.\nThe restoration of the equilibrium increases the entropy and causes the dissipation of energy.\n\n\nThe viscous damping scale is compared to collisionless cutoff scale (Eq.\\ref{landauk}) in Fig.\\ref{mfp} {\\it left}. \nAs shown there, both viscous damping and collisionless damping are important in WIM. Viscous damping is dominant for small $\\theta$ and\n collisionless damping takes over for large $\\theta$ except for $\\theta=90^\\circ$.\nThis is because collisionless damping increases with $\\theta$ much faster than the viscous damping. For sufficiently small wave pitch angles, the viscous damping is too small to prevent the fast modes to cascade down to scales smaller than the mean free path $l_{mfp}$. Because of the similar quasi-slab structure on small scales, \nEq.(\\ref{lbgyro}) can be also applied in WIM. The results are illustrated in Fig.\\ref{fastcompr}. Compared to the case in halo, we see that the qualitative difference stands in the gyroresonance. This is because gyroresonance is sensitive to the quasi-slab modes whose damping differs in halo and WIM.  \n\n\\subsection{Other phases}\n\nIn hot ionized medium (HIM), the plasma is also in collisionless regime, but the density is higher and the plasma beta is larger than 1. The damping by protons thus becomes substantial especially at small pitch angles. The damping truncates the turbulence at much larger scales than the gyroscales of the CRs of the energy range we consider. No gyroresonance can happen and some other mechanisms are necessary to prevent CRs streaming freely along the field. The turbulence injected from small scales might play an important role (see \\S6). \n\n\nIn partially ionized gas one should take into account an additional damping that arises from ion-neutral collisions \\citep[see][]{Kulsrud_Pearce, LG01, LVC04}. In the latter work a viscosity-damped regime of turbulence was predicted at scales less the scale $k_{c, amb}^{-1}$ at which the ordinary magnetic turbulence is damped by ionic viscosity. The corresponding numerical work, e.g., \\cite{CLV_newregime} testifies that for the viscosity-damped regime the parallel scale stays equal to the scale of the ambipolar damping, \ni.e., $k_{\\|}=k_{c, amb}$, while $k_{\\bot}$ increases. In that respect, the scattering by such magnetic fluctuations is analogous to the scattering induced by the weak turbulence (see \\S 2.3, \\citealt{YL08}). The difference stems from the spectrum\nof $k_{\\bot}$ is shallower than the spectrum of the weak turbulence. The predicted values of the spectrum for the viscosity-damped turbulence $E(k_\\bot)\\sim\nk_\\bot^{-1}$ \\citep{LVC04} are in rough agreement with simulations. More detailed studies of scattering in partially ionized gas will be necessary.    \n\n\\section{Perpendicular transport}\n\nIn this section we deal with the diffusion perpendicular\nto {\\it mean} magnetic field. \n\nPropagation of CRs perpendicular to the mean magnetic field is another important problem in which QLT encounters serious difficulties.\nCompound diffusion, resulting from the convolution of diffusion along the magnetic field line and diffusion of field line perpendicular to mean field direction, has been invoked to discuss transport of cosmic rays in the Milky Way \\citep*{Getmantsev, Lingenfelter:1971,Allan:1972}. The role of compound diffusion in the acceleration\n of CRs at quasi-perpendicular shocks were investigated by \\cite{Duffy:1995} and \\cite{Kirk:1996}.  \n\nIndeed, the idea of CR transport in the direction perpendicular to the mean magnetic field being dominated by the field line random walk \n(FLRW, \\citealt{Jokipii1966, Jokipii_Parker1969, Forman1974}) can be easily justified\nonly in a restricted situation where the turbulence perturbations are small and CRs do not scatter backwards to retrace their trajectories. If the latter is not true, the particle motions are subdiffusive, \ni.e., the squared distance diffused growing as not as $t$ but as $t^{\\alpha}$, $\\alpha<1$, e.g., $\\alpha=1\/2$ \\citep{Kota_Jok2000, Mace2000, Qin2002}.\nIf true, this could indicate a substantial shift in the paradigm of CR transport, a shift that surely dwarfs a modification of magnetic turbulence model from the 2D+slab to a more simulation-motivated model that we deal here.\n\n \nIt was also proposed that with substantial transverse structure, {\\it i.e.}, transverse displacement of field lines, perpendicular diffusion is recovered \\citep{Qin2002}. Is it the case of the MHD turbulence models we deal with? \n\n\nHow realistic is the subdiffusion in the presence of turbulence? The answer for this question apparently depends on the models of turbulence chosen. \n\nCompound diffusion happens when particles are restricted to the magnetic field lines and perpendicular transport is solely due to the random walk of field line wandering \\citep[see][]{Kota_Jok2000}. \nIn the three-dimensional turbulence, field lines are diverging away due to shearing by the Alfv\\'en modes \\citep[see][]{LV99, Narayan_Medv, Lazarian06}.\n Since the Larmor radii of CRs are much larger than the minimum scale of eddies $l_{\\bot, min}$, field lines within the CR Larmor orbit are effectively diverging away owing to shear by the Alfv\\'enic turbulence.\nThe cross-field transport thus results from the deviations of field lines at small scales, as well as field line random walk at large scale ($>{\\rm min}[L\/M^3_A,L]$).\n\nBoth observation of Galactic CRs and solar wind indicate that the diffusion of CRs perpendicular to magnetic field is normal diffusion \\citep[]{Giacalone_Jok1999, Maclennan2001}. Why is that?\n\nMost recently the diffusion in magnetic fields was considered for thermal particles in Lazarian (2006), for cosmic rays in \\cite{YL08}. In what follows we present the results based on the studies in  \\cite{YL08}.\n\n\\subsection{Perpendicular diffusion on large scale}\n\nFor perpendicular diffusion, the important issue is the reference frame. We emphasize that we consider the diffusion perpendicular to the {\\emph mean} field direction in the global reference of frame.  \n\n{\\it High $M_A$ turbulence}: High $M_A$ turbulence corresponds to the field that is easily bended by\nhydrodynamic motions at the injection scale as the hydro energy at the\ninjection scale is much larger than the magnetic energy, i.e.\n$\\rho V_L^2\\gg B^2$. In this case\nmagnetic field becomes dynamically important on a much smaller scale, i.e. the \nscale $l_A=L\/M_A^3$ \\citep[see][]{Lazarian06}. If the parallel mean free path of CRs $\\lambda_\\|\\ll l_A$, the stiffness of B field is negligible so that the perpendicular diffusion coefficient is the same as the parallel one, i.e., $D_\\bot=D_\\|\\sim 1\/3 \\lambda_{\\|} v$. If $\\lambda_\\|\\gg l_A$, the\n diffusion is controlled by the straightness of the field lines, and $\nD_\\bot=D_{\\|}\\approx 1\/3l_Av.\n\\label{dbb}\n$ The diffusion is isotropic if scales larger than $l_A$ are\nconcerned. \n\n{\\it Low $M_A$ turbulence}: In the magnetically dominated case, i.e. the field that cannot be easily bended at\nthe turbulence injection scale, individual magnetic field lines are aligned\nwith the mean magnetic field. The diffusion in this case is anisotropic.\nIf turbulence is injected at scale $L$ it stays \nweak for the scales larger than $LM_A^2$ and it is \nstrong at smaller scales. Consider first the case of $\\lambda_\\|>L$.\nThe time of the individual step is $L\/v_\\|$, then $D_\\perp\\approx 1\/3Lv M_A^4.$\nThis is similar to the case discussed in the FLRW model (Jokipii 1966). However, we obtain the dependence of $M_A^4$ instead of their $M_A^2$ scaling. In the opposite case of $\\lambda_\\|<L$, the perpendicular diffusion coefficient is $\nD_{\\bot}\\approx D_{\\|}M_A^4,\n\\label{diffx}$ which coincides with the result\nobtained for the diffusion of thermal electrons in magnetized plasma \\citep{Lazarian06}. This is due to the anisotropy of the Alfv\\'enic turbulence.\n\n\\subsection{Superdiffusion on small scales}\n\nThe diffusion of CR on the scales $\\ll L$ is different and it is determined by how fast field lines are diverging away from each other. The mean deviation of a field in a distance $\\delta x$ is proportional to $[\\delta z]^{3\/2}$  \\citep{LV99, Lazarian06}, same as Richardson diffusion in the case of hydrodynamic turbulence \\cite[see][]{Eyink2011}. Following the argument, we showed in \\cite{YL08} that the cosmic ray perpendicular transport is superdiffusive. The reason is that there is no random walk on small scales up to the injection scale of strong MHD turbulence ($LM_A^2$ for $M_A < 1$\n and $l_A$ for $M_A>1$). This can well explain the recently observed super-diffusion in solar wind \\citep{Perri2009}. Superdiffusion can have important implications for shock acceleration as discussed in details in \\cite{LY13}.\n\n\n\\subsection{Is there subdiffusion?}\nThe diffusion coefficient $D_{\\|}M_A^4$ we obtained in the case of $M_A<1$, means that the transport\nperpendicular to the dynamically strong magnetic field is a normal diffusion, rather\nthan the subdiffusion as discussed in a number of recent papers. This is also supported by test particle simulations (\\citealt*{BYL2011, Xu_Yan}, see Fig.\\ref{xx_yy} {\\it right}). Let us\nclarify this point by obtaining the necessary conditions for the subdiffusion\nto take place.\n\nThe major implicit assumption in subdiffusion (or compound diffusion) is that the particles trace back \ntheir \ntrajectories in x direction on the scale $\\delta z$. When is it possible to talk about retracing of particles? In the case of random motions at a single scale {\\it only}, the distance over \nwhich the particle\ntrajectories get uncorrelated is given by the \\cite{RR1978}\nmodel. Assuming that the damping scale of the turbulence  is larger\nthat the CR Larmor radius, the \\cite{RR1978}\nmodel, when generalized to anisotropic turbulence provides \\citep{Narayan_Medv, Lazarian06} $L_{RR}=l_{\\|, min}\\ln(l_{\\bot, min}\/r_{Lar})$\nwhere $l_{\\|, min}$ is the parallel scale of the cut-off of turbulent motions, \n$l_{\\bot, min}$ is the corresponding perpendicular scale, $r_{Lar}$ is the\nCR Larmor radius. The assumption of $r_{Lar}<l_{\\bot, min}$ can be\n valid, for instance, for Alfv\\'enic motions in partially ionized gas.\nHowever, it is easy to see that, even in this case,\n the corresponding scale is rather\nsmall and therefore subdiffusion is not applicable for the transport\nof particles in Alfv\\'enic turbulence over scales $\\gg l_{\\|,min}$.\n\nIf  $r_{Lar}>l_{\\bot, min}$, as it is a usual case for Alfv\\'en motions in the\nphase of ISM with the ionization larger than $\\approx 93\\%$, where the\nAlfv\\'enic motions go to the thermal particle gyroradius \n\\citep[see estimates in][]{LG01, LVC04}, \nthe subdiffusion of CR is not an applicable concept for Alfv\\'enic turbulence.  \nThis does\nnot preclude subdiffusion from taking place\nin particular models of magnetic perturbations,\ne.g. in the slab model considered in \\cite{Kota_Jok2000}, but we believe in the omnipresence of Alfv\\'enic turbulence in interstellar gas \\citep[see][]{Armstrong95}.\n\n\\section{Streaming Instability in the Presence of Turbulence}\n\n\\begin{table}\n\\caption{The notation we used in this section}\n\\label{notations}\n\\begin{tabular}{|c|r|}\n\\hline\nA  & normalized wave amplitude $\\delta B\/B_0$\\\\\na& hardening of the CR spectrum at the shock front\\\\\n$B_0$ & mean magnetic field at the shock in the later Sedov phase\\\\\n$B_{cav}$ &inercloud magnetic field strength\\\\\n$\\delta B$& wave amplitude\\\\\nc & light speed\\\\\nd& distance of the molecular cloud from observer\\\\\nD& diffusion coefficient of CRs\\\\\nE& CR energy\\\\\n$E_{SN}$& supernova explosion\\\\\nf& distribution function of CRs\\\\\n$f_\\pi$& fraction of energy transferred from parent protons to pions\\\\\nk& wave number\\\\\nK(t) & Normalization factor of CR distribution function\\\\\nL& the injection scale of background turbulence\\\\\nm& proton rest mass\\\\\n$M_c$ & cloud mass\\\\ \nn& intercloud number density\\\\ \n$N_\\gamma$ & $\\gamma$ ray flux\\\\\np& CR's momentum\\\\\n$p_{max}$& the maximum momentum accelerated at the shock front\\\\\n$P_{CR}$ & CR pressure\\\\\nq & charge of the particle\\\\\nr& distance from SNR centre\\\\\n$R_c$& the distance of the molecular cloud from the SNR centre\\\\\n$r_g$ & Larmor radius of CRs\\\\\n$R_d$ & diffusion distance of CRs\\\\\n$R_{sh}$ & shock radius\\\\\n$R_{esp},\\,t_{esp}$& the escaping distance\/time of CRs\\\\ \ns& 1D spectrum index of CR distribution\\\\\nt& time since supernova explosion\\\\\n$t_{age}$ & the age of SNR\\\\\n$t_{sed}$ & the time at which SNR enters the Sedov phase\\\\\nU& shock  speed\\\\\n$U_i$ &initial shock velocity\\\\\nv& particle speed\\\\\n$v_s$ & streaming speed of CRs\\\\\nW& wave energy\\\\\n$\\alpha$& power index of D with respect to particle momentum p\\\\\n$\\chi$&reduction factor of D with respect to $D_{ISM}$\\\\\n$\\delta$& power index of $p_{max}$ with respect to t\\\\\n $\\eta$&fraction of SN energy converted into CRs\\\\\n $\\eta_A$& a numerical factor in Eq.\\ref{pp}\\\\\n$\\Gamma_{cr}$& the growth rate of streaming instability\\\\\n$\\Gamma_d$& wave damping rate\\\\\n $ \\kappa$&ratio of diffusion length to shock radius\\\\\n $\\Omega_0$ & the Larmor frequency of non-relativistic protons\\\\\n $\\sigma_{pp}$ &cross section for pp collision\\\\\n $\\xi$& the ratio of CR pressure to fluid ram pressure\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n                \nWhen cosmic rays stream at a velocity much larger than Alfv\\'{e}n\nvelocity, they can excite by gyroresonance MHD modes which in turn scatter\ncosmic rays back, thus increasing the amplitude of the resonant mode. This \nrunaway process is known as streaming instability. It was claimed\nthat the instability could provide confinement for cosmic rays with\nenergy less than $\\sim 10^2$GeV \\citep{Cesarsky80}. However, this was calculated\nin an ideal regime, namely, there was no background MHD turbulence.\nIn other words, it was thought that the\nself-excited modes would not be appreciably damped in fully ionized gas\\footnote{We neglect the nonlinear Landau damping, which is suppressed in turbulence due to decrease of mean free path.}. \n\nThis is not true for turbulent medium, however. \\citet{YL02}\npointed out that the streaming instability is partially suppressed\nin the presence of background turbulence \\citep[see more in][]{LCY02}. More recently, detailed calculations of the \nstreaming instability in the presence of background Alfv\\'enic turbulence \nwere presented in \\cite{FG04}. The growth rate of the modes of wave number $k$ is \\citep{Longairbook}.\n\\begin{equation}\n\\Gamma_{cr}(k)=\\Omega_0\\frac{N(\\geq E)}{n_{p}}(-1+\\frac{v_{stream}}{V_{A}}),\n\\label{instability}\n\\end{equation}\n where $N(\\geq E)$ is the number density of cosmic rays with energy\n$\\geq E$ which resonate with the wave,\n$n_{p}$ is the number density of charged particles in the medium.\nThe number density of cosmic rays near the sun is $N(\\geq E)\\simeq 2\\times10^{-10}(E\/$GeV$)^{-1.6}$\ncm$^{-3}$sr$^{-1}$ \\citep{Wentzel74}.\n\nInteraction with fast modes was considered by \\citet{YL04}. \nSuch an interaction happens at the rate \n$\\tau_k\\sim (k\/L)^{-1\/2}V_{ph}\/V^2$. \nBy equating it with the growth rate Eq.(\\ref{instability}), we can find that the streaming instability\nis only applicable for particles with energy less than\n\\begin{equation}\n\\gamma_{max}\\simeq1.5\\times 10^{-9}[n_{p}^{-1}(V_{ph}\/V)(Lv\\Omega_0\/V^2)^{0.5}]^{1\/1.1},\n\\end{equation}\nwhich for HIM, provides $\\sim 20$GeV if taking the injection speed to be $V\\simeq 25$km\/s. Similar result was obtained with Alfv\\'en modes by \\citet{FG04}. \n\n\nMagnetic field itself is likely to be amplified through an inverse \ncascade of magnetic energy at which perturbations created at a particular\n$k$ diffuse in $k$ space to smaller $k$ thus inducing inverse cascade. \nAs the result, the magnetic perturbations at smaller $k$ get larger than the \nregular field. Consequently, even if the instability is suppressed\nfor the growth rate given by Eq. (\\ref{instability}) it gets efficient\ndue to the increase of perturbations of magnetic field stemming from the\ninverse cascade. The precise picture of the process depends on yet not completely\nclear details of the inverse cascade of magnetic field. \n\nBelow, we present the application of the current understanding of the interaction between the streaming instability and the background turbulence to the modeling of the gamma ray emission from molecular clouds near SNRs \\citep[see more details in][]{YLS12}. We shall treat the problem in a self-consistent way by comparing the streaming level that is allowed by the preexisting turbulence and the required diffusion for the CRs. \n\n\\subsection{Application to CR acceleration at the shocks }\n\\label{pmax}\nDiffusive shock acceleration of energetic CR particles relies on the crucial process of amplification of MHD turbulence so that particles can be trapped at the shock front long enough to be accelerated to the high energy observed. One of the most popular scenarios that has been adopted in the literature is the streaming instability generated by the accelerated particles. However, in the highly nonlinear regime the fluctuations of magnetic field arising from the streaming\ninstability get large and the classical treatment of the streaming instability is not applicable. We circumvent the\nproblem by proposing that the field amplification we consider does not arise from the streaming\ninstability, but is achieved earlier through other processes, e.g. the interaction of the shock precursor with density perturbations preexisting in the interstellar medium \\citep*{BJL09}. Due to the\nresonant nature of the streaming instability, the perturbations $\\delta B$ arising from it are more efficient\nin scattering CRs compared to the large scale fluctuations produced by non-resonant mechanisms, e.g.\nthe one in \\citet{BJL09}. Therefore in this chapter, we limit our discussions to the regime of $\\delta B \\sim< B_0$, where $B_0$ is the magnetic field that has already been amplified in the precursor region\\footnote{The effective $B_0$ is therefore renormalized and can be much larger than the typical field in ISM (see, e.g., \\citealt{Diamond_Makov}).}.  \n\nWhen particles reach the maximum energy at a certain time, they escape and the growth of the streaming instability stops. Therefore we can obtain the maximum energy by considering the stationary state of the evolution. The steady state energy density of the turbulence $W(k)$ at the shock is determined by\n\n\\begin{equation}\n(U\\pm v_A)\\nabla W(k) = 2 (\\Gamma_{cr}-\\Gamma_d)W(k),\n\\label{wave}\n\\end{equation}\nwhere $U$ is the shock speed, and the term on the l.h.s. represents the advection of turbulence by the shock flow. $v_A\\equiv B_0\/\\sqrt{4\\pi nm}$ and $n$ are the Alfv\\'en speed and the ionized gas number density of the precursor region, respectively. The plus sign represents the forward propagating Alfv\\'en waves and the minus sign refers to the backward propagating Alfv\\'en waves. The terms on the r.h.s. describes the wave amplification by the streaming instability and damping with $\\Gamma_d$ as the corresponding damping rate of the wave.  The distribution of accelerated particles at strong shocks is $f(p)\\propto p^{-4}$. If taking into account the modification of the shock structure by the accelerated particles, the CR spectrum becomes harder. Assume the distribution of CRs at the shock is $f_0(p)\\propto p^{-4+a}$. The nonlinear growth was studied by \\citet{Ptuskin:2005}. \n\nThe generalized growth rate of streaming instability is\n\n\\begin{eqnarray}\n\\Gamma_{cr}&=&\\frac{12\\pi^2 q^2v_A\\sqrt{1+A^2}}{c^2k}\\nonumber\\\\\n&\\times& \\int^\\infty_{p_{res}} dp p\\left[1-\\left(\\frac{p_{res}}{p}\\right)^2\\right]D\\left|\\frac{\\partial f}{\\partial x}\\right|, \n\\label{general_growth}\n\\end{eqnarray}\nwhere $q$ is the charge of the particle, c is the light speed, $p_{res}=ZeB_0\\sqrt{1+A^2}\/c\/k_{res}$ is the momentum of particles that resonate with the waves.  $A=\\delta B\/B_0$ is wave amplitude normalized by the mean magnetic field strength $B_0$.\n\\begin{equation}\nD=\\sqrt{1+A^2}v r_g\/3\/A^2(>k_{res})\n\\label{crdiff}\n\\end{equation}\nis the diffusion coefficient of CRs, $v$ and $r_g$ are the velocity and Larmor radius of the CRs.  \nIn the planar shock approximation, one gets the following growth rate of the upstream forward moving wave at x=0,\n\\begin{equation}\n\\Gamma_{cr}(k)=\\frac{C_{cr}\\xi U^2(U+v_A)k^{1-a}}{(1+A^2)^{(1-a)\/2}cv_A\\phi(p_{max})r_0^a} \n\\label{growth}\n\\end{equation}\nwhere $C_{cr}=4.5\/(4-a)\/(2-a)$, $r_0=m c^2\/q\/B_0$, where $\\xi$ measures the ratio of CR pressure at the shock and the upstream  momentum flux entering the shock front, $m$ is the proton rest mass, and $p_{max}$ is the maximum momentum accelerated at the shock front. $H(p)$ is the Heaviside step function.\n\nThe linear damping is negligible since the medium should be highly ionized. In fully ionized gas, there is nonlinear Landau damping, which, however, is suppressed due to the reduction of particles' mean free path in the turbulent medium \\citep[see][]{YL11}. We therefore neglect this process here. Background turbulence itself can cause nonlinear damping to the waves \\citep{YL02}. Unlike hydrodynamical turbulence, MHD turbulence is anisotropic with eddies elongated along the magnetic field. The anisotropy increases with the decrease of the scale \\citep{GS95}. Because of the scale disparity, $k_\\| > k_\\bot \\gg k^t_\\|$, the nonlinear damping rate in MHD turbulence is less than the wave frequency $k_\\| v_A$, and it is given by \\citep{FG04, YL04}\n\n\\begin{equation}\n\\Gamma_d \\sim \\sqrt{k\/L} v_A,\n\\label{damping}\n\\end{equation}\nwhere L is the injection scale of background turbulence, and the $k$ is set by the resonance condition $k \\sim k_\\| \\sim 1\/r_L$.\n\nThere are various models for the diffusive shock acceleration. We consider here the escape-limited acceleration. In this model, particles are confined in the region near the shock where turbulence is generated. Once they propagate far upstream at a distance $l$ from the shock front, where the self-generated turbulence by CRs fades away, the particles escape and the acceleration ceases.  The characteristic length that particles penetrate into the upstream is $D(p)\/U$. The maximum  momentum is reached when $D(p)\/U\\simeq l\/4$\\footnote{The factor 1\/4 arises from the following reason. As pointed out by \\cite{Ostrowski:1996}, the spectrum is steepened for small l, i.e., $l U\/D(p) \\sim< 4$}. Assuming $l=\\kappa R_{sh}$, where $\\kappa<1$ is a numerical factor, one can get\n\\begin{equation}\n\\frac{p_{max}}{mc} = \\frac{3\\kappa A^2 U R_{sh}}{\\sqrt{1+A^2}v r_0}.\n\\label{gmax}\n\\end{equation}\n\nIn particular, for $A<1$\n\\begin{eqnarray}\\frac{p_{max}}{mc}&=&\\left[\\left(-v_A\\sqrt{\\frac{1}{r_0 L}}+\\sqrt{\\frac{v_A^2}{r_0 L}+\\frac{2C_{cr}a\\xi U^3(U+v_A)}{\\kappa r_0R_{sh}cv_A}}\\right) \\left(\\frac{\\kappa R_{sh}}{U}\\right)\\right]^2,\\nonumber\\\\\nA&=&\\frac{p_{max}r_0}{\\sqrt{18}\\kappa mU R_{sh}}\\sqrt{1+\\sqrt{1+36 \\left(\\frac{\\kappa mU R_{sh}}{p_{max}r_0}\\right)^2}},\n\\label{gmax_general}\n\\end{eqnarray}\n\nIn the limit of low shock velocity, \n\\begin{eqnarray\nv_A\\ll U\\ll  c\\left[\\left(\\frac{v_A}{c}\\right)^3\\frac{\\kappa R_{sh}}{2 L C_{cr}a\\xi}\\right]^{1\/4},\n\\label{lowshockU}\n\\end{eqnarray}\nwe get \n\\begin{eqnarray\n\\frac{p_{max}}{mc}&=&(C_{cr}\\xi U^3)^2\\frac{a^2 L}{r_0c^2v_A^4}\n\\label{gmax_solution}\n\\end{eqnarray}\nfor the Sedov phase ($t>t_{sed}\\equiv 250(E_{51}\/(n_0U_9^5))^{1\/3}$yr), where $E_{51}=E_{SN}\/10^{51}$erg and $U_9=U_i\/10^9$cm\/s are the total energy of explosion and the initial shock velocity. In Fig.\\ref{Emax}, we plot the evolution of $p_{max}\/(mc)$ during the Sedov phase. The solid line represents the results from Eqs.(\\ref{gmax_general}). As we see, at earlier epoch when advection and streaming instability are both important, the evolution of $p_{max}$ does not follow a power law. For comparison, we also put a power law evolution in the same figure as depicted by Eq.(\\ref{gmax_solution})  (dashed line). \nOur result is also larger than that obtained by \\cite{Ptuskin:2005} since the wave dissipation rate is overestimated in their treatment.\n\n\\begin{figure}\n\\includegraphics[width=0.47\\textwidth]{Emax.eps}\n\\includegraphics[width=0.47\\textwidth]{spectrum0.eps}\n \\caption{\\small {\\em Left}: The energy of CRs that are released at the shock at time t in the Sedov phase. Our result shows that the often assumed power law solution \\citep[see][]{Gabici:2009, Ohira:2010} is only realized in asymptotic regime as described in Eqs.(\\ref{lowshockU},\\ref{gmax_solution}). It is also larger than the earlier result (dotted line) in Ptuskin \\& Zirakashvili (2005) where the damping of the waves by background turbulence is overestimated. {\\em Right}: The spectrum of CRs at a distance $r=12$ pc after 1800 (solid line), 6000 (dotted line), 12000 (dashdot line), 50000 years (cross line). The Galactic mean is plotted as a reference (dashed line). From \\cite{YLS12}.}\n\\label{Emax} \n\\end{figure}\n\n\\subsection{Enhanced scattering and streaming instability near SNRs}\n\\label{nearby}\n\nThe result from \\cite{YLS12} show that the local scattering of CRs has to be enhanced by an order of magnitude $\\chi =0.05$ in order to produce the amount of $\\gamma$ ray emission observed. A natural way to increase the scattering rate is through the streaming instability. The enhanced flux of the CRs are demonstrated to generate strong enough instability to overcome nonlinear damping by the background turbulence \\citep{YLS12}. The growth rate in the linear regime is\n\n\\begin{equation}\n\\Gamma_{gr}=\\Omega_0\\frac{N(\\geq E)}{n}\\left(\\frac{v_s}{v_A}-1\\right),\n\\end{equation}\nwhere $v_s$ is the streaming speed of CRs. The growth rate should overcome the damping rate (eq.\\ref{damping}) for the instability to operate. The condition $\\Gamma_{gr}>\\Gamma_d$ leads to\n\\begin{equation}\nv_s > v_A \\left(1+\\frac{n v_A}{N \\Omega_0\\sqrt{r_gL}}\\right)\n\\end{equation}\n\nThe spatial diffusion coefficient adopted here, $D \\approx v_s L = \\chi D_{ISM}$, satisfies  this requirement. The growth and damping rates are compared in Fig.\\ref{rates} {\\em right}. We see that the streaming instability works in the energy range needed to produce the observed $\\gamma$ ray emission, proving that our results are self-consistent.\n\nNote that the case we consider here is different from the general interstellar medium discussed in \\cite{YL04} and \\cite{FG04}, namely, the local cosmic ray flux near SNRs is much enhanced (see Fig.\\ref{Emax} {\\em right}). Consequently, the growth rate of the streaming instability becomes high enough to overcome the damping rate by the preexisting turbulence in the considered \nenergy range.\n\n\\begin{figure}\n\\includegraphics[width=0.47\\textwidth]{gmray.eps}\n\\includegraphics[width=0.47\\textwidth]{stream.eps}\n \\caption{\\small {\\em Left}: The spectrum of Gamma ray emission from W28. The Fermi data are shown as dotted points \\citep{Abdo:2010}, and the H.E.S.S. data are plotted as 'x' points \\citep{Aharonian:2008} with error bars. Solid line is our result. {\\em Right}: The growth and nonlinear damping rates of streaming instability. With the locally enhanced flux, the growth rate of streaming instability becomes much larger than the mean Galactic value so that it can overcome the nonlinear damping by turbulence for a wide energy range. This is consistent with our earlier treatment in which streaming instability plays an essential role in the cosmic ray diffusion near SNRs. From \\cite{YLS12}.}\n \\label{rates}\n\\end{figure}\n\n\\begin{table}\n\\caption{Model parameters adopted}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\na&$\\chi$&$\\eta$&$ \\kappa$&$\\xi$&$\\alpha$\\\\\n\\hline\n0.1$\\sim 0.3$&$\\sim$0.05&$\\sim 0.3$&$0.04\\sim 0.1$& $0.2\\sim$0.4& 0.5\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Gyroresonance Instability of CRs in Compressible Turbulence}\n\n\\begin{figure}\n\\includegraphics[width=0.55\\columnwidth,\n  height=0.25\\textheight]{feedback.eps}\n\\includegraphics[width=0.4\\columnwidth,\n  height=0.25\\textheight]{YL_fig1c.eps}\n\\caption{{\\em Left}: The spectral energy density of slab waves that is transferred from the large scale compressible turbulence via the gyroresonance instability of CRs. In the case that the instability grows up to the maximum energy rate allowed by the turbulence cascade, large scale turbulence is truncated at $\\lambda_{fb}$ and the wave amplitude $E(k)dk\\sim \\epsilon_N^u$ is given by Eq.(\\ref{energy}). Note that the picture is different from LB06, namely, the feedback on the large scale turbulence occurs only in some cases when the scattering is not sufficient to prevent the waves from growing to the maximum values \\citep{YL11}; {\\em right}: CR scattering is dominated by compressible modes. For high energy CRs ($>\\sim$10GeV), the scattering is due to direct interaction with fast modes; For low energy CRs, the interaction is mainly due to the gyroresonance instability induced by compression of magnetic fields.}\n\\label{feedback_fig}\n\\end{figure}\n\nUntil recently, test particle approximation was assumed in most of earlier studies in which turbulence cascade is established from large scales and no feedback of CRs is included. This may not reflect the reality as we know the energy of CRs is comparable to that in turbulence and magnetic field \\citep[see][]{Kulsrudbook}. It was suggested by \\cite{LB06} that the gyroresonance instability of CRs can drain energy from the large scale turbulence and cause instability on small scales by the turbulence compression induced anisotropy on CRs (see Fig.\\ref{feedback_fig} {\\em left}). And the wave generated on the scales, in turn, provides additional scattering to CRs. In \\cite{YL11}, quantitative studies was provided based on the nonlinear theory of the growth of the instability and the feedback of the growing waves on the distributions of CRs.\n\nIn the presence of background compressible turbulence, the CR distribution is bound to be anisotropic because of the conservation of the first adiabatic invariant $\\mu\\equiv v_\\bot^2\/B$. Such anisotropic distribution is subjected to various instabilities. Waves are generated through the instabilities, enhancing the scattering rates of the particles, their distribution will be relaxed to the state of marginal state of instability even in the collisionless environment.  While the hydrodynamic instability requires certain threshold, the kinetic instability can grow very fast with small deviations from isotropy. Here, we focus on the gyroresonance instability. Both the qualitative and quantitative studies in \\cite{YL11} show that the isotropization rate is roughly $\n\\tau^{-1}_{scatt} \\sim \\frac{\\Gamma_{gr}\\epsilon_N}{\\beta_{CR} A}\\label{nu_est}$, where $\\Gamma_{cr}, \\epsilon_N$ are the instability growth rate and the wave energy normalized by magnetic energy, respectively. $\\beta_{CR}$ is the ratio of CR pressure to magnetic pressure, $A$ is the degree of anisotropy of the CR momentum distribution.\n\nBy balancing the rate of decrease in anisotropy due to scattering and the growth due to compression, one can get\n\\begin{equation}\n\\epsilon_N\\sim \\frac{ \\beta_{CR}\\omega\\delta v}{\\Gamma_{gr} v_A },~~~\\lambda_{CR}=r_p\/\\epsilon_N.\n\\label{epsilon_est},\n\\end{equation}\nwhere $v_A$ is the Alfv\\'en speed, $\\omega, \\delta v$ are the wave frequency and amplitude at the scale that effectively compresses the magnetic field and create anisotropy in CRs' distribution \\citep{YL11}. Since the growth rate decreases with energy, the instability only operates for low energy CRs ($\\sim<$ 100GeV, see Fig.\\ref{feedback_fig} {\\em right}) due to the damping by the preexisting turbulence \\citep{YL11} .\n  \n\\subsection{Bottle-neck for the growth of the instability and feedback on turbulence}\n\\label{feedback}\nThe creation of the slab waves through the CR resonant instability is another channel to drain the energy of large scale turbulence. This process, on one hand, can damp the turbulence. On the other hand, it means that the growth rate is limited by the turbulence cascade. The energy growth rate  cannot be larger than the turbulence energy cascading rate, which is $1\/2 \\rho V_L^4\/v_A\/L$ for fast modes in low $\\beta$ medium and $\\rho v_A^3\/l_A$ for slow modes in high $\\beta$ medium. This places a constraint on the growth, thus the upper limit of wave energy is given by\n\\begin{eqnarray}\n\\epsilon^u_N=\\cases{ M_A^2 L_i\/(L A)\\gamma^{\\alpha-1},& $\\beta<1$ \\cr\n  L_i\/(l_A A)\\gamma^{\\alpha-1}, & $\\beta>1$, \\cr}\n\\label{energy}\n\\end{eqnarray}\nwhere $\\gamma$ is the Lorentz factor and $L_i\\simeq 6.4\\times 10^{-7}(B\/5{\\rm \\mu G})(10^{-10}{\\rm cm}^3\/n_{cr})$ pc. The growth is induced by the compression at scales $\\sim< \\lambda_{CR}$. Therefore, in the case that $\\Gamma_{gr} \\epsilon$ reaches the energy cascading rate, fast modes are damped at the corresponding maximum turbulence pumping scale $\\lambda_{fb}=r_p\/\\epsilon_N$ (see Fig.\\ref{feedback_fig} {\\em left}).  If $\\lambda_{fb}$ is larger than the original damping scale $l_c$, then there is a feedback on the large scale compressible turbulence. This shows that test particle approach is not adequate and feedback should be included in future simulations.\n\n\n\\section{Summary}\n\nIn this chapter, we reviewed recent development on cosmic ray transport theories based on modern understanding of MHD turbulence. The main conclusions from both analytical study and test particle simulations in MHD turbulence are:\n\\begin{itemize}\n\\item Compressible fast modes are most important for CR scattering. CR transport therefore varies from place to place.\n\\item Nonlinear mirror interaction is essential for pitch angle scattering (including 90 degree).\n\\item Cross field transport is diffusive on large scales and super-diffusive on small scales.\n\\item Subdiffusion does not happen in 3D turbulence.\n\\item Self - generated waves are subject to damping by preexisting turbulence \n\\item Small scale waves can be generated in compressible turbulence by gyroresonance instability. Feedback of CRs on turbulence need to be included in future simulations. \n\\end{itemize}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}