Hydrodynamic modes for a granular gas from kinetic theory

Hydrodynamic modes for a granular gas from kinetic theory J. Javier Brey Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain James W. Dufty Department of Physics, University of Florida, Gainesville, Florida 32611, USA 共Received 7 February 2005; published 18 July 2005兲 Small perturbations of the homogeneous cooling state for a low density granular gas are described by means of the linearized Boltzmann equation. The spectrum of the generator for this dynamics is shown to contain points corresponding to hydrodynamic excitations. The corresponding eigenvectors and eigenvalues are calculated to Navier-Stokes order and shown to agree with those obtained by the Chapman-Enskog method. The conditions for the hydrodynamic excitations to dominate all other excitations are discussed. DOI: 10.1103/PhysRevE.72.011303 PACS number共s兲: 45.70.⫺n, 05.20.Dd, 51.10.⫹y I. INTRODUCTION The use of hydrodynamic equations to describe granular fluids in rapid flow has been in practice for many years 关1兴. The justification for this fluidlike description and prediction of the transport coefficients appearing in these equations has been the focus of attention for some time as well 关2–4兴.In recent years, an accurate derivation of Navier-Stokes order hydrodynamics has been given from the Boltzmann equation for granular gases using an adaptation of the Chapman- Enskog method for normal gases 关4,5兴. The expressions for the transport coefficients as a function of the degree of inelasticity have been confirmed by both Monte Carlo and molecular dynamics simulation 关6兴. Successful application of these Navier-Stokes equations to a number of different states also supports their validity 关7–9兴. However, the context of the hydrodynamic equations remains uncertain. What are the relevant space and time scales? How much inelasticity can be described in this way? Such questions can be addressed for gases using the Boltzmann kinetic equation to describe the complete dynamics for properties of interest. Then it can be asked under what conditions do the hydrodynamic excitations emerge as the dominant dynamics. The analysis of this problem for normal gases is quite complete and precise 关10,11兴. The objective here is to initiate a similar formulation of the problem for granular gases 关12兴. First, solutions to the Boltzmann equation are considered for states that deviate from spatial homogeneity only by small perturbations. The dynamics in this case is governed by the linear Boltzmann operator for spatially inhomogeneous states. The spectrum of this operator determines all possible excitations on all space and time scales, and for all degrees of inelasticity. The first problem is to identify points in this spectrum corresponding to hydrodynamics. This is one of the main results reported here. Both the hydrodynamic eigenvalues and eigenfunctions are calculated for long wavelength excitations corresponding to Navier-Stokes order hydrodynamics. Their agreement with the corresponding results from the Chapman-Enskog method is established. Next, the issue of conditions for the dominance of the hydrodynamic excitations, or modes, is considered. This entails showing that the hydrodynamic eigenvalues are smaller than all other parts of the spectrum, such that there is a long enough time scale for the latter to decay relative to the hydrodynamic modes. There are two new difficulties for granular gases. First, the hydrodynamic eigenvalues cannot be made arbitrarily small since they do not all scale with the inverse wavelength of the perturbation, as for normal gases. Second, there is a new characteristic frequency, the cooling rate, in addition to the collision frequency to set the scale of the spectrum. The cooling rate can be relatively large or small depending on the inelasticity of the gas particle collisions. Analysis of the nonhydrodynamic spectrum of the Boltzmann equation for a granular gas remains a difficult unsolved problem. Instead, we consider here a simpler kinetic model 关13兴that retains the exact hydrodynamic spectrum and allows identification of the entire spectrum as well. It is found that the time scale for the hydrodynamic excitations is longer than that for all other excitations, for any degree of inelasticity. A Brief Report of these results has been given in Ref. 关12兴, with the details and elaboration given below. The plan of the paper is as follows. In the next section, a short summary of the Boltzmann equation is given, and the Navier-Stokes equations derived from it by the Chapman- Enskog procedure are recalled. More details are given in Appendix A. Also, the hydrodynamic equations are linearized and the hydrodynamic modes to second order in the wave vector are identified. In Sec. III, the structure of the linear Boltzmann equation is discussed, emphasizing the relevance of the spectrum of the operator generating the linear dynamics to determine the existence and validity of hydrodynamics. The first of the above issues is addressed in Sec. IV, where the kinetic theory hydrodynamic modes are identified in the long wavelength limit. A technical point associated with the non Hermitian character of the operator is discussed in Appendix B. The results are extended to Navier-stokes order in Sec. V, and the obtained expressions are shown to agree with those derived in Sec. II from the hydrodynamic equations. Details of the calculations are given in Appendix C. The possibility of a description in terms of the hydrodynamic modes is studied in Sec. VI, by means of a model kinetic equation. It is estab- PHYSICAL REVIEW E 72, 011303 共2005兲 1539-3755/2005/72共1兲/011303共16兲/$23.00 ©2005 The American Physical Society011303-1 lished that there are length and time scales on which only the excitations associated to the hydrodynamic modes persist. The mathematical details are given in Appendix D. Finally, the last section contains a short summary of the main points and conclusions in the paper. II. BOLTZMANN EQUATION AND HYDRODYNAMIC MODES In all of the following, the simplest model of a granular fluid is considered: a low density gas of smooth and inelastic hard spheres 共d=3兲or disks 共d=2兲of mass mand diameter ␴ . The binary collision rule given below is characterized by a constant coefficient of restitution ␣ , defined in the interval 0⬍ ␣ 艋1, and measuring the loss of energy in each collision. At sufficiently low density, the distribution function f共r,v,t兲 is determined from the Boltzmann equation 关4,14,15兴 共 ⳵ t+v·⵱兲f=J关f,f兴,共1兲 where Jis the inelastic Boltzmann collision operator defined by J关X,Y兴⬅ ␴ d−1 冕 dv1 冕 d ␴ ˆ⌰共 ␴ ˆ·g兲 ␴ ˆ·g ⫻关 ␣ −2X共r,v⬘,t,兲Y共r,v1 ⬘,t兲−X共r,v,t兲Y共r,v1,t兲兴, 共2兲 for arbitrary functions X共v兲and Y共v兲. Here, ␴ ˆis a unit vector along the line joining the centers of the colliding pair, ⌰ is the Heaviside step function, and g=v−v1. The primes on the velocities denote the initial values 兵v1 ⬘,v2 ⬘其that lead to 兵v1,v2其following a “restituting” binary collision v⬘=v−1 2共1+ ␣ −1兲共 ␴ ˆ·g兲 ␴ ˆ,v1 ⬘=v1+1 2共1+ ␣ −1兲共 ␴ ˆ·g兲 ␴ ˆ. 共3兲 The usual Boltmann collision operator is recovered from Eq. 共2兲in the elastic limit ␣ =1. The macroscopic variables of interest are the hydrodynamic fields, i.e., the density n共r,t兲, the flow velocity u共r,t兲, and the 共granular兲temperature T共r,t兲. They are defined as moments of the solution to the Boltzmann equation 冢 n共r,t兲 n共r,t兲u共r,t兲 d 2n共r,t兲T共r,t兲 冣 = 冕 dv 冢 1 v 1 2m 共 v−u 兲 2 冣 f共r,v,t兲.共4兲 An exact set of equations for these variables is obtained from the following properties of the collision operator 冕 dv 冢 1 v 1 2m共v−u兲2 冣 J关f,f兴= 冢 0 0 −d 2nT ␨ 冣 .共5兲 The first two equations follow from conservation of mass and momentum in the particle collisions. The last equation reflects the loss of energy in collisions due to the inelasticity. This appears through the “cooling rate” ␨ 共r,t兲艌0, defined by this equation. The macroscopic balance equations resulting from the above properties are ⳵ tn+⵱·共nu兲=0, 共6兲 共 ⳵ t+u·⵱兲u+共mn兲−1关⵱共nT兲+⵱·⌸兴=0, 共7兲 共 ⳵ t+u·⵱+ ␨ 兲T+2 nd共nT ⵱·u+⌸:⵱u+⵱·q兲=0. 共8兲 The functionals giving the dissipative part of the pressure tensor ⌸and the heat flux qare also moments of the solution to the Boltzmann equation ⌸共r,t兲= 冕 dvm 冉 VV −V2 d I 冊 f共r,v,t兲,共9兲 q共r,t兲= 冕 dv 冉 m 2V2−d+2 2T 冊 Vf共r,v,t兲,共10兲 where Iis the unit tensor of dimension dand V⬅v−uthe so-called peculiar velocity. Equations 共6兲–共8兲are the basis for a hydrodynamic description, once ⌸共r,t兲,q共r,t兲, and ␨ 共r,t兲are specified. These can be obtained from their definitions using a solution of the Boltzmann equation generated by the Chapman-Enskog method. This method assumes the existence of a solution whose space and time dependence is given entirely through the hydrodynamic fields and their gradients 共i.e., a “normal” solution兲. As a result, ⌸共r,t兲and q共r,t兲are given in terms of these variables, and Eqs. 共6兲–共8兲 become a closed set of hydrodynamic equations. The primary results of this method are recalled in Appendix A. To leading order in the spatial gradients, the dissipative fluxes are found to be given by 关5,6兴 ⌸ij共r,t兲=− ␩ 冉 ⵱iuj+⵱jui−2 d ␦ ij ⵱·u 冊 ,共11兲 q共r,t兲=− ␬ ⵱T− ␮ ⵱n.共12兲 For the cooling rate the result is ␨ 共r,t兲=n ␴ d−1 冉 2T m 冊 1/2 ␨ 0 *+ ␨ 1ⵜ2T+ ␨ 2ⵜ2n + bilinear in ⵱n,⵱T,⵱uterms, 共13兲 with ␨ 0 *being a dimensionless positive constant proportional to 共1− ␣ 2兲. The explicit forms for the shear viscosity ␩ , the thermal conductivity ␬ , and the new transport coefficients ␮ , ␨ 1, and ␨ 2, peculiar to granular gases, are given in Appendix A. The nonlinear contributions to ␨ 共t兲indicated in Eq. 共13兲play no role in the following linear analysis and will be not discussed further in this paper. Equations 共6兲–共8兲together with the “constitutive relations” 共11兲–共13兲are the Navier-Stokes hydrodynamic equa- J. J. BREY AND J. W. DUFTY PHYSICAL REVIEW E 72, 011303 共2005兲 011303-2 tions for a granular gas. A special solution of these equations is that for a homogeneous state, characterized by a constant density nHCS, a vanishing velocity flow uHCS=0, and a uniform temperature THCS共t兲determined from ⳵ tTHCS共t兲=−nHCS ␴ d−1 冉 2 m 冊 1/2 ␨ 0 *THCS 3/2 共t兲.共14兲 This is referred to as the homogeneous cooling state 共HCS兲, since the only macroscopic dynamics is the monotonic decrease in the temperature field. It is easily seen that the HCS as defined above is also a solution of the exact balance equations, taking into account that ␨ HCS⬀nHCS ␴ d−1共THCS/m兲1/2 on dimensional grounds. In the following, the solution to the Boltzmann equation will be considered for small perturbations of the HCS, and it will be useful to have the corresponding results from Navier-Stokes hydrodynamics. The linearized Navier-Stokes equations have time dependent coefficients, since the distribution function of the reference HCS depends on time. This complication can be overcome by the introduction of suitable dimensionless space and time scales, as well as scaled hydrodynamic fields. Thus we define r*and sby r*=r ᐉ,ds =vHCS共t兲 ᐉdt.共15兲 Here, vHCS共t兲⬅关2THCS共t兲/m兴1/2 is the thermal velocity in the HCS, and ᐉ⬅共nHCS ␴ d−1兲−1 is proportional to the mean free path of the particles. The dimensionless fields ␦ yj共r*,s兲are chosen as ␳ 共r*,s兲=n共r,t兲−nHCS nHCS , ␪ 共r*,s兲=T共r,t兲−THCS共t兲 THCS共t兲, ␻ 共r*,s兲=u共r,t兲 vHCS共t兲.共16兲 In addition, since the equations are linear and the reference state is homogeneous, it is sufficient to consider a single Fourier mode ␦ yj共r*,s兲=eik·r* ␦ y ˜ j共k,s兲.共17兲 The velocity components are chosen as a longitudinal component relative to k, ␻ ˜ 储 共k,s兲=k ˆ· ␻ ˜ 共k,s兲, and d−1 transverse components ␻ ˜ ⬜,i共k,s兲=e ˆi· ␻ ˜ 共k,s兲, where 兵k ˆ⬅k/k,e ˆ共i兲,i =1,…,d−1其are a set of dpairwise orthogonal unit vectors . The dimensionless linear Navier-Stokes equations then become a system of ordinary differential equations with constant coefficients that can be expressed in the compact form 关5兴 ⳵ s ␦ y ˜ 共k,s兲+K共k兲· ␦ y ˜ 共k,s兲=0, 共18兲 where we use a 共d+2兲dimensional space representation with ␦ y ˜ 共k,s兲being the vector ␦ y ˜ 共k,s兲= 冢 ␳ ˜ 共k,s兲 ␪ ˜ 共k,s兲 ␻ ˜ 储 共k,s兲 ␻ ˜ ⬜共k,s兲 冣 ,共19兲 and ␻ ˜ ⬜共k,s兲denoting the vector formed by the d−1 components ␻ ˜ ⬜,i共k,s兲. The matrix K共k兲is block diagonal, expressing the decoupling of transverse and longitudinal modes K共k兲= 冉 K10 0K2 冊 ,共20兲 K1共k兲= 冢 00 ik ␨ 0 *+共 ␮ *− ␨ 2 *兲k2 ␨ 0 * 2+共 ␬ *− ␨ 1 *兲k22i dk i 2ki 2k− ␨ 0 * 2+2共d−1兲 d ␩ *k2 冣 ,共21兲 K2共k兲=− 冉 ␨ 0 * 2− ␩ *k2 冊 I.共22兲 In the above expressions, K2and Iare matrices of dimension d−1. The dimensionless transport coefficients used in the above equations are defined by ␩ *= ␩ ᐉmnHCSvHCS , ␬ *=2 ␬ dᐉnHCSvHCS , ␮ *=2 ␮ dᐉTHCSvHCS , 共23兲 ␨ 1 *=THCS ␨ 1 ᐉvHCS , ␨ 2 *=nHCS ␨ 2 ᐉvHCS .共24兲 The formal solution to the initial value problem 共18兲is ␦ y ˜ 共k,s兲=e−K共k兲s· ␦ y ˜ 共k,0兲.共25兲 The eigenvalues and eigenvectors of the generator Kfor this dynamics define the d+2 Navier-Stokes order hydrodynamic modes. They are given by the solutions of the equation HYDRODYNAMIC MODES FOR A GRANULAR GAS FROM…PHYSICAL REVIEW E 72, 011303 共2005兲 011303-3 K共k兲· ␸ j共k兲=␭j共k兲 ␸ j共k兲,j=1,…,d+2. 共26兲 A simple calculation provides the expressions for the eigenvalues to order k2. They are given by ␭1共k兲=k2 ␨ 0 *,␭2共k兲= ␨ 0 * 2− 冉 d+1 d ␨ 0 *− ␬ *+ ␨ 1 * 冊 k2, ␭ 储 共k兲=− ␨ 0 * 2+ 冋 1 d ␨ 0 *+2共d−1兲 ␩ * d 册 k2,␭⬜共k兲=− ␨ 0 * 2+ ␩ *k2, 共27兲 the eigenvalue ␭⬜共k兲being 共d−1兲-fold degenerate. The corresponding eigenvectors to leading order in kare ␸ 1共k兲= 冢 1 −2 0 0 冣 , ␸ 2共k兲= 冢 0 1 0 0 冣 , ␸ 储 共k兲= 冢 0 0 1 0 冣 , ␸ ⬜,i共k兲= 冢 0 0 0 i ˆ 冣 .共28兲 Here, 0=0, and i ˆ=1 ˆ=1, for d=2,0= 共 0 0 兲 ,1 ˆ= 共 1 0 兲 , and 2 ˆ= 共 0 1 兲 , for d=3. The first of these modes is excited by the condition ␪ ˜ 共k,0兲=−2 ␳ ˜ 共k,0兲at zero flow velocity. The second is produced by a temperature perturbation at constant density and also zero velocity, while the third one corresponds to a longitudinal velocity perturbation at constant temperature and density. There is a 共d−1兲-fold degeneracy for the shear modes of eigenvalue ␭⬜. These diffusive modes are excited by a perturbation of the velocity field in the transverse plane orthogonal to k. It should be noted that while the above analysis is restricted to the Navier-Stokes equations, derived from the Boltzmann equation by the Chapman-Enskog method, the eigenvalues and eigenvectors to order kfollow more generally from the exact macroscopic balance equations and do not depend on the approximate constitutive equations 共11兲 and 共12兲. The hydrodynamic modes sought by kinetic theory in the subsequent sections can therefore be defined as those excitations due to small perturbations which agree with the above in the long wavelength limit. Analyticity then allows extension of that identification to shorter wavelengths. A consistency check of the Chapman-Enskog method is agreement with the above results at order k2. However, the concept of hydrodynamic modes in this context does not require the validity of the Chapman-Enskog method nor the limitation to the Navier-Stokes approximation. III. LINEAR BOLTZMANN EQUATION A more complete and accurate description of the response to small spatial perturbations of the density, temperature, and flow velocity is obtained directly from the Boltzmann equation. Consider first an isolated system. As already indicated in the previous section, the exact balance equations have a solution describing the HCS, with a monotonically decreasing temperature obeying Eq. 共14兲. The solution to the Boltzmann equation corresponding to this macroscopic state is characterized by the scaling form 关4兴 fHCS共v,t兲=nHCSvHCS −d共t兲 ␾ 共V*兲,共29兲 with V*=V vHCS共t兲=v−uHCS vHCS共t兲.共30兲 Substitution into the Boltzmann equation gives ␨ HCS 2 ⳵ ⳵ V·共VfHCS兲=J关fHCS,fHCS兴.共31兲 For later convenience, a constant velocity uHCS for the system as a whole has been included, although this can always be removed by means of a Gallilean transformation. Then, in the following it will be considered that uHCS=0, unless it be explicitly established otherwise. An exact and explicit solution of Eq. 共31兲is not known yet, but the behavior of ␾ at large and small velocities has been determined 关14兴and the results obtained by the direct simulation Monte Carlo method strongly supports the existence of such a scaling form 关16兴. The HCS distribution function is a “universal” homogeneous solution in the same sense as the Maxwellian for elastic collisions. An arbitrary homogeneous state is expected to approach the HCS after a few collisions. Therefore, in discussing response of any homogeneous state to small spatial perturbations, it is sufficient to consider the HCS as the reference state. All the other cases will simply induce additional short time transients. Consider then small spatial perturbations of the HCS f共r,v,t兲=fHCS共v,t兲关1+⌬共r,v,t兲兴,兩⌬共r,v,t兲兩 Ⰶ1. 共32兲 To linear order in ⌬共r,v,t兲, the Boltzmann equation becomes 关 ⳵ t+v·⵱+L共t兲兴共fHCS⌬兲=0, 共33兲 where L共t兲is the linearized Boltzmann collision operator given by L共t兲X共v兲=−J关fHCS,X兴−J关X,fHCS兴,共34兲 for arbitrary X共v兲. Just as for the analysis of the Navier- Stokes equations in the previous section, the above linear kinetic equation takes a simpler form when expressed in terms of the dimensionless variables defined in Eq. 共15兲, and considering a single Fourier mode ⌬共r,v,t兲=eik·r*⌬ ˜ 共k,v*,s兲,共35兲 where v*=v/vHCS共t兲. Then, Eq. 共33兲becomes 共 ⳵ s+ik·v*+L*兲⌬ ˜ 共k,v*,s兲=0. 共36兲 The dimensionless operator L*is now time independent and it is given by J. J. BREY AND J. W. DUFTY PHYSICAL REVIEW E 72, 011303 共2005兲 011303-4 L*X⬅ ␨ 0 * 2 ␾ −1 ⳵ ⳵ v*·共v* ␾ X兲+L*X,共37兲 where L*is the dimensionless linear Boltzmann collision operator L*X⬅− ␾ −1共J*关 ␾ , ␾ X兴+J*关 ␾ X, ␾ 兴兲,共38兲 J*关X,Y兴⬅ 冕 dv1 * 冕 d ␴ ˆ⌰共 ␴ ˆ·g*兲 ␴ ˆ·g*关 ␣ −2X共v*⬘兲Y共v1 *⬘兲 −X共v*兲Y共v1 *兲兴.共39兲 Here, v*⬘and v1 *⬘are related with v*and v1 *by Eqs. 共3兲. The operator L*differs from the linearized Boltzmann collision operator L*by terms representing the cooling effects of the inelastic collisions. The latter arise because the derivative with respect to sis taken at constant v*rather than v. Solutions to the dimensionless, linear kinetic equation 共36兲for ⌬ ˜ are sought in a Hilbert space whose scalar product is defined by 共X,Y兲= 冕 dv* ␾ 共v*兲X†共v*兲Y共v*兲,共40兲 with the dagger denoting complex conjugation. The formal solution to the kinetic equation is ⌬ ˜ 共k,v*,s兲=1 2 ␲ i 冖 dz e−zsR共z兲⌬*共k,v*,0兲,共41兲 R共z兲⬅共z−ik·v*−L*兲−1,共42兲 where the contour encloses the entire spectrum of ik·v* +L*, both point and residual, counterclockwise. It is important to realize that all the linear excitations of the granular gas are determined from this spectrum. This formulation of the problem for small spatial perturbations provides a precise context for addressing many questions regarding hydrodynamics for a granular gas. The existence of the hydrodynamic modes and their role relative to other dynamical processes are determined by the characterization of the above spectrum. To see how this occurs, it is useful to recall briefly the status of the corresponding problem for gases with elastic collisions 关10,11兴. In that case, it has been proved that the hydrodynamic modes exist as d+2 poles located at the origin in the long wavelength limit and corresponding to the local conserved quantities. Furthermore, the remainder of the spectrum is bounded away from these poles, and the spectrum is analytic in kabout k=0, so this isolation of the hydrodynamic modes is preserved at finite wavelengths. IV. EXISTENCE OF HYDRODYNAMIC MODES The spectrum of ik·v*+L*is expected to be quite complex, based on the special case of elastic collisions, with points spectra, continua, and limit points. The hydrodynamic excitations, whenever they exist, are part of the point spectrum so in order to investigate them it suffices to consider the eigenvalue problem 共ik·v*+L*兲⌿i=␭i共k兲⌿i.共43兲 The search for hydrodynamic excitations can be carried out by assuming they are analytic in kand looking first for the k=0solutions of Eq. 共43兲. The practical issue of constructing these modes at finite kis addressed in the next section. The central idea for constructing the hydrodynamic eigenvalues and eigenvectors at k=0 is the note that the HCS is parametrized by the hydrodynamic fields nHCS,THCS, and uHCS, which is now considered different from zero, as discussed at the beginning of the previous section. Therefore, differentiating the Boltzmann equation for the distribution function of the HCS, Eq. 共31兲, with respect to these fields gives exact properties of the linearized Boltzmann collision operator. For example, ⳵ ⳵ nHCS 再 ␨ HCS 2 ⳵ ⳵ V·共VfHCS兲−J关fHCS,fHCS兴 冎 =0 共44兲 gives directly 1 2 ⳵␨ HCS ⳵ nHCS ⳵ ⳵ V· 共 VfHCS 兲 + ␨ HCS 2 ⳵ ⳵ V· 冉 V ⳵ fHCS ⳵ nHCS 冊 +L ⳵ fHCS ⳵ nHCS =0. 共45兲 The derivatives of ␨ HCS and fHCS are easily calculated from the properties ␨ HCS⬀nHCS and fHCS⬀nHCS. In terms of the dimensionless variables, and setting uHCS=0, Eq. 共45兲be- comes L*1=− ␨ 0 * 2 ␾ −1 ⳵ ⳵ v*·共v* ␾ 兲.共46兲 Next, calculate the derivative of Eq. 共31兲with respect to THCS, 1 2 ⳵␨ HCS ⳵ THCS ⳵ ⳵ V·共VfHCS兲+ ␨ HCS 2 ⳵ ⳵ V· 冉 V ⳵ fHCS ⳵ THCS 冊 +L ⳵ fHCS ⳵ THCS =0. 共47兲 Since fHCS has the scaling form 共29兲,itis ⳵ fHCS ⳵ THCS =− fHCS 2THCS 冉 d+V· ⳵ ln fHCS ⳵ V 冊 ,共48兲 and, taking into account that ␨ HCS⬀THCS 1/2 , Eq. 共47兲becomes ␨ HCS 2 ⳵ ⳵ V·共VfHCS兲− ␨ HCS 2 ⳵ ⳵ V· 冋 VfHCS 冉 d+V· ⳵ ln fHCS ⳵ V 冊 册 −L 冋 fHCS 冉 d+V· ⳵ ln fHCS ⳵ V 冊 册 =0. 共49兲 Setting uHCS=0and transforming to dimensionless variables, the above equation yields L* 冉 d+v*· ⳵ ln ␾ ⳵ v* 冊 = ␨ 0 * 2 冉 d+v*· ⳵ ln ␾ ⳵ v* 冊 .共50兲 Finally, differentiating Eq. 共31兲with respect to uHCS, afterwards setting uHCS=0, and introducing dimensionless variables leads in a similar way to HYDRODYNAMIC MODES FOR A GRANULAR GAS FROM…PHYSICAL REVIEW E 72, 011303 共2005兲 011303-5 L* 冉 ⳵ ln ␾ ⳵ v* 冊 =− ␨ 0 * 2 冉 ⳵ ln ␾ ⳵ v* 冊 .共51兲 The d+2 equations 共46兲,共50兲, and 共51兲, provide exact properties of L*. In fact, Eqs. 共50兲and 共51兲are of the form of the eigenvalue problem to be solved at k=0, with eigenvalues given by ␨ 0 */2 and − ␨ 0 */2, respectively. It is straightforward to construct linear combinations of Eqs. 共46兲and 共50兲 to obtain an additional eigenvalue and eigenvector. The results can then be expressed as 关12兴 L*⌿i共0兲=␭i共0兲⌿i共0兲,i=1,…,d+2, 共52兲 with 兵␭i共0兲其 = 再 0, ␨ 0 * 2,− ␨ 0 * 2 冎 ,共53兲 兵⌿i共0兲其 = 再 d+1+v*· ⳵ ln ␾ ⳵ v*,−d−v*· ⳵ ln ␾ ⳵ v*,−k ˆ· ⳵ ln ␾ ⳵ v*, − ⳵ ln ␾ ⳵ v⬜ * 冎 .共54兲 The eigenvalue − ␨ 0 */2 is d-fold degenerated. For convenience for the perturbation calculation to be carried out in the following section, the subspace associated to it has been rearranged. The velocity v*has been decomposed into its component in the direction of k,v 储 *=k ˆ·v*, and the remaining d−1 ones, forming with it a set of dpairwise orthogonal components, i.e., v⬜ *is defined by the d−1 components v*·e ˆ共i兲. When appropriate, the set of the deigenfunctions associated with the eigenvalue − ␨ 0 */2 will be denoted by ⌿3共0兲. Clearly, the above are the long wavelength limit of the hydrodynamic modes defined by Eq. 共27兲. This is a the first primary result of the analysis developed here, demonstration of the existence of hydrodynamic excitations in the spectrum of the linearized Boltzmann equation. The results are exact and apply for arbitrary degree of dissipation. These long wavelength eigenfunctions are determined from the HCS distribution which depends only on the magnitude of v*, so the terms on the right side of Eq. 共54兲are all determined from ⳵ ln ␾ 共v*兲/ ⳵ v*. In the elastic limit, ␾ 共v*兲 becomes Gaussian, and the hydrodynamic eigenfunctions become linear combinations of 1, v*, and v*2, i.e., of the summational invariants for the conservation laws of mass, momentum, and energy, as expected. For inelastic collisions, the eigenfunctions are quite different, particularly at large v*due to an overpopulation in the HCS distribution relative to the Gaussian. This is illustrated in Figs. 1 and 2, where ⳵ ln ␾ 共v兲/ ⳵ v*has been obtained from direct simulation Monte Carlo solution of the Boltzmann equation for the HCS 关17兴. The set of eigenfunctions ⌿i共0兲span a 共d+2兲- dimensional subspace of the Hilbert space, but they are not orthogonal. Consistently, it is easily verified that the operator L*is not Hermitian. Therefore, it is useful to introduce a biorthogonal set of functions ⌽i,i=1,…,d+2, with the requirement „⌽i,⌿j共0兲…= ␦ ij.共55兲 To identify the appropriate set of functions, first note that 共1,L*X兲=0, 共v*,L*X兲=− ␨ 0 * 2共v*,X兲,共56兲 as a consequence of the number of particles and momentum conservation in the moment equations 共5兲. Therefore, 1 and v*are eigenfunctions of the adjoint of L*,L*†, with eigenvalues 0 and − ␨ 0 */2, respectively. This gives the set 兵⌽i其=兵1,⌽2,k ˆ·v*,v⬜ *其.共57兲 The final choice of ⌽2does not appear to be unique. This is discussed further in Appendix B. For the purposes of the next section, it suffices to make the choice FIG. 1. Plot of 共ln ␾ 兲⬘⬅ ⳵ ln ␾ / ⳵ v*as a function of v*for d =3 and ␣ =0.95. The circles are the numerical derivative of the DSMC results and the solid line is the Gaussian. Quantities are measured in the dimensionless units defined in the text. FIG. 2. The same as Fig. 1 but for ␣ =0.6. J. J. BREY AND J. W. DUFTY PHYSICAL REVIEW E 72, 011303 共2005兲 011303-6 兵⌽i其= 再 1, v*2 d+1 2,k ˆ·v*,v⬜ * 冎 .共58兲 The function ⌽2is not an eigenfunction of the adjoint operator L*†, but it is easily verified that the biorthogonality conditions 共55兲are satisfied. V. NAVIER-STOKES ORDER MODES In the previous section, the hydrodynamic modes were identified in the long wavelength limit. Assuming analyticity, their existence at finite wavevectors can be inferred. Furthermore, their explicit construction is possible by perturbation theory. This construction provides a critical test of the internal consistency of other quite different approaches 共e.g., the Chapman-Enskog method discussed above and in Appendix A兲. In particular, the detailed form of the eigenvalues and the dependence of the associated transport coefficients on the restitution coefficient should be exactly the same. This is demonstrated to Navier-Stokes order in this section. Return now to the eigenvalue problem 共43兲and consider the case of kⰆ1. Look for solutions with the expansion 共a more complete characterization of the conditions for this perturbation expansion is given below兲, ⌿i共k兲=⌿i共0兲+k⌿i 共1兲+k2⌿i 共2兲+¯,共59兲 ␭i共k兲=␭i共0兲+k␭i 共1兲+k2␭i 共2兲+¯.共60兲 The reference eigenfunctions ⌿i共0兲and eigenvalues ␭i共0兲are taken to be the long wavelength hydrodynamic results of Eqs. 共53兲and 共54兲. As already indicated, there is a d-fold degeneracy for the eigenvalue ␭共0兲=− ␨ 0 */2. However, the d-dimensional subspace spanned by its eigenvectors was naturally partitioned by symmetry into the longitudinal and transverse components. The eigenvectors for the transverse modes decouple from the remaining three modes even at finite wave vector for the same symmetry reasons, so there are no complications of degenerate perturbation theory. In the longitudinal subspace, all the eigenvalues are distinct, except in the elastic limit ␣ =1, where there is a threefold degeneracy in this subspace. Thus the two cases of unperturbed reference states with ␣ =1 and ␣ ⬍1 must be distinguished. In the former case, the eigenvalues behave as ␭i共k, ␣ 兲=关␭i共0兲+k␭i 共1兲+k2␭i 共2兲+¯兴 ␣ =1 + terms of order 共1− ␣ 兲,共61兲 and are regular in 共1− ␣ 兲. This occurs when the degree of dissipation because of inelasticity is small relative to the effects of the spatial variation. The corresponding eigenvalues are then similar to those of a normal gas, with d−1 shear diffusion modes, two sound modes, and a heat diffusion mode. Here attention is restricted to the more interesting and relevant second case of fixed ␣ ⬍1 with small spatial perturbations. It will be seen that the modes are now qualitatively different, consistently with the results reported in Sec. II, since the degeneracy of the elastic limit is lifted at the outset by the finite dissipation. To set up the perturbation expansion, projection operators Pifor the biorthogonal set 兵⌽i,⌿i共0兲其 are defined by PiX=⌿i共0兲共⌽i,X兲,共62兲 for an arbitrary element Xin the Hilbert space. The eigenvalue problem 共43兲can be rearranged as 关L*−␭i共0兲兴⌿i共k兲=关␭i共k兲−␭i共0兲−ik·v*兴⌿i共k兲.共63兲 Then, operating on both sides of this equation with Qi ⬅1−Pi, and using the property 关L*−␭i共0兲兴Pi=0, this becomes Qi关L*−␭i共0兲兴Qi⌿i共k兲=Qi关␭i共k兲−␭i共0兲−ik·v*兴⌿i共k兲. 共64兲 By construction, the right side of the above equation is orthogonal to the null space for the adjoint of 关L*−␭i共0兲兴 , and the Fredholm alternative assures solutions to this equation 关18兴. The eigenvalue problem for ⌿i共k兲is determined only up to an overall scale factor, amounting to the choice of normalization. It is convenient to choose „⌽i,⌿i共k兲…=1, 共65兲 implying 共⌽i,⌿i 共n兲兲=0 for n艌1. With this, Eq. 共64兲gives two sets of equations for the eigenvalues and eigenvectors ␭i共k兲=␭i共0兲+ 共 ⌽i,ik·v*⌿i共k兲 兲 + 共 ⌽i,关L*−␭i共0兲兴Qi⌿i共k兲 兲 , 共66兲 Qi⌿i共k兲=兵Qi关L*−␭i共0兲兴Qi其−1Qi关␭i共k兲−␭i共0兲 −ik·v*兴⌿i共k兲.共67兲 To zeroth order in k, these equations give the hydrodynamic modes of the last section in the long wavelength limit, consistently. To first order in k, the eigenvectors are ⌿i 共1兲共k兲=Qi⌿i 共1兲共k兲=兵Qi关L*−␭i共0兲兴其−1Qi共␭i 共1兲 −ik ˆ·v*兲⌿i共0兲=−兵Qi关L* −␭i共0兲兴其−1Qiik ˆ·v*⌿i共0兲.共68兲 The first equality in the above transformations is a consequence of the normalization condition 共65兲. For the first order eigenvalues it is found ␭i 共1兲= 共 ⌽i,ik ˆ·v*⌿i共0兲 兲 + 共 ⌽i,L*Qi⌿i 共1兲 兲 =0. 共69兲 Now the last equality follows from the fact that each ⌽i and ⌿i共0兲have the same definite parity under the change v*→−v*and the distribution function of the HCS, ␾ 共v*兲, defining the scalar product is invariant under this change. Thus the first term on the right side vanishes. The second term also vanishes for similar reasons, since ⌽iand ⌿i 共1兲 have opposite parity and L*is invariant under the change in sign of the velocity. To second order in kthe eigenvectors and eigenvalues are given by Qi⌿i 共2兲=−兵Qi关L*−␭i共0兲兴其−1Qiik ˆ·v*⌿i 共1兲,共70兲 HYDRODYNAMIC MODES FOR A GRANULAR GAS FROM…PHYSICAL REVIEW E 72, 011303 共2005兲 011303-7 ␭i 共2兲=共⌽i,ik·v*⌿i 共1兲兲+共⌽i,L*Qi⌿i 共2兲兲.共71兲 The above expressions for the eigenvalues are evaluated in Appendix C. The results have the same forms as given in Eqs. 共27兲. Furthermore, the reduced transport coefficients ␩ *, ␬ *, and ␨ 1 *are determined from the same integral equations as following from the Chapman-Enskog solution summarized in Appendix A. This confirms that the hydrodynamic modes determined from the spectrum of the linearized Boltzmann equation are consistently determined to Navier- Stokes order by both methods. VI. “AGEING” TO HYDRODYNAMICS The existence of hydrodynamic excitations only assures that there is a hydrodynamic contribution to the dynamics of small perturbations of the HCS. To establish a description in terms of these hydrodynamic excitations alone, it is necessary to characterize the rest of the excitations in the spectrum. For gases with elastic collisions, it has been shown that for sufficiently small kthe hydrodynamic excitations are smaller in magnitude than all other excitations, and bounded away from them 关10,11兴. Consequently, there is a time scale beyond which only the hydrodynamic excitations persist. It is on this space and time scales that hydrodynamics in the usual sense applies. Typically, the conditions are wavelengths larger than the mean free path and times later than the mean free time. This leaves a large domain of macroscopic space and time scales for hydrodynamics. The extension of this concept of “ageing to hydrodynamics” for granular gases is expected, but its verification is not so straightforward. The mathematical analysis for elastic collisions does not transfer to the granular gas due to the significant differences in the linear collision operator. There are qualitative differences in the hydrodynamic modes. For example, the fact that energy is not conserved means that the hydrodynamic excitations cannot be made arbitrarily small simply by making ksmall. The fastest decaying hydrodynamic modes is that with eigenvalue ␨ 0 */2 at long wavelengths. This has its maximum value at large dissipation, and the question arises as to whether the time scale for this mode can become comparable to or exceed those of the nonhydrodynamic modes at strong dissipation. A. Model kinetic equation Current analysis of the spectrum of the linearized Boltzmann operator for granular gases appears to be limited to the hydrodynamic excitations discussed here, with no characterization of the rest of the spectrum as yet. Consequently, in the remainder of this presentation these questions are addressed in the context of a model kinetic equation. This model 关19兴is an extension of the familiar Bhatnager, Gross, Krook 共BGK兲single relaxation time model for normal gases 关20兴. The Boltzmann equation can be formally written in the form 冉 ⳵ ⳵ t+v·⵱ 冊 f=− ␯ 共f−g兲.共72兲 There are two significant differences of the model considered here with respect to the original Boltzmann equation. First, the collision frequency is replaced by a velocity independent function of the local density and temperature, ␯ = ␯ 共n,T兲. Second, the gain term of the Boltzmann equation is replaced by ␯ g, where gis taken to be a Gaussian function of the velocity g共r,v,t兲=n 冋 b共T兲 ␲ 册 d/2 e−b共T兲V2.共73兲 Here Vis the peculiar velocity defined below Eq. 共10兲. The parameter bis chosen to be function of the local density and temperature so as to enforce the moment conditions 共5兲 above. This leads to the identification b共T兲=m 2T共1− ␨ / ␯ 兲.共74兲 In this way, it is assured that the exact macroscopic balance equations 共6兲–共10兲are preserved by the model. Then the Chapman-Enskog method leads to the same Navier-Stokes hydrodynamic equations as for the Boltzmann equation, with only the transport coefficients being different. In the following, it will be considered that the expression for the collision frequency ␯ is chosen as scaling with nT1/2, in order to mimic the hard sphere behavior. Dimensional analysis then implies the same scaling for the cooling rate ␨ . Moreover, note that consistency of the model kinetic equation requires that ␨ ⬍ ␯ for all values of ␣ . A possible choice for the cooling rate is to be the same as obtained from the Boltzmann equation by using a local Maxwellian for the distribution function. In the same spirit, the collision frequency can be fixed by fitting one of the transport coefficients of the model to that obtained from the Boltzamnn equation by the Chapman-Enskog procedure in the first Sonine approximation. If the shear viscosity ␩ is used, the above choices lead to 关21兴 ␯ 共r,t兲=共3−3 ␣ +2d兲共1+ ␣ 兲 4d ␯ 0共r,t兲,共75兲 ␨ 共r,t兲=共2+d兲共1− ␣ 2兲 4d ␯ 0共r,t兲,共76兲 where ␯ 0共r,t兲is an average local collision frequency ␯ 0共r,t兲=8 ␲ 共d−1兲/2n ␴ d−1 共2+d兲⌫共d/2兲 冉 T m 冊 1/2 .共77兲 The above expressions yield ␨ 共r,t兲 ␯ 共r,t兲=共2+d兲共1− ␣ 兲 3−3 ␣ +2d艋2+d 3+2d⬍1, 共78兲 in agreement with the model consistency requirement. This kinetic model reduces to the BGK model for normal gases at ␣ =1 关20兴. Otherwise it reproduces all of the qualitative features of the granular Boltzmann equation, including a nontrivial HCS and the same hydrodynamic excitations discussed in the sections above. In some respects, the model kinetic equation is more complex than the Boltzmann equation since the collision operator is a nonlinear functional of f through the dependence of gon Tand u. However, the lin- J. J. BREY AND J. W. DUFTY PHYSICAL REVIEW E 72, 011303 共2005兲 011303-8 earized model kinetic equation for small perturbations of the HCS is considerably simpler than that for the Boltzmann equation, as shown below. B. Model HCS and linear model kinetic equation The HCS equation 共31兲for this model becomes in dimensionless form ␨ 0 * 2 ⳵ ⳵ v*·共v* ␾ 兲+ ␯ 0 * ␾ = ␯ 0 *关 ␲ 共1− ␨ 0 */ ␯ 0 *兲兴−d/2exp 冉 −v*2 1− ␨ 0 */ ␯ 0 * 冊 , 共79兲 where ␨ 0 *⬅ᐉ ␨ HCS/vHCS共t兲, as in the preceding sections and, consistently, ␯ 0 *⬅ᐉ ␯ HCS/vHCS共t兲. The solution to the above equation is ␾ 共v*兲= ␯ 0 *关 ␲ 共1− ␨ 0 */ ␯ 0 *兲兴−d/2 冕 0 ⬁ ds ⫻exp 冋 − 冉 d ␨ 0 * 2+ ␯ 0 * 冊 s 册 exp 冉 −e− ␨ 0 *sv*2 1− ␨ 0 */ ␯ 0 * 冊 .共80兲 It is easily verified that this distribution exhibits algebraic decay for large velocities ␾ 共v*兲⬃p ␲ −d/2 2 冉 p p−2 冊 −p/2 ⌫ 冉 p+d 2 冊 v*−共p+d兲,共81兲 with p=2 ␯ 0 */ ␨ 0 *. Therefore, moments of degree por greater do not exist. The linearized kinetic model equation for small perturbations of the HCS is obtained in Appendix D, with the result, in the dimensionless variables of the previous sections, 共 ⳵ s+ik·v*+Lm *兲⌬ ˜ 共k,v*,s兲=0. 共82兲 The linear collision operator in this case is Lm *=兺 i ␭i共0兲Pi+Q 冉 ␯ HCS *+ ␨ 0 * 2 ␾ −1 ⳵ ⳵ v*·v* ␾ 冊 Q.共83兲 The projection operators Piare the same as defined in Eq. 共62兲and P⬅兺 i Pi,Q⬅1−P.共84兲 In the above expressions, the summations are over the hydrodynamic modes. The first term on the right-hand side of Eq. 共83兲is the projection onto the hydrodynamic eigenfunctions, while the second one is orthogonal to this subspace. Consequently, the spectrum of Lm *for the model kinetic equation has the same k=0 hydrodynamic eigenfunctions and eigenvalues as the Boltzmann equation, i.e., it is Lm *⌿i共0兲=␭i共0兲⌿i共0兲,i=1,…,d+2. 共85兲 Furthermore, the structure of Lm *decomposes into operators defined in the hydrodynamic subspace and its orthogonal complement. This allows more detailed analysis of the nonhydrodynamic spectrum. The associated eigenfunctions lie in the orthogonal complement Q. Consider the general form for ⌿Q=Q⌿ 冉 ⌿Q,关L*−兺 i ␭i共0兲Pi兴⌿Q 冊 = 冠 ⌿Q, 冉 ␯ 0 *+ ␨ 0 * 2 ␾ −1 ⳵ ⳵ v*·v* ␾ 冊 ⌿Q 冡 = 冠冉 ␯ 0 *− ␨ 0 * 2v*· ⳵ ⳵ v* 冊 ⌿Q,⌿Q 冡 .共86兲 The second term in the scalar product above can be simplified as ␨ 0 * 2 冉 v*· ⳵ ⳵ v*⌿Q,⌿Q 冊 = ␨ 0 * 4 冉 1,v*· ⳵ ⳵ v*⌿Q 2 冊 =− ␨ 0 * 4 冉 ␾ −1 ⳵ ⳵ v*·共v* ␾ 兲,⌿Q 2 冊 = ␯ 0 * 2共⌿Q,⌿Q兲− ␯ 0 * 2共 ␾ −1g0 *,⌿Q 2兲, 共87兲 where g0 *共v*兲=nHCS −1 vd共t兲gHCS共v,t兲艌0. 共88兲 Then Eq. 共86兲gives the desired inequality 冠 ⌿Q,共Lm *−兺 i ␭i共0兲Pi兴⌿Q 冡 共⌿Q,⌿Q兲= ␯ 0 * 2 冋 1+共⌿Q, ␾ −1g0 *⌿Q兲 共⌿Q,⌿Q兲 册 ⬎ ␯ 0 * 2.共89兲 Therefore, the nonhydrodynamic spectrum of Lm *consists of points 共or continuum兲with real parts larger than ␯ 0 */2. The fastest decaying hydrodynamic eigenvalue is at ␨ 0 */2. The hydrodynamic excitations are isolated from the rest of the spectrum for ␨ 0 *⬍ ␯ 0 *, as is always the case. Assuming analyticity in k, the hydrodynamic modes at finite wavelength also will be isolated from the rest of the spectrum for sufficiently small k. In summary, the kinetic model considered here illustrates the expected behavior for the Boltzmann equation. The linearized kinetic equation for small perturbations of the HCS characterizes the complete complex response. Among the excitations there are d+2 hydrodynamic modes. At long wavelengths, these modes are isolated from and have smaller eigenvalues than the rest of the spectrum. Hence, there is a sufficiently long time scale on which only the hydrodynamic excitations persist. This kinetic model can be used to explore the role of hydrodynamics in great detail as the linear equation can be solved exactly 关21兴. For example, it has been shown that the hydrodynamic modes extend to very short wavelengths far beyond the validity of the Navier-Stokes approximation. HYDRODYNAMIC MODES FOR A GRANULAR GAS FROM…PHYSICAL REVIEW E 72, 011303 共2005兲 011303-9 关18兴N. Dunford and J. Schwartz, Linear Operators 共Interscience Publishers, New York, 1967兲. 关19兴J. J. Brey, F. Moreno, and J. W. Dufty, Phys. Rev. E 54 ,445 共1996兲. 关20兴C. Cercignani, Theory and Applications of the Boltzmann Equation 共Elsevier, New York, 1975兲. 关21兴J. W. Dufty, A. Baskaran, and L. Zogaib, Phys. Rev. E 69, 051301 共2004兲; A. Baskaran and J. W. Dufty, cond-mat/ 0410084. J. J. BREY AND J. W. DUFTY PHYSICAL REVIEW E 72, 011303 共2005兲 011303-16