Breakdown of hydrodynamics in the inelastic Maxwell model of granular gases

Breakdown of hydrodynamics in the inelastic Maxwell model of granular gases J. Javier Brey, M. I. García de Soria, and P. Maynar Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain 共Received 13 April 2010; published 18 August 2010兲 Both the right and left eigenfunctions and eigenvalues of the linearized homogeneous Boltzmann equation for inelastic Maxwell molecules corresponding to the hydrodynamic modes are calculated. Also, some nonhydrodynamic modes are identified. It is shown that below a critical value of the parameter characterizing the inelasticity, one of the kinetic modes decays slower than one of the hydrodynamic ones. As a consequence, a closed hydrodynamic description does not exist in that regime. Some implications of this behavior on the formally computed Navier-Stokes transport coefficients are discussed. DOI: 10.1103/PhysRevE.82.021303 PACS number共s兲: 45.70.⫺n, 51.10.⫹y I. INTRODUCTION Granular gases provide an appropriate context in which a number of fundamental issues related with nonequilibrium systems can be addressed. Primary among these is the existence of a hydrodynamic description, i.e., of closed, deterministic equations for the hydrodynamic fields, identified as the number density n, the flow velocity u, and the 共granular兲 temperature T, in the case of one-component systems. A theoretical framework for deriving macroscopic equations from the underlying microscopic dynamics and investigating its range of validity is kinetic theory. In the low density limit, the Boltzmann equation for smooth and inelastic hard spheres or disks has been used to derive hydrodynamic equations for granular gases since many years ago 关1–3兴. Presently, explicit expressions for the transport coefficients appearing in the analogous of the Navier-Stokes equations are available, and they have been confirmed via direct Monte Carlo simulations 关4,5兴. Nevertheless, the methods used are formal and do not address either the existence or the context of the hydrodynamic description itself, although internal mathematical consistency is accomplished at the level of the Navier-Stokes approximation, i.e., first order in the gradients of the hydrodynamic fluxes. For elastic hard spheres or disks, the problem has been satisfactorily solved by analyzing the spectrum of the linearized Boltzmann operator 关6兴.Onthe other hand, for the inelastic Boltzmann equation 共IBE兲, although the hydrodynamic eigenfunctions and eigenvalues have been identified in the limit of large wave vectors 关7,8兴, almost nothing is known about the kinetic, nonhydrodynamic part of the spectrum. Consequently, the dominance of the hydrodynamic modes has not been established for the IBE, although it has been proven for some single-time relaxation model equations 关8兴. In this paper, the validity of a hydrodynamic description for granular gases will be addressed using a simplified IBE. Specifically, the inelastic Maxwell model 共IMM兲will be employed 关9–14兴. This kinetic equation is obtained from the IBE for hard spheres of disks by replacing the velocity dependent collision rate by an effective average value proportional to the thermal velocity. Although other choices have also been considered 关9,11兴, here the effective collision rate will be assumed to be also independent of the collision angle. Moreover, attention will be restricted to the modes at asymptotically long wavelengths, i.e., to perturbations occurring also in homogeneous systems. This will suffice to establish that the hydrodynamic spectrum is not isolated from the rest at strong inelasticity, contrary to what is required for the existence of hydrodynamics. Let f共v1,t兲be the one-particle distribution function of particles of mass mwith velocity v1at time t. The homogenous Boltzmann equation for the the IMM in d-dimensions reads ⳵ tf共v1,t兲=JM关v1,t兩f,f兴,共1兲 JM关v1,t兩f,g兴⬅d+2 2 ␯ 共t兲 n⍀d 冕 d ␴ ˆ 冕 dv2共 ␣ −1b ␴ −1 −1兲 ⫻f共v1,t兲g共v2,t兲.共2兲 Here ␯ 共t兲is an effective collision frequency. Its explicit form will not be needed, being enough to know that ␯ ⬀nT1/2. Moreover, ⍀d⬅2 ␲ d/2/⌫共d/2兲is the total solid angle element, d ␴ ˆis the solid angle element around the direction of the unit vector ␴ ˆ, and b ␴ −1 is an operator changing all the velocities v1and v2to its right into their precollisional values given by b ␴ −1v1,2=v1,2 ⴱ⬅v1⫿共1+ ␣ 兲v12· ␴ ˆ ␴ ˆ/2 ␣ , with v12⬅v1−v2. The coefficient of normal restitution ␣ is defined in the interval 0⬍ ␣ ⱕ1. This model differs from the one considered in ref 关15兴, where some properties of a onedimensional IMM with a constant collision frequency ␯ are investigated. The hydrodynamic fields are defined in terms of f共v1,t兲in the usual way. Then, by taking velocity moments in Eq. 共1兲it is seen that ⳵ n ⳵ t= ⳵ u ⳵ t=0, ⳵ T ⳵ t=− ␨ 共t兲T,共3兲 with ␨ being the cooling rate given by ␨ 共t兲=d+2 4d共1− ␣ 2兲 ␯ 共t兲.共4兲 As the IBE for hard spheres and disks, Eq. 共1兲has a similarity solution describing the homogeneous cooling state 共HCS兲, PHYSICAL REVIEW E 82, 021303 共2010兲 1539-3755/2010/82共2兲/021303共4兲©2010 The American Physical Society021303-1 fH共v1,t兲=nHv0 −d共t兲 ␹ 共c1兲,v0共t兲⬅ 冋 2TH共t兲 m 册 1/2 ,共5兲 where ␹ 共c1兲is an isotropic function of the scaled velocity c1⬅v1/v0共t兲. Substitution of Eq. 共5兲into Eq. 共1兲, using Eq. 共3兲yields ␨ ˜ 2 ⳵ ⳵ c1 ·关c1 ␹ 共c1兲兴 =J ˜ M关c1兩 ␹ , ␹ 兴.共6兲 In the above expression, ␨ ˜ ⬅ ␨ H共t兲/ ␯ H共t兲and J ˜ Mis the dimensionless version of the collision operator JM. The index His used to characterize quantities computed in the HCS. Next, Eq. 共1兲is particularized for small 共homogeneous兲 perturbations of the HCS. A distribution ␦ ␹ is defined by f共v1,t兲=nHv0 −d共t兲关 ␹ 共c1兲+ ␦ ␹ 共c1,t兲兴, with the velocities scaled relative to v0共t兲. Moreover, the time is expressed in terms of ␶ given by d ␶ = ␯ H共t兲dt. Retaining terms up through linear order in ␦ ␹ it is obtained ⳵ ␶ ␦ ␹ 共c1, ␶ 兲=⌳共c1兲 ␦ ␹ 共c1, ␶ 兲,共7兲 ⌳共c1兲 ␦ ␹ 共c1, ␶ 兲=J ˜ M关c1兩 ␹ , ␦ ␹ 兴+J ˜ M关c1兩 ␦ ␹ , ␹ 兴 − ␨ ˜ 2 ⳵ ⳵ c1 ·关c1 ␦ ␹ 共c1, ␶ 兲兴 + ␨ ˜ 4 ⳵ ⳵ c1 ·关c1 ␹ 共c1兲兴 冕 dc2 冉 2c2 2 d−1 冊 ␦ ␹ 共c2, ␶ 兲. 共8兲 Linearization of the balance Eqs. 共3兲around the HCS leads to ⳵␳ ⳵ ␶ =0, ⳵␻ ⳵ ␶ = ␨ ˜ 2 ␻ , ⳵ ␪ ⳵ ␶ =− ␨ ˜ 2 ␪ − ␨ ˜ ␳ ,共9兲 with ␳ ⬅共n−nH兲/nH, ␻ ⬅u/v0共t兲, and ␪ ⬅共T−TH兲/TH. The spectrum of the above equations is given by three points, ␭1=0, ␭2= ␨ ˜ /2, and ␭3=− ␨ ˜ /2, ␭2being d-fold degenerate. Their perturbation for finite wave vectors 共gradients兲in the context of the inhomogeneous IMM defines the hydrodynamic modes more generally. These eigenvalues are the same as for the IBE for hard spheres or disks 关7,8兴.Inthe elastic limit ␣ →1, the well known result of a 共d+2兲-fold degenerate point at zero eigenvalue is recovered. The operator ⌳is expected to have the corresponding hydrodynamic modes, the remaining part of the spectrum being referred to as the kinetic modes. Of course, this terminology does not preclude any mathematical difference or separation between both parts of the spectrum. Then, the eigenproblem ⌳共c1兲 ␰ i共c1兲=␭i ␰ i共c1兲共10兲 is considered. First, attention is focused on the hydrodynamic part of the spectrum that can be determined as follows. Define the function F共c1, ␳ , ␻ , ␥ 兲⬅ ␳␹ 共 ␥ C1兲, with C1=c1− ␻ . Using Eq. 共6兲it follows that ␳␨ ˜ 2 ⳵ ⳵ c1 ·共C1F兲= ␥ dJ ˜ M关c1兩F,F兴.共11兲 Differentiating this equation with respect to ␳ , ␻ , and ␥ , taking afterwards the limit ␳ = ␥ =1, ␻ =0, it is straightforward to show that ⌳has the hydrodynamic eigenvalues ␭i, i=1,2,3, given above, with the eigenfunctions ␰ 1共c1兲=共d+1兲 ␹ 共c1兲+c1· ⳵␹ 共c1兲 ⳵ c1 , ␰ 2共c1兲=− ⳵␹ 共c1兲 ⳵ c1 , ␰ 3共c1兲=− ⳵ ⳵ c1 ·关c1 ␹ 共c1兲兴.共12兲 The technical details of the calculations are fully similar to those discussed in 关7,8兴for the IBE for hard spheres or disks and they will not be reproduced here. As for the hydrodynamic equations, the eigenvalue ␭2is d-fold degenerate. The eigenfunctions ␰ ihere are the same functionals of ␹ as for the IBE 关7兴. Solutions to Eq. 共7兲are sought in a Hilbert space defined by the scalar product 具g兩h典⬅ 冕 dc ␹ −1共c兲g†共c兲h共c兲,共13兲 with the dagger denoting complex conjugate. The hydrodynamic eigenfunctions span a d+2 dimensional subspace of the Hilbert space. They are not orthogonal, as a manifestation of ⌳being non Hermitian. Then, the left hand eigenproblem ⌳+共c1兲 ␰ ¯ i共c1兲=␭ ¯ i ␰ ¯ i共ci兲,共14兲 has to be considered. In this equation, ⌳+is the adjoint of ⌳, defined through 具g兩⌳h典†=具h兩⌳+g典, ⌳+共c1兲g共c1兲=d+2 2⍀d 冕 dc2 冕 d ␴␹ 共c1兲 ␹ 共c2兲共b ␴ −1兲 ⫻关 ␹ −1共c1兲g共c1兲+ ␹ −1共c2兲g共c2兲兴 + ␨ ˜ 4 ␹ 共c1兲 冉 2c1 2 d−1 冊 冕 dc2 ␹ −1共c2兲g共c2兲 ⳵ ⳵ c2 ⫻关c2 ␹ 共c2兲兴 + ␨ ˜ 2 ␹ 共c1兲c1· ⳵ ⳵ c1 关 ␹ −1共c1兲g共c1兲兴, 共15兲 b ␴ being the operator inverse of b ␴ −1. Although there seems to be no simple relationship between the functions ␰ iand ␰ ¯ i, the hydrodynamic part of the spectrum of ⌳+has been found by simple inspection. The functions ␰ ¯ 1共c1兲= ␹ 共c1兲, ␰ ¯ 2共c1兲=c1 ␹ 共c1兲, ␰ ¯ 3共c1兲= 冉 c1 2 d+1 2 冊 ␹ 共c1兲, 共16兲 are solutions of Eq. 共14兲, corresponding to the eigenvalues ␭ ¯ i=␭i,i=1,2,3, respectively. Again, the eigenvalue ␨ ˜ /2is d-fold degenerate. Moreover, it is easily verified that BREY, GARCÍA DE SORIA, AND MAYNAR PHYSICAL REVIEW E 82, 021303 共2010兲 021303-2 具 ␰ ¯ i兩 ␰ j典= ␦ ij,共17兲 for the hydrodynamic eigenfunctions given by Eqs. 共12兲and 共16兲. It is also possible to get some information about the remaining, kinetic part of the spectrum of ⌳+. Here attention will be restricted to two eigenfunctions which have a clear physical interpretation. A direct evaluation gives ⌳+共c1兲c1xc1y ␹ 共c1兲=−1 4共1+ ␣ 兲2c1xc1y ␹ 共c1兲.共18兲 It follows that ␰ ¯ 4共c1兲=c1xc1y ␹ 共c1兲is an eigenfunction of ⌳+ with eigenvalue ␭ ¯ 4=−共1+ ␣ 兲2/4. Note that ␰ ¯ 4is proportional to the dynamical variable whose velocity average provides the xy component of the pressure tensor 关6兴. It is important to check whether the kinetic eigenvalue ␭ ¯ 4is actually separated from the hydrodynamic part of the spectrum, as required for the validity of the hydrodynamic description. Note that if it is assumed that any function of the Hilbert space can be expanded in terms of the eigenfunctions of ⌳, then the left eigenvalues are also right eigenvalues, i.e., there is a solution of Eq. 共10兲with ␭4=␭ ¯ 4. Consequently, the general solution of Eq. 共7兲contains a contribution proportional to exp共␭ ¯ 4 ␶ 兲 ␰ 4共c1兲, where ␰ 4共c1兲is the 共unknown兲eigenfunction of ⌳with eigenvalue ␭4=␭ ¯ 4, or a linear combination of eigenfunctions in the case of degeneracy. The validity of hydrodynamics requires that the above contribution decay faster than all the hydrodynamic modes, i.e., it must be 兩␭ ¯ 4兩⬎兩␭3兩= ␨ ˜ /2. Taking into account the exact result given by Eq. 共5兲, this condition is equivalent to ␣ ⬎ ␣ 1with ␣ 1=−共d−2兲/共3d+2兲. For d=3 this expression leads to an unphysical negative value of ␣ 1, while for d=2 it is ␣ 1=0. The above result suggests to check whether there is another left eigenfunction proportional to the dynamical variable providing the heat flux. One reason to consider these fluxes as candidates to be eigenfunctions of ⌳+is that they are known to be orthogonal to the hydrodynamic modes ␰ i 关7兴. Explicit evaluation gives that ␰ ¯ 5共c1兲= 冉 c1 2−d+2 2 冊 c1x ␹ 共c1兲共19兲 is a solution of Eq. 共14兲with ␭ ¯ 5=−共d−1兲共1+ ␣ 兲2 4d.共20兲 The condition for the validity of hydrodynamics following from the existence of this kinetic mode is that ␣ ⬎ ␣ 2=共4−d兲/3d. Therefore, there is a finite value of the coefficient of normal restitution below which there is no time scale separation between the hydrodynamic and the kinetic parts of the distribution function. For d=2 it is ␣ 2=1/3 and for d=3 ␣ 2=1/9. In Fig. 1the two kinetic eigenvalues ␭ ¯ 4 and ␭ ¯ 5are compared with the slowest hydrodynamic mode ␭3. The lack of dominance of the hydrodynamic modes over the mode related with the heat flux for long times, reflects itself in the calculation of the Navier-Stokes heat conductivity of an inelastic gas of Maxwell molecules. As already mentioned, the form of the hydrodynamic spectrum for IMM reported here is the same as for the IBE for hard spheres. The only difference is in the explicit expression of the distribution function of the HCS and in the value of the cooling rate. As a consequence, the formal expressions for the transport coefficients derived in 关7,16兴from the IBE, can be directly applied to the case of Maxwell molecules. Then, the timedependent heat conductivity ␬ is given by ␬ 共 ␶ 兲=nH v0 2共t兲 ␯ 共t兲 冕 0 ␶ d ␶ ⬘ ␬ ˜ 共 ␶ ⬘兲,共21兲 ␬ ˜ 共 ␶ 兲=1 2 冕 dc1 冉 c1 2−d+2 2 冊 c1xe ␶ 共⌳+ ␨ ˜ /2兲 ␰ 3共c1兲c1x =e ␶ 共␭5+ ␨ ˜ /2兲 2 冕 dc1 ␰ 3共c1兲 冉 c1 2−d+2 2 冊 c1x 2.共22兲 The existence of hydrodynamics to Navier-Stokes order requires that the correlation function ␬ ˜ 共 ␶ 兲decay to zero for ␶ Ⰷ1. This clearly implies that 兩␭5兩⬎ ␨ ˜ /2, i.e., ␣ ⬎ ␣ 2. Consistently with this analysis, it is found that the heat conductivity for inelastic Maxwell molecules obtained by the Chapman-Enskog procedure also diverges for ␣ → ␣ 2关17兴. To put the results reported here in a proper context some comments seem appropriate. 共i兲It has been proven that the solution of the Boltzmann equation for IMM tends to the HCS for arbitrary initial conditions 关18兴. The lack of time scale separation discussed here does not contradict this general property, since the obtained kinetic eigenvalues are negative, therefore leading to decaying in time contributions. 共ii兲It is worth to stress that the failure of hydrodynamics in IMM should not be understood as limited to the Navier- Stokes approximation, but to any closed description of the system in terms of the hydrodynamic fields. 共iii兲Of course, the value of ␣ for which the scale separation actually fails is not known, but only a lower bound has been determined. It seems sensible to expect higher limiting values of ␣ associated to eigenfunctions involving higher velocity powers. 共iv兲 On the other hand, there is no reason to expect a similar -1 -0.8 -0.6 -0.4 - 0.2 0 0 0.2 0.4 0.6 0.8 1 α -1 -0.8 -0.6 -0.4 - 0.2 0 d=2 d=3 FIG. 1. The two kinetic eigenvalues ␭ ¯ 4共dotted lines兲and ␭ ¯ 5 共dashed lines兲, and the slowest hydrodynamic mode ␭3共solid lines兲 versus the coefficient of normal restitution ␣ for a system of inelastic Maxwell molecules. Note that the three modes are negative. BREAKDOWN OF HYDRODYNAMICS IN THE INELASTIC…PHYSICAL REVIEW E 82, 021303 共2010兲 021303-3 behavior for the IBE for hard spheres or disks. On the contrary, there are some indications that this is not the case. The expressions derived for the Navier-Stokes transport coefficients are regular functions of ␣ for 0⬍ ␣ ⱕ1关19兴, and the two kinetic modes considered here are not left eigenfunctions of the linearized inelastic Boltzmann operator. Actually, other relevant deep differences between the IBE and the IMM have been found, as for instance the asymptotic decay of the distribution function of the HCS for large velocities 关14,20兴. This research was partially supported by the Ministerio de Educación y Ciencia 共Spain兲through Grant No. FIS2008-01339 共partially financed by FEDER funds兲. 关1兴C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy, J. Fluid Mech. 140, 223 共1984兲. 关2兴J. T. Jenkins and M. W. Richman, Arch. Ration. Mech. Anal. 87, 355 共1985兲. 关3兴N. Sela and I. Goldhirsch, J. Fluid Mech. 361,41共1998兲. 关4兴J. J. Brey, J. W. Dufty, C. S. Kim, and A. Santos, Phys. Rev. E 58, 4638 共1998兲. 关5兴J. J. Brey and D. Cubero, in Granular Gases, Lecture Notes in Physics Vol. 564 edited by T. Pöschel and S. Luding 共Springer, Berlin, 2001兲. 关6兴J. A. McLennan, Introduction to Nonequilibrium Statistical Mechanics 共Prentice-Hall, Englewood Cliffs, NJ, 1989兲. 关7兴J. J. Brey, J. W. Dufty, and M. J. Ruiz-Montero, in Granular Gas Dynamics, edited by T. Pöschel and N. Brilliantov 共Springer-Verlag, Berlin, 2003兲. 关8兴J. W. Dufty and J. J. Brey, Phys. Rev. E 68, 030302共R兲共2003兲. 关9兴A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, J. Stat. Phys. 98, 743 共2000兲. 关10兴E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 61,R5共2000兲. 关11兴J. A. Carrillo, C. Cercignani, and I. M. Gamba, Phys. Rev. E 62, 7700 共2000兲. 关12兴A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi, EPL 58,14共2002兲. 关13兴E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 66, 011309 共2002兲. 关14兴M. H. Ernst and R. Brito, J. Stat. Phys. 109, 407 共2002兲. 关15兴G. Costantini, U. Martini Bettolo Marconi, and A. Puglisi, J. Stat. Mech. 共2007兲, P08031. 关16兴J. W. Dufty and J. J. Brey, J. Stat. Phys. 109, 433 共2002兲. 关17兴A. Santos, Physica A 321, 442 共2003兲. 关18兴A. V. Bobylev, C. Cercignani, and G. Toscani, J. Stat. Phys. 111, 403 共2003兲. 关19兴J. J. Brey, M. J. Ruiz-Montero, P. Maynar, and M. I. García de Soria, J. Phys.: Condens. Matter 17, S2489 共2005兲. 关20兴S. E. Esipov and T. Pöschel, J. Stat. Phys. 86, 1385 共1997兲. BREY, GARCÍA DE SORIA, AND MAYNAR PHYSICAL REVIEW E 82, 021303 共2010兲 021303-4