Towards an H-theorem for granular gases

J. Stat. Mech. (2015) P11009 Towards an H -theorem for granular gases María Isabel García de Soria1, Pablo Maynar1, Stéphane Mischler2, Clément Mouhot3, Thomas Rey4 and Emmanuel Trizac5 1 Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain 2 CEREMADE, CNRS UMR 7534, Université Paris-Dauphine, Paris Cedex 16, F-75775 France 3 DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 4 Laboratoire P. Painlevé, CNRS UMR 8524, Université Lille 1, 59655 Villeneuve d’Ascq, France 5 LPTMS, CNRS UMR 8626, Université Paris-Sud, 91405 Orsay, France Received 27 July 2015 Accepted for publication 27 September 2015 Published 13 November 2015 Online at stacks.iop.org/JSTAT/2015/P11009 doi:10.1088/1742-5468/2015/11/P11009 Abstract. The H-theorem, originally derived at the level of the Boltzmann nonlinear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional H related to the original H, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic direct simulation Monte Carlo technique, and molecular dynamics) indicate that it is also nonincreasing. Both driven and unforced cases are investigated. Keywords: kinetic theory of gases and liquids M I G de Soria et al Towards an H-theorem for granular gases Printed in the UK P11009 JSMTC6 © 2015 IOP Publishing Ltd and SISSA Medialab srl 2015 15 J. Stat. Mech. JSTAT 1742-5468 10.1088/1742-5468/15/11/P11009 PaPers 11 Journal of Statistical Mechanics: Theory and Experiment © 2015 IOP Publishing Ltd and sIssa Medialab srl ournal of Statistical Mechanics: J Theory and Experiment IOP 1742-5468/15/P11009+20$33.00 Towards an H -theorem for granular gases 2 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 I. Introduction In 1872, Boltzmann published one of his most important papers [1]. This contribution can arguably be considered as the eective birth of kinetic theory, a domain pioneered by Bernoulli, Joule and Maxwell to name but a few, and that has turned into an active field of research in mathematics [2], physics [3], and engineering [4]. The interest of reference [1] is twofold. First, the time evolution of the velocity distribution function f tv, () for a dilute gas far from equilibrium was derived, under the essential assumption of molecular chaos (Stosszahlansatz), that is, assuming the absence of correlations between the precollisional velocities of colliding partners [5, 6]. For the sake of the discussion, we omit here the spatial dependence of the velocity distribution. Second, and remarkably, Boltzmann also introduced a functional ∫ = Ht ffv lo gd () of the instantaneous and time dependent distribution function f , that can be seen as a nonequilibrium entropy: the corresponding H-theorem [5, 6] states that this functional has a negative production, and vanishes only in equilibrium [7]6,7. It is therefore a Contents I. Introduction 2 II. N -particle description 4 II.A. The unforced system ................................. 4 II.B. The driven system .................................. 7 III. One-particle description 8 IV. The conjectured Lyapunov function 10 V. Conclusions 15 Appendix A. Evolution equation for the temperature in the free-cooling case 16 Appendix B. Evaluation of ss ss dd dd NN H () 17 Appendix C. Consistency of Kac’s equation in the stochastic thermostat case 18 References 19 6 It should be noted here that Boltzmann’s functional is clearly a forerunner of Shannon’s information measure, introduced more than 70 years later [45]. This measure is of paramount importance in data and signal analysis or transmission. It should also be stressed that the H-theorem today is still a remarkable source of inspiration to attack many mathematical problems (see e.g. [46]), sociophysics questions (see e.g. [47]) or to find new classes of time dependent exact solutions to the Boltzmann equationunder confinement [48]. 7 The H-theorem essentially holds for dilute hard core systems (with negligible internal energy compared to the kinetic energy), see e.g. Jaynes [7] for a discussion of the ‘violations’ occurring in real gases. See also Garrido etal [7] for a more recent and precise account. Towards an H -theorem for granular gases 3 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 Lyapunov-like functional. This explains, in this particular context, how molecular collisions irreversibly lead to equilibrium. Given that the underlying equations of motion are time-reversible (with elastic collisions), this finding ignited a long lasting and heated debate, an account of which is not the purpose of the present paper, and can be found in [8, 9] (see also [10] and references therein for a more technical discussion). In essence, the statistical nature of the H-theorem was not fully recognized in the early days. Ironically, when dissipative collisions are considered—as e.g. is the case between macroscopic grains for which the equations of motion are irreversible [11, 12]—, a well defined time’s arrow is present at the level of interactions, but no H-like theorem could be derived so far [13, 14]. The aim of this paper is to present strong hints that a simple generalization of Boltzmann’s original functional can be constructed, that exhibits monotonous behavior with time and tends to zero. More specifically, we shall be interested in the dynamical behavior of a spatially homogeneous granular gas, made up of a collection of a large number of grains undergoing dissipative collisions. The grains are treated as inelastic hard spheres, see e.g. [11, 15] for reviews of the rich phenomenology of this class of systems. We will consider two cases depending on the dynamics of the grains between collisions: the free-cooling case on the one hand, in which the grains move freely, and the stochastic thermostat case on the other hand, in which the grains are driven by some random external force. In the first case, it is known from particle simulations that the system reaches an autosimilar regime in which all the time dependence in the oneparticle distribution function goes through the instantaneous temperature (defined as the second velocity moment of the distribution) [16–18]. In the second case, a stationary state is reached in which the energy lost in collisions is compensated by the energy injected by the thermostat [19]. We first study an N-particle model where the dynamics of the particles’ velocities are treated as a Markov process. A Lyapunov function can be identified exactly implying, under plausible conditions, that the thermostated system reaches the stationary state in the long-time limit. In the free-cooling case, the consequence is that all the time dependence in the N-particle probability distribution is encoded in the instantaneous temperature. This scaling is similar to the one proposed in [20] at the level of the Liouville equation and, of course, implies the one at the level of the one-particle distribution. Inspired by the previous model, we propose a Lyapunov functional for the inelastic homogeneous nonlinear Boltzmann equation that describes the dynamics of the system in the low density limit and for ∞N→ [12, 20]. The functional is measured with three independent and complementary numerical techniques, with the result that it is always a nonincreasing function of time. These results point in the same direction as those of [21], which focussed on thermostated systems with an emphasis on simplified collision models or kernels. The structure of the paper is as follows. In section II, the N-particle model is introduced and the corresponding H-functional identified. This is then analyzed in section III at the level of the one-particle distribution function to in turn define, in section IV, the candidate Lyapunov function that is a central object in our work. In section IV, simulation results are also shown. Finally, the last section contains some summarizing remarks. Towards an H -theorem for granular gases 4 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 II. N -particle description We consider a system of N inelastic particles of mass m, diameter σ and spatial dimension d. We will assume that a microstate is specified by giving the velocities of the N particles, ≡ = ttVv i i N 1 () {()} , at a given time t. The dynamics is Markovian, generated by the following rule: two particles i and j are chosen at random, together with a unit, center-to-center vector σ ˆ , from which the postcollisional velocities ′′ vv , ij follow σσ α =− + ⋅ ′ vv v 1 2 , i ii j ˆˆ () (1) σσ α =+ + ⋅ ′ vv v 1 2 . j ji j ˆˆ () (2) Here, =−vvv ij ij , and α is the coecient of normal restitution that will be considered velocity-independent. It fulfills α<0 1⩽ and collisions are elastic for α=1 . In the hard particle model, the collision frequency is proportional to σ⋅ v ij ˆ () . Nevertheless, we will consider a more general model in which this frequency is proportional to σ⋅γ v ij ˆ () where γ is a fixed positive parameter ( γ 0 ⩾ ) [22]. Then, the hard particles model is obtained for γ=1 , while Maxwell molecules are for γ= 0. Two cases shall be considered, depending on the dynamics of the grains between collisions. II.A. The unforced system Let us first address the free-cooling case. With the dynamics specified above, trajectories are generated and the state of the system is described in terms of the N-particle distribution function, ρ tV, N() . The evolution equation for this distribution is the generalization for inelastic collisions of Kac’s equation [23] ∑ ρρ ∂ ∂= < ttK N LtVvvV,,,, N ij ij N () () () (3) where we have introduced the operator ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∫ σσ ρ α ρ=⋅ − σ γ γ+− Lb ij tvv Vv V,1,1 ,d, ij Nij N 1 1 ˆˆ ()() () () () ˆ (4) and K is a parameter of the model with the only restriction that σ⋅γ Kv ij ˆ () has dimensions of inverse of time (in the hard sphere case it is σ=− Kn d1, where n is the density). The operator σ − bi j,1 () ˆ acts on any function of V replacing v i and v j by the precollisional velocities, i.e. =… …… σ −−+−+ bi jf fVvvvvvvv v,,,, *,,,, *,, ,, iiij j jN 111111 ()() () ˆ (5) found from inverting the law (1)–(2) σσ α α =− + ⋅vv v *1 2 , i ii j ˆˆ () (6) Towards an H -theorem for granular gases 5 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 σσ α α =+ + ⋅vv v *1 2. j ji j ˆˆ () (7) It can be seen that equation (3) admits a special solution in which all the time dependence in the distribution function is subsumed in the granular temperature, T(t), defined as ∫∑ρ= = dNT tm vtvV 2d2 ,. i N iN 1 2 () () (8) By dimensional analysis, this means that ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ρ ϕ=tvt vt VV ,1 , N HdN N H () () () (9) where ⎡ ⎣ ⎤ ⎦ ≡ vt H Tt m 2 1/2 () () is the thermal velocity. In appendix A, a consistent equation for ϕN is obtained and the equation for the temperature is analyzed. For γ> 0, the temperature behaves for long times as ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ γ ≈∞ γ− Tt B tt 2,for , 2/ () ¯ → (10) where B ¯ is a constant. This means that, for γ> 0, the temperature ‘forgets’ the initial condition in the long time limit [24, 25]. This important property suggests working in the following dimensionless variables ∫ == ′′ γ sK tv tvt CV d, , t 00 0 () () (11) where we have introduced an ‘eective’ thermal velocity (proportional at long times to vH) ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ γ = ∼ γ− vt m Bt 2 2 , 0 1/2 1/ () (12) with ∼ B an arbitrary constant. Then, the actual thermal velocity is proportional to the eective one in the long time limit. The evolution equation for the distribution function in the new variables, φ sC, N() , φ ρ= sv ttCV,,, N dN N 0 () () () (13) is ∑ φφφ ∂ ∂ =− ∂ ∂⋅ < ssNLs BsCccC CCC,1,, 2 ,, N ij ij NN () ()() () (14) where = ∼ BBK / is a dimensionless constant. This equation is equivalent to the one of an inelastic system (with a collision rule given by equations (1)–(2)) whose particles Towards an H -theorem for granular gases 6 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 are accelerated with a force proportional to the velocity, i.e. =c s B i cd d2 i, and is usually called Gaussian thermostat. Let us analyze equation (14) to establish the conditions under which a stationary state is reached in the long time limit. General results pertaining to master equations do apply to the probability distribution, φ sC, N() , see e.g. [26]. Let us assume that there exists a stationary solution of equation (14), φ C N st () , which fulfills ∑ φφ = ∂ ∂⋅ < NLB cc CC CC 1,2 . ij ij NN st st ()() () (15) Then, we consider a convex-up function h(x) ( ″ h x0 ()⩾ ), bounded from below and defined for x0⩾ , from which the N H -functional follows as ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∫φφ φ = sh s CC C C d, . NN N N st st () () () () H (16) It is shown in appendix B that ⎪ ⎪ ⎪ ⎪ ⎧ ⎨ ⎩ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎫ ⎬ ⎭ ∫∫ σσφφ φ φ φ φ φ φ φ φ φ =−⋅ −+ − ′ ′ ′′ ′ ′ ′ γ s s Nhs hshss s Cc CC C C C C C C C C C d d 1 2dd , ,, ,, , N N N N N N N N N N N N 12 st st st st st st ˆˆ () ()() () () () () () () () () () () H (17) where we have introduced the notation =… ′′′ Cc cc c,, ,N 12 3 {} , and use has been made of the invariance of φ sC, N() under the change of labels c c ij ↔ for any i and j. As h(x) is a convex function, the integrand is negative and N H decreases monotonically in time. On the other hand, as N H is bounded from below, it must reach at long times a limit in which =0 s s d d N () H . In this limit, the distribution is φ φ∞≡ ∞ s CC ,lim , NsN () () → that fulfills σ φ φ φ φ ∞ = ∞ ′ ′ C C C CC ,, ,for all, . N N N N st st ˆ () () () ()  (18) Note that, due to the invariance property of φ sC, N() alluded to above, equation (18) is also valid when ′ C is the postcollisional velocity vector for any pair of particles, not necessarily the pair 1 and 2. Then, if for any C and U such that >CU 22 and ∑=∑= == cu0 i Ni i Ni 11 , there exists a sequence of collisions that links C with U , we can conclude that φ φ∞= CC , N N st () () . This result is important because a stationary state in the s-variable is related to a scaling of the form given by equation (9) in the original t-variable. Then, if the dynamics of equation (14) is such that, for any initial condition φ C,0 N() , the system reaches a stationary state in the long time limit, the dynamics of equation (3) will be such that, for any initial condition, ρ V,0 N() , the system will reach the auto-similar regime with a scaling of the form given by equation (9). Moreover, for γ>0 , this holds independently of the auxiliary parameter B. For γ=0 the situation is dierent, Towards an H -theorem for granular gases 7 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 but it suces that there exists one B for which the stationary solution of equation (14) exists. II.B. The driven system We next treat the case in which, between collisions, the grains are heated by a stochastic force modeled by a white noise. This model is referred to as the stochastic thermostat model and has been extensively studied in the literature [27–30]8. More specifically, the jump moments of the particles’ velocities, βγ B ij,; ,, due to the thermostat are assumed to verify [31] ξδδξ δδ ≡∆∆ ∆ =+ − − βγ βγ βγ βγ ∆ B vv tN lim 11, ij t ij ij ij,; ,0 ,, noise 0 20 2 ⟨⟩ () → (19) for =…ij N,1 ,, and βγ=… d,1 ,, . We have introduced the notation ∆≡+∆− ββ β vvtt vt ii i,, , ()() , β vt i, () being the β-component of the particle i at time t. The parameter ξ0 2 is the amplitude of the noise and …noise ⟨ ⟩ denotes the average over dierent realizations of the noise. The nondiagonal terms are introduced to conserve total momentum. Let us remark that, as discussed in appendix C, if total momentum is not conserved, a stationary state is not possible (indeed, the center-of-mass velocity follows then a standard Brownian motion, see also [32]). The evolution equation for the N-particle distribution function, ρ tV, N() , is ∑ ρρρ ∂ ∂ =+ ∂ ∂| < ttK NLt ttVvvV V,, ,,, N ij ij NN noise () ()() () (20) where the first term of the right hand side of the equation gives the collisional contribution (Lvv, ij () is given by equation (4)), and the second term gives the contribution of the thermostat. If the jumps due to the thermostat are small compared with the velocity scale in which ρ tV, N() varies, the usual conditions to derive Fokker–Planck equations are fulfilled and the thermostat contribution can be approximated by [26] ∑∑ ρρ ∂ ∂|≈ ∂ ∂ ∂ ∂ βγ βγ βγ == ttB vv t VV,, , N ij N ij ij N noise ,1,1 d ,; , ,, () () (21) with βγ B ij,; , given by equation (19), independently of the specific probability distribution of the jumps. Then, in this limit, the evolution equation for ρ tV, N() is ∑ ρρ ξρ ∂ ∂ =+ < ttK NLt tVvvV VV ,, ,,, N ij ij NN 0 2 () ()() () () T (22) where the operator T is defined as T () ∑∑ =∂ ∂−− ∂ ∂⋅∂ ∂ =< VvN vv 1 2 1 1. i N iij ij 1 2 2 (23) As in the free-cooling case, it is convenient to work with dimensionless variables. We introduce 8 See also, Puglisi [30] for a slight variant of the model. Towards an H -theorem for granular gases 8 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ξ === γ γ+ sK vt vvK CV ,, , s s s0 21 2 (24) the latter having dimensions of a velocity. For the sake of readability, similar names as for unforced systems have been employed. In these units, the evolution equation for the distribution function reads ∑ φφ ξφ ∂ ∂ =+ < ssNLs sCccC CC ,1,, ,, N ij ij NN 2 () ()() () () T (25) where ξ=ξ γ+ Kv 2 s 0 2 2 is the dimensionless amplitude of the noise. Performing a similar analysis to in the free-cooling case, and under the same hypothesis, it can be shown that the function ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∫φφ φ = sh s CC C C d, , NN N N st st () () () () H (26) decays monotonically in time and that, for any initial condition, a stationary distribution ρ V N st () is reached in the long time limit. III. One-particle description In the previous section, we provided a description at the N-particle level. Here, we consider the ∞N→ limit case, where the problem is expected to admit a closed description in terms of the velocity distribution function, f sc, () , φ≈ sf sN cc,,fo r1, N,1 11 ()() (27) that we take normalized to unity, where the jth marginal of φN is defined by ∫ φ φ…≡ … + cc ccC,, dd . Nj jjNN, 11 () () (28) Integrating equations (14) and (25) over …c c,, N2 and assuming the chaos property φ≈ sf sf sN cc cc,, ,,,for 1, N,2 12 12 ()()() (29) the homogeneous Boltzmann equation for the two cases is obtained (resp. in the unforced and driven cases) ∫ ∂ ∂ =− ∂ ∂⋅ sfs Lfsf sB fs cccc cc ccc,d,,,2 ,, 1212 12 1 11 () ()()() () (30) ∫ξ ∂ ∂ =+ ∂ ∂ sfs Lfsf sf scccc cc cc,d,,,2 ,. 1212 12 22 1 21 () ()()() () (31) Towards an H -theorem for granular gases 9 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 It is worth emphasizing that the above ‘chaos’ notion has been introduced by Kac in order to formalize the idea of asymptotic independence of particles in the limit ∞N→ . Until now, no Lyapunov functional for equations (30) and (31) has been identified. Nevertheless, the analysis made in the previous section at the N-particle level suggests the following. Consider a specific example of N H discussed above. Taking =h xxxlog () , which is bounded from below by −e−1 for x0⩾ , we get ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∫φφ φ =ss s CC C C d,log, . NN N N st () () () () H (32) In addition, if we assume that the N-particle distribution factorizes for all times in terms of the one-particle probability distribution, i.e. φ=… sfsf sCc c,, , N N1 ()()() (33) and φ=… ff Cc c , N N st st 1st () () () (34) where fst is the stationary solution of the Boltzmann equation, equations (30) or (31), the functional N H becomes extensive and transforms into a functional of the one-particle distribution function f sc, () , ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∫ = Nsfs fs f cc c c 1d,log, . Nst () () () () H (35) Let us remark that the factorization form given by equation (33) and equation (34) should be understood in the sense of equation (29): it can represent a good approximation at least in the ∞N→ limit. In the elastic limit, the above argument can be completely justified and it has been shown that ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∫ = ∞Nsfs fs f cc c c lim 1d,log, , NNst () () () () → H (36) a property called ‘entropic chaos’. The above limit was first established for a particular class of well-prepared time independent sequences of N-particle densities φN by Kac in [23] (see also [33]) and more recently for any sequence of solutions to the elastic Kac’s equation (3) in [34] (see also [35, 36]). The most important diculty in establishing (36) lies in the proof of the convergence ∫∫ φ… N ff ff Cc cCcc c 1dlog dlog NN 1st 11 st 1 () () ()→ ( )() (37) that one can deduce from a careful use of an accurate version of the central limit theorem. It is worth mentioning that the limit (36) is not the only possible scenario. Consider for instance the N-particle McKean–Vlasov model )() ((()) () ∑∑ φφ φ ∂ ∂ =+ ∆ == ttA tt CC CC ,div ,, N i N Ni N i N N cc 1 , 1 ii (38) Towards an H -theorem for granular gases 16 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 It should be noted that the asymptotic steady state f st enters the definition (42), so that we cannot deduce the form of the nonequilibrium steady or scaling solution from our functional. This is at variance with the reversible dynamics case (elastic collisions, corresponding to α=1 above), and is the price for losing the energy conservation law. We emphasize that we have not been able to prove analytically our central result, which requires us to work at Boltzmann equation level. The situation is simpler at N-body level, where a counterpart of the H-theorem can be shown. We nevertheless believe our result is conceptually important; monotonicity of t () H with time is a strong statement, the derivation of which has been the focus of some eort in the past. In the particular free cooling case, we have shown that the N-particle velocity distribution function reaches an auto-similar regime in which all its time dependence is encoded in the instantaneous temperature (a similar scaling has already been used in the context of the Liouville equation (with spatial dependence) [20]). A comparable analysis can be put forward for mixtures, binary or polydisperse, where the existence of a scaling solution implies the coincidence of the dierent cooling rates. Appendix A. Evolution equation for the temperature in the free-cooling case Assuming the scaling form of the main text, equation (9), the evolution equation for the temperature can be straightforwardly obtained by taking the second velocity moment in equation (3). The result is =− γ + Tt t BT t d d, 21 () ¯() (A.1) where we have introduced the time-independent coecient ⎜⎟ ⎛ ⎝ ⎞ ⎠α =− γ γ + B KA c dm4 2 1, 12 2/2 2 ¯⟨⟩ () (A.2) with ∫∫∫ σ σϕ == γ γγ + ++ Ac ccc ccd, dd ,. x2 12 212 12 2 2 12 ˆˆ⟨⟩ () (A.3) The function ϕ2 is just the integrated scaled distribution, ∫∫ ϕ ϕ=…cc ccC,ddNN212 3 () () . Equation (A.1) can be readily integrated, leading to ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γγ=+ > γ γ − Tt T Bt 1 02,for 0, /2 2 () () ¯ (A.4) and γ == − Tt Te 0f or 0. Bt () ()  ¯ (A.5) Then, for γ> 0 the temperature ‘forgets’ the initial condition in the long time limit and behaves as Towards an H -theorem for granular gases 17 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ γ ≈∞ γ− Tt B tt 2,for . 2/ () ¯ → (A.6) We point out now that the scaling given by equation (9) is compatible with Kac’s equation, equation (3). By substituting equation (9) into equation (3), the following equation for ϕN is obtained ∑ ϕϕ ∂ ∂⋅+ = γ+< Kv vt tN L CCC cc C 1d d 1 ,0, H H N ij ijN 1 () () ()() (A.7) where =vtCV /H () . As a consequence of equation (A.1), the term γ+ Kv vt t 1d d H H 1 () is timeindependent, and thus, equation (A.7) is consistent. Appendix B. Evaluation of ss ss dd dd NN H () Before evaluating s s d d N () H , let us note that equation (14) can be written as a master equation ∫∫ φφφ ∂ ∂ =| −| s sssCUCU UUUC C,d ,d ,, NNN () ()() () () WW (B.1) with the transition probabilities per unit time, |UC() W , given by |= |+ | UC UC UC , CT () () () WWW (B.2) where ∫ ∑σσδδ δ|= ⋅−−Π − ′′ γ < ≠ N UC cucu cu c 1 d, C ij ij iijjkijk k, ˆˆ () ()()() () W (B.3) is the contribution due to collision and δ|=−⋅ ∂ ∂ − B UC C UUC 2, T() () W (B.4) the contribution of the thermostat. Following the same steps as van Kampen [26], it is obtained that s sxsxshx s hx shxs CUCU UU CC CU d ddd ,, , ,,, N N st HW () ()(){[ ()()][ () ] [( )] [( )]} ∫∫ φ=| − +− ′ (B.5) where φ φ =x s s CC C ,, . N N st () () () (B.6) It can be seen that the contribution coming from the thermostat vanishes. Indeed, Towards an H -theorem for granular gases 18 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 xsxshx shxs hx s Bxsxshx s hx shxs CUCU UU C CCU CUUC UU CUC C CU dd ,, ,,, 2dd ,, , ,,0. TN N st st W ()(){[ ()()][()][()][()]} ()() {[ ()()][ () ] [( )] [( )]} ∫∫ ∫∫ φ δφ |−+− =−⋅∂ ∂− +− = ′ ′ (B.7) The collisional counterpart can be written as xsxshx shxs hx s N hx shxs hx sx sx s CUCU UU C CCU Cc C CCCC C dd ,, ,,, 1 2dd ,, ,, ,, CN N st 12 st W ˆˆ ()(){[ ()()][()][()][()]} ()() {[()][()][()][ ()()]} ∫∫ ∫∫ σσ φ φ |−+− =−⋅ ×−+− ′ ′′ ′′ γ (B.8) where we have introduced the notation =… ′′′ Cc cc c,,,, N 12 3 {} and we have used that, due to the invariance of φ sC, N() under the change of labels c c ij ↔ for any i and j, the N(N − 1)/2 collisional terms are equal. By substituting equations (B.7) and (B.8) into (B.5), the expression of the main text is obtained. Appendix C. Consistency of Kac’s equation in the stochastic thermostat case In this appendix, it is shown that the conservation of the total momentum assumed by the thermostat is essential to ensure that equation (22) be compatible with the existence of a stationary state. We assume that a stationary solution of equation (22) exists, ρ V N st () . It then fulfills ∑ρξρ += < K N LT vv VVV ,0. ij ij NN st 0 2st ()() () () (C.1) Let us multiply the equation by v1 2 and integrate. The collisional contribution is ∫ ∫∫ ∫∫ ∑ σσ σσ ρ ρ αρ =−⋅−+ =− − −⋅ σ γ γ < + K NvL KN N bv v KN N vvvV vV v vV v d, 1 2dd 1 1 4 1d d, ij ijN N N 1 2st st 12 1 22 2 2st 12 2 ˆˆ ˆˆ ()() () () ()() () () () () () ˆ (C.2) where use has been made of σ α −+=− − ⋅ σ bvvv11 2 . 1 22 22 12 2 ˆ ()() () ˆ (C.3) The thermostat contribution is ∫∫ ξ ρ ξ ρξ =∂ ∂=vT vvvVVvVd2 dd , NN 0 21 2st0 2 st 2 1 21 2 0 2 () () () (C.4) Towards an H -theorem for granular gases 19 doi:10.1088/1742-5468/2015/11/P11009 J. Stat. Mech. (2015) P11009 where the cross-terms do not contribute. The obtained equation is ∫∫ σσαρ ξ −−⋅ = γ+ KN N vV v 1 4 1d dd. N 2st 12 20 2 ˆˆ () () () () (C.5) Keeping in mind the above ‘steady-state constraint’, let us now multiply equation (C.1) by ⋅vv 12 and integrate. The collisional contribution is K NL K Nb K N vv vvvV vV vv v vV v d, dd 1 4 1d d, ij ij N N N 12 st st 12 12 2st 12 2 ˆˆ ˆˆ ()() () () () () () () ˆ ∫ ∫∫ ∫∫ ∑ σσ σσ ρ ρ αρ ⋅ =⋅−⋅ =− ⋅ σ γ γ < + (C.6) where it has been used that σ α −⋅= − ⋅ σ bvv v11 4. 12 2 12 2 ˆ () () ˆ (C.7) From the conservation of total momentum in a collision, we further have −+= σ bvv 10 12 2 ()() ˆ , so that −⋅=− −+ σσ bbvv vv1 1 21, 12 1 22 2 () () () ˆˆ (C.8) from equations (C.3) and (C.7). The thermostat contribution is ∫ ∫ ξρ ξρξ ⋅ =− − ∂ ∂ ∂ ∂⋅=−− T NN vv vV V vV vv vv d 1dd 1, N N 0 212 st 0 2 st 12 12 0 2 () () () (C.9) where only the cross-terms contribute. The obtained equation is equivalent to equation (C.5) but, as indicated above, only because of the presence of the cross terms (second term on the right hand side of equation (19)). 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