Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles

Abstract

The time dependence of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastically colliding spheres is investigated by kinetic theory. We determine the full time dependence of the coefficients of an expansion around the Gaussian state in Generalized Laguerre polynomials. Approximating this system of equations to sixth order, we find that the asymptotic state, where the mean energy T follows Haff's law with time independent cooling rate, is reached within a few collisions per particle. Two-dimensional molecular dynamics simulations confirm our results and show exponential behavior in the high-energy tails.

The second and third Sonine coefficients of a freely cooling granular gas revisited

“…In particular, if Eq. (14) for p = 2 is rewritten as µ 4 / c 4 = 2µ 2 / c 2 and then a linear approximation is applied the result is given again by Eq. (15) but with (λ 0 , λ 1 , λ 2 ) = (25+24d, 8d −57, −2), implying (λ 0 , λ 1 , λ 2 ) = (97, −33, −2) for d = 3.…”

“…A different truncation scheme was followed by Huthmann et al [14], who assumed that a k = O(ε k ), where ε ∼ |a 2 | 1/2 was treated as a small parameter. The solution to order k ≥ 2 was obtained by taking Eq.…”

“…On the other hand, It is known that in the high-inelasticity region α 0.6 (i.e., once a 2 becomes positive) the higherorder Sonine coefficients are no longer negligible [8,14,17], so that the linear approximations based on the neglect of nonlinear terms and of a k with k ≥ 3 or k ≥ 4 are not a priori reliable. This is made evident by the lack of selfconsistency of different linear approximations used to estimate a 2 from the first non-trivial equation of the moment hierarchy, as shown in Fig.…”

“…(21), in contrast to the two approaches "a", which are based on linearization of Eq. (14). Given that the class-I estimate a Ib 2 = a Ih 2 (where a 3 is neglected) is much simpler than the class-II estimate a IIb 2 (where a 3 is retained), the former is preferable to the latter.…”

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“…Overpopulated tails in free IHS fluids [9] and driven ones [10] have also been studied by molecular dynamics simulations of inelastic hard spheres. The observed overpopulations in IHS systems are mainly stretched exponentials exp[−Ac b ] with b = 1 [11,4,7,8] in systems without energy input, and b = 3/2 [4,8] in driven systems, but for some forms of driving [8,5] b = 2 has been observed.…”