The granular phase diagram

Abstract

The kinetic energy distribution function satisfying the Boltzmann equation is studied analytically and numerically for a system of inelastic hard spheres in the case of binary collisions. Analytically, this function is shown to have a similarity form in the simple cases of uniform or steady-state flows. This determines the region of validity of hydrodynamic description. The latter is used to construct the phase diagram of granular systems, and discriminate between clustering instability and inelastic collapse. The molecular dynamics results support analytical results, but also exhibit a novel fluctuational breakdown of mean-field descriptions.Granular media such as sand provide an attractive opportunity to revisit a number of topics in classical physics, and contribute new angles. In this paper we describe the granular phase diagram which we hope will be helpful to a broad community given the raising interest in granular systems [1]. The researchers interested in diluted granular gases, such as in astrophysical applications [2], and researchers who study, say, compaction of sand [3] use different approaches. The phase diagram may represent a ground for communication.The phase of granular system depends on inelasticity of collisions, r, (restitution coefficient approximation, [4, 5, 6]), particle density, ρ, particle size, a, system size, * J. Stat. Phys. (in press)

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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. For hard sphere systems there is in general good agreement between the analytic predictions and numerical or Monte Carlo solutions of the nonlinear Boltzmann equation.…”

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“…As explained above, the appearance of density clusters can even be further delayed, or all together suppressed by decreasing the system size [27,28,29,23] This implies,…”

“…The present theoretical understanding of these instabilities [27,28,29,23,25] is essentially based on a linear stability analysis of the hydrodynamic fluctuations in the density, δn = n −n, temperature, δT = T −T , and flow velocity u. This is done by using the rescaled…”

“…Moreover, these arguments also suggest that the pressure fluctuations and the corresponding gradients remain substantially smaller than those in density and temperature. The shape of the dispersion relations for the growth rates also explains the suppression of instabilities through a reduction of the system size [27,28,29,23]. Let k 0 (N ) = 2π/L be the smallest wave number, allowed in a system of linear extent L at fixed density and with periodic boundary conditions.…”

“…The search for the proper macroscopic description of unstable granular fluids has been pursued by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13]18,20,[22][23][24][25][26][27][28][29][30][31][32]. Recently, two new points of view have been presented, namely by Ben-Naim et al [11] and by Soto et al [12].…”