Using fluctuating hydrodynamics we describe the slow buildup of long range spatial correlations in a freely evolving fluid of inelastic hard spheres. In the incompressible limit, the behavior of spatial velocity correlations (including r 2d behavior) is governed by vorticity fluctuations only and agrees well with two-dimensional simulations up to 50 to 100 collisions per particle. The incompressibility assumption breaks down beyond a distance that diverges in the elastic limit. [S0031-9007(97) In the characterization of granular matter as an unusual solid, fluid, or gas by Jaeger et al. [1], this Letter addresses the granular gas regime, controlled by inelasticity, clustering [2], and collapse [3]. Clustering is a long wavelength, low frequency (hydrodynamic) phenomenon and inelastic collapse a short wavelength, high frequency (kinetic) phenomenon. In the granular gas regime, also called rapid granular flows, the dynamics is dominated by inelastic collisions. Here the methods of nonequilibrium statistical mechanics, molecular dynamics, kinetic theory, and hydrodynamics are most suitable for describing the observed average macroscopic behavior [2][3][4][5][6][7] and the fluctuations around it.The lack of energy conservation makes the granular gas, whether driven or freely evolving, behave very differently from molecular fluids. The essential physical processes and detailed dynamics are described in [2,3] and references therein: the similarities and differences with molecular fluids; lack of separation of microscales and macroscales, not only because the grains themselves are macroscopic, but also because of the existence of intermediate intrinsic scales which are controlled by the inelasticity and are only well separated when the system is nearly elastic. A simple model which incorporates the inelasticity of the granular collisions consists of inelastic hard spheres (IHS), taken here of unit mass and diameter, with momentum conserving dynamics. The energy loss in a collision is proportional to the inelasticity parameter e 1 2 a 2 where a is the coefficient of normal restitution.For an understanding of what follows, we recall two important properties of the undriven granular gas: (i) the existence of a homogeneous cooling state (HCS) and (ii) its instability against spatial fluctuations. The hydrodynamic equations for an IHS fluid, started in a uniform equilibrium state with temperature T 0 , admit an HCS solution (see, e.g., [2,3,7]) with a homogeneous temperature T ͑t͒, described by ≠ t T 22g 0 vT. Here the collision frequency is v͑T͒ ϳ p T͞l 0 with a mean free path l 0 ,given by the Enskog theory [8] for a dense system of hard disks or spheres (d 2, 3) and g 0 e͞2d. Then T ͑t͒ T 0 ͓͞1 1 g 0 v͑T 0 ͒t͔ 2 T 0 exp͑22g 0 t͒, where t is the average number of collisions suffered per particle within a time t. It is found by integrating dt v͑T ͑t͒͒dt. Moreover, this HCS solution is linearly unstable once the linear extent L of the system exceeds some dynamic correlation length, which increases with decreasing e, and is ...

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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.

An initially homogeneous freely evolving fluid of inelastic hard spheres develops inhomogeneities in the flow field u(r, t) (vortices) and in the density field n(r, t) (clusters), driven by unstable fluctuations, δa = {δn, δu}. Their spatial correlations, δa(r, t)δa(r ′ , t) , as measured in molecular dynamics simulations, exhibit long range correlations; the mean vortex diameter grows as ξ(t) ∝ √ ln t; there occur transitions to macroscopic shearing states, etc. The Cahn-Hilliard theory of spinodal decomposition offers a qualitative understanding and quantitative estimates of the observed phenomena. When intrinsic length scales are of the order of the system size, effects of physical boundaries and periodic boundaries (finite size effects in simulations) are important.

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