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The nonequilibrium steady state of a granular fluid, driven by a random external force, is demonstrated to exhibit long-range correlations, which behave as ϳ1/r in three and ϳln(L/r) in two dimensions. We calculate the corresponding structure factors over the whole range of wave numbers, and find good agreement with two-dimensional molecular dynamics simulations. It is also shown by means of a mode coupling calculation, how the mean field values for the steady-state temperature and collision frequency, as obtained from the Enskog-Boltzmann equation, are renormalized by long wavelength hydrodynamic fluctuations.

In irreversible aggregation processes without a gelation transition the cluster size distribution approaches a scaling form, c"(t) -s 2@(k/s). Usking Smoluchowski's coagulation equation we determine the exponents in the mean cluster size s(t) -t' (t~) and in the small-and large-x behavior of the scaling function @(x). Depending on certain characteristics of the coagulation coefficients, @(x) -x ' (x 0) or $(x) -exp( -x") (x 0) with p, some negative constant. In aggregation processes with gelation a similar scaling form is obtained as t approaches the gel point.PACS numbers: 64.60. -i, 05.50. +q, 64.75. +g, 82.35.+t To study the kinetics of irreversible aggregation and clustering phenomena, in particular the time evolution of the cluster size distribution ck(t), Smoluchowski s coagulation equation is one of the few available, and also one of the most widely used, theoretical tools in many fields of physics, astronomy, polymer physics, colloid chemistry, atmospheric physics, biology, and technology. ' 5 It reads Ck= 2 g K(l,J)ctcJ Ck XK(k,J) Jc, i+ j=k j=l where the coagulation kernel K (i j ) represents the rate coefficient for a specific clustering mechanism between clusters of sizes i and j. We distinguish gelling and nongelling mechanisms.In the former the mean cluster size s(t) diverges as t approaches the gel point t, ; in the latter s (t) keeps increasing with time.It is known from exact solutions, ' coagulation experiments, and computer simulations that the size distribution approaches a scaling form, c"(t) -s '@(k/s), as soon as s(t) becomes large compared to the characteristic size at the initial time. The important point is that the k and t dependence of ck (t) is given through a universal function of a single variable, k/s(t), that does not depend on the initial distribution. For a limited number of coagulation mechanisms, all belonging to class III (see below), Friedlander's theory of selfpreserving spectra (SPS theory) gave a satisfactory explanation of the experimental observations on Brownian coagulation in the hydrodynamic and molecular regime, although the experimental data at large k and t are rather poor.By generalizing the SPS theory we can give a unifying description of the scaling behavior occurring in gelling and nongelling systems, described by Smoluchowski's equation.Our generalization covers all coagulation kernels K (i j) that are homogeneous functions of i and j, and includes large classes of models, for which the original SPS theory is not valid, e.g. , K(ij) =i +j.Since Smoluchowski's equation with a homogeneous kernel is invariant under a group of similarity transformations, it admits exact similarity or scaling solutions, 2 3 that can be solved from a nonlinear integral equation and whose properties are analyzed in this Letter. The basic assumption of our method is that the solutions for general initial distributions indeed approach the special similarity solution. With the help of the integral equation we can determine scaling functions and related exponents, analytically or nu...

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The algorithm for the dissipative particle dynamics ͑DPD͒ fluid, the dynamics of which is conceptually a combination of molecular dynamics, Brownian dynamics, and lattice gas automata, is designed for simulating rheological properties of complex fluids on hydrodynamic time scales. This paper calculates the equilibrium and transport properties ͑viscosity, self-diffusion͒ of the thermostated DPD fluid explicitly in terms of the system parameters. It is demonstrated that temperature gradients cannot exist, and that there is therefore no heat conductivity. Starting from the N-particle Fokker-Planck, or Kramers equation, we prove an H theorem for the free energy, obtain hydrodynamic equations, and derive a nonlinear kinetic equation ͑the FokkerPlanck-Boltzmann equation͒ for the single-particle distribution function. This kinetic equation is solved by the Chapman-Enskog method. The analytic results are compared with numerical simulations.

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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