In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau-Ginzburg (LG) models for critical and unstable fluids (e.g. spinodal decomposition). The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field.Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.

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Using the Hamilton-Jacobi method, we solve chemical Fokker-Planck equations within the Gaussian approximation and obtain a simple and compact formula for a conditional probability distribution. The formula holds in general transient situations, and can be applied not only for a steady state but also for a oscillatory state. By analyzing the long time behavior of the solution in the oscillatory case, we obtain the phase diffusion constant along the periodic orbit and the steady distribution perpendicular to it. A simple method for numerical evaluation of these formulas are devised, and they are compared with Monte Carlo simulations in the case of Brusselator as an example. Some results are shown to be identical to previously obtained expressions.

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We study inelastic collapse in a one-dimensional N-particle system when the system is driven from below under gravity. We investigate the hard-sphere limit of inelastic soft-sphere systems by numerical simulations to find how the collision rate per particle n(coll) increases as a function of the elastic constant of the sphere k when the restitution coefficient e is kept constant. For systems with large enough N>/~20, we find three regimes in e depending on the behavior of n(coll) in the hard-sphere limit: (i) an uncollapsing regime for 1≥e>e(c1), where n(coll) converges to a finite value, (ii) a logarithmically collapsing regime for e(c1)>e>e(c2), where n(coll) diverges as n(coll)~logk, and (iii) a power-law collapsing regime for e(c2)>e>0, where n(coll) diverges as n(coll)~k(α) with an exponent α that depends on N. The power-law collapsing regime shrinks as N decreases and seems not to exist for the system with N=3, while, for large N, the size of the uncollapsing and the logarithmically collapsing regime decreases as e(c1)=/~1-2.6/N and e(c2)=/~1-3.0/N. We demonstrate that this difference between large and small systems exists already in the inelastic collapse without external drive and gravity.

In this paper we consider a chemical process, for which the reagent of interest decays shortly after it is created. For such a system the standard theory of nonequilibrium spatial correlations in the density fluctuations, which leads to Ornstein–Zernike type of expression, does not work. We compare results of molecular simulations with another theoretical description based on the assumption that the lifetime of the reagent is short enough to treat its motion as a ballistic rather than diffusive one.

We propose a novel approach based on a Langevin equation for fluctuating motion of the center of mass of granular media fluidized by energy injection from a bottom plate. In this framework, the analytical solution of the Langevin equation is used to derive analytic expressions for several macroscopic quantities and the power spectrum for the center of mass. In order to test our theory, we performed event-driven molecular dynamics simulations for one-and two-dimensional systems. Energy is injected from a vibrating bottom plate in the one-dimensional case and from a thermal wall at the bottom in the two-dimensional case. We found that the theoretical predictions are in good agreement with the results of those simulations under the assumption that the fluctuation-dissipation relation holds in the case of nearly elastic collisions between particles. However, as the inelasticity of the interparticle collisions increases, the power spectrum for the center of mass obtained by the simulations gradually deviates from the prediction of theoretical curve. Connection between this deviation and violation of the fluctuation-dissipation relation is discussed.