Energy fluctuations in the homogeneous cooling state of granular gases

Driven inelastic Maxwell gases

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates τ(c)(-1) and τ(w)(-1), respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution r. The velocity change of a particle with velocity v, due to driving, is taken to be Δv=-(1+r(w))v+η, where r(w)∈[-1,1] and η is Gaussian white noise. For r(w)∈(0,1], this driving mechanism mimics the collision with a randomly moving wall, where r(w) is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by dv/dt=-Γv+η. We show that while the equations for the n-particle velocity distribution functions (n=1,2,...) do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for r(w)≠-1, the system goes to a steady state. Also we obtain the exact tail of the velocity distribution in the steady state. On the other hand, for r(w)=-1, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with Γ≠0, whereas for the purely diffusive driving (Γ=0), the system does not have a steady state.

Internal energy fluctuations of a granular gas under steady uniform shear flow

The stochastic properties of the total internal energy of a dilute granular gas in the steady uniform shear flow state are investigated. A recent theory formulated for fluctuations about the homogeneous cooling state is extended by analogy with molecular systems. The theoretical predictions are compared with molecular dynamics simulation results. Good agreement is found in the limit of weak inelasticity, while systematic and relevant discrepancies are observed when the inelasticity increases. The origin of this behavior is discussed.

Fluctuating Navier-Stokes equations for inelastic hard spheres or disks

Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients.

Fluctuating hydrodynamics for dilute granular gases: a Monte Carlo study

We investigate hydrodynamic noise in a dilute granular gas during the homogeneous cooling state, by means of a proper application of the direct simulation Monte Carlo (DSMC) algorithm. The DSMC includes a source of randomization which is not present in molecular dynamics (MD) for inelastic hard disks. Notwithstanding this difference, a fair quantitative agreement is found, including a violation of the fluctuation-dissipation relation for the noise amplitude of the same order observed in MD. This study suggests that deterministic collision dynamics is not an essential ingredient to reproduce, up to a good degree of approximation, hydrodynamic fluctuations in dilute granular gases.

Volume fluctuations and compressibility of a vibrated granular gas

The volume fluctuations in the steady state reached by a vibrated granular gas of hard particles confined by a movable piston on the top are investigated by means of event-driven simulations. Also, a compressibility factor, measuring the response in volume of the system to a change in the mass of the piston, is introduced and measured. From the second moment of the volume fluctuations and the compressibility factor, an effective temperature is defined by using the same relation as obeyed by equilibrium molecular systems. The interpretation of this effective temperature and its relationship with the granular temperature of the gas, and also with the velocity fluctuations of the movable piston, is discussed. It is found that the ratio of the temperature based on the volume fluctuations to the temperature based on the piston kinetic energy obeys simple dependencies on the inelasticity and on the piston-particle mass ratio.

Fluctuating hydrodynamics for dilute granular gases

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white-noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular-dynamics simulation results.

Dynamics of annihilation. II. Fluctuations of global quantities

We develop a theory for fluctuations and correlations in a gas evolving under ballistic annihilation dynamics. Starting from the hierarchy of equations governing the evolution of microscopic densities in phase space, we subsequently restrict our attention to a regime of spatial homogeneity, and obtain explicit predictions for the fluctuations and time correlation of the total number of particles, total linear momentum, and total kinetic energy. Cross correlations between these quantities are worked out as well. These predictions are successfully tested against molecular dynamics and Monte Carlo simulations. This provides strong support for the theoretical approach developed, including the hydrodynamic treatment of the spectrum of the linearized Boltzmann operator. This paper makes use of the spectral analysis reported in the preceding paper [Phys. Rev. E 77, 051127 (2008)].

Mesoscopic theory of critical fluctuations in isolated granular gases

Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas of inelastic hard spheres near the threshold of the clustering instability. They are shown to be governed only by vorticity fluctuations that also lead to a renormalization of the average total energy. The theory predicts a power-law divergent behavior of the scaled second moment of the fluctuations, and a scaling property of their probability distribution, both in agreement with simulations results. A more quantitative comparison between theory and simulation for the critical amplitudes and the form of the scaling function is also carried out.

Scaling and universality of critical fluctuations in granular gases

The total energy fluctuations of a low-density granular gas in the homogeneous cooling state near the threshold of the clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the symmetric of the one found in several molecular equilibrium and nonequilibrium systems.

Simulation study of the Green-Kubo relations for dilute granular gases

The Green-Kubo relations for dilute granular gases are employed to compute their transport coefficients by means of the direct simulation Monte Carlo method. This requires not only to follow the dynamics of the system, but also to identify some modified fluxes appearing in the time-correlation functions. The results are compared with those obtained from the Boltzmann equation by means of the Chapman-Enskog procedure in the first Sonine approximation. A good agreement is found for the shear viscosity over a wide range of inelasticities. Nevertheless, for the two transport coefficients associated with the heat flux, significant discrepancies appear for strong inelasticity. Their origin is discussed, showing that they are partially due to the presence of velocity correlations in the homogeneous cooling state of a dilute granular fluid.