P. Maynar

Fluctuating hydrodynamics for driven granular gases

Abstract We study a granular gas heated by a stochastic thermostat in the dilute limit. Starting from the kinetic equations governing the evolution of the correlation functions, a Boltzmann-Langevin equation is constructed. The spectrum of the corresponding linearized Boltzmann-Fokker-Planck operator is analyzed, and the equation for the fluctuating transverse velocity is derived in the hydrodynamic limit. The noise term (Langevin force) is thus known microscopically and contains two terms: one coming from the thermostat and the other from the fluctuating pressure tensor. At variance with the free cooling situation, the noise is found to be white and its amplitude is evaluated.

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.

Hydrodynamic modes, Green–Kubo relations, and velocity correlations in dilute granular gases

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spheres can be identified, and calculated in the long wavelength limit. Assuming they dominate at long times, formal expressions for the Navier–Stokes transport coefficients are derived. They can be expressed in a form that generalizes the Green–Kubo relations for molecular systems, and it is shown that they can also be evaluated by means of N-particle simulation methods. The form of the hydrodynamic modes to zeroth order in the gradients is used to detect the presence of inherent velocity correlations in the homogeneous cooling state, even in the low density limit. They manifest themselves in the fluctuations of the total energy of the system. The theoretical predictions are shown to be in agreement with molecular dynamics simulations. Relevant related questions deserving further attention are pointed out.

Breakdown of the fluctuation-dissipation relations in granular gases

A numerical molecular dynamics experiment measuring the two-time correlation function of the transversal velocity field in the homogeneous cooling state of a granular gas modeled as an ensemble of inelastic hard particles is reported. By measuring the decay rate and the amplitude of the correlations, the accuracy of the Landau-Langevin equation of fluctuating hydrodynamics is checked. The results indicate that although a Langevin approach can be valid, the fluctuation-dissipation relation must be modified, since the viscosity parameter appearing in it differs from the usual hydrodynamic shear viscosity.

Breakdown of hydrodynamics in the inelastic Maxwell model of granular gases.

Both the right and left eigenfunctions and eigenvalues of the linearized homogeneous Boltzmann equation for inelastic Maxwell molecules corresponding to the hydrodynamic modes are calculated. Also, some nonhydrodynamic modes are identified. It is shown that below a critical value of the parameter characterizing the inelasticity, one of the kinetic modes decays slower than one of the hydrodynamic ones. As a consequence, a closed hydrodynamic description does not exist in that regime. Some implications of this behavior on the formally computed Navier-Stokes transport coefficients are discussed.

Linear hydrodynamics for driven granular gases.

We study the dynamics of a granular gas heated by a stochastic thermostat. From a Boltzmann description, we derive the hydrodynamic equations for small perturbations around the stationary state that is reached in the long time limit. Transport coefficients are identified as Green-Kubo formulas obtaining explicit expressions as a function of the inelasticity and the spatial dimension.

Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions.

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)].

Towards an H-theorem for granular gases

The H-theorem, originally derived at the level of Boltzmann non-linear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional H related to the original H, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic Direct Simulation Monte Carlo technique, and Molecular Dynamics) indicate that it is also non-increasing. Both driven and unforced cases are investigated.

Scaling and aging in the homogeneous cooling state of a granular fluid of hard particles

The presence of the aging phenomenon in the homogeneous cooling state (HCS) of a granular fluid composed of inelastic hard spheres or disks is investigated. As a consequence of the scaling property of the N-particle distribution function, it is obtained that the decay of the normalized two-time correlation functions slows down as the time elapsed since the beginning of the measurement increases. This result is confirmed by molecular dynamics simulations for the particular case of the total energy of the system. The agreement is also quantitative in the low density limit, for which an explicit analytical form of the time correlation function has been derived. Moreover, the reported results provide support for the existence of the HCS as a solution of the N-particle Liouville equation.