J. Javier Brey

Dissipative Dynamics for Hard Spheres

The dynamics for a system of hard spheres with dissipative collisions is described at the levels of statistical mechanics, kinetic theory, and simulation. The Liouville operator(s) and associated binary scattering operators are defined as the generators for time evolution in phase space. The BBGKY hierarchy for reduced distribution functions is given, and an approximate kinetic equation is obtained that extends the revised Enskog theory to dissipative dynamics. A Monte Carlo simulation method to solve this equation is described, extending the Bird method to the dense, dissipative hard-sphere system. A practical kinetic model for theoretical analysis of this equation also is proposed. As an illustration of these results, the kinetic theory and the Monte Carlo simulations are applied to the homogeneous cooling state of rapid granular flow.

A kinetic model for a multicomponent gas

A kinetic model for a dilute multicomponent gas system is proposed. It is constructed by replacing the Boltzmann collision operator with a relaxation‐time term, in the same manner as in the Bhatnagar–Gross–Krook (BGK) model for a single gas. The model contains several parameters that are determined by keeping some of the main properties of the Boltzmann description. In contrast to previous works, the BGK equation is recovered when mechanically identical particles are considered. Thus the model can be expected to apply to systems in which masses are comparable. The transport properties to the Navier–Stokes level are studied and Onsager’s reciprocal relations are found to hold.

Kinetic theory of simple granular shear flows of smooth hard spheres

Steady simple shear flows of smooth inelastic spheres are studied by means of a model kinetic equation and also of a direct Monte Carlo simulation method. Both approaches are based on the Enskog equation and provide for each other a test of consistency. The dependence of the granular temperature and of the shear and normal stresses on both the solid fraction and the coefficient of restitution is analysed. Quite a good agreement is found between theory and simulations in all cases. Also, simplified expressions based on the analytical solution of the model for small dissipation are shown to describe fairly well the simulation results even for not small inelasticity. A critical comparison with previous theories is carried out

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.

Efficient schemes to compute diffusive barrier crossing rates

The formulation of the classical barrier-crossing problem is reviewed in the context of numerical simulations, with the focus on barrier crossing problems where the reaction coordinate depends in a non-trivial way on the Cartesian coordinates of many particles. Often it is convenient to measure the barrier height using constrained dynamics. Such a calculation requires a knowledge of the Jacobian for the coordinate transformation between Cartesian and generalized (‘reaction’) coordinates, and it is shown that the calculation of this Jacobian can be simplified. The conventional expression for the crossing rate is found to become computationally inefficient when the barrier crossing is diffusive. An alternative formulation of the barrier-crossing rate is given that leads to much better statistical accuracy in the computed crossing rates.

Self-diffusion in freely evolving granular gases

A self-diffusion equation for a freely evolving gas of inelastic hard disks or spheres is derived starting from the Boltzmann–Lorentz equation, by means of a Chapman–Enskog expansion in the density gradient of the tagged particles. The self-diffusion coefficient depends on the restitution coefficient explicitly, and also implicitly through the temperature of the system. This latter introduces also a time dependence of the coefficient. As in the elastic case, the results are trivially extended to the Enskog equation. The theoretical predictions are compared with numerical solutions of the kinetic equation obtained by the direct simulation Monte Carlo method, and also with molecular dynamics simulations. An excellent agreement is found, providing mutual support to the different approaches.

Thermodynamic description in a simple model for granular compaction

A simple model for the dynamics of a granular system under tapping is studied. The model can be considered as a particularization for short taps of a more general one-dimensional lattice model with facilitated dynamics. The steady state reached by the system is discussed and the results are shown to be consistent with the thermodynamic granular theory developed by Edwards and coworkers. In particular, the basic assumption of the theory, i.e., that the probability distribution depends only on the volume of the configuration, is verified.

Transport Coefficients for dense hard-disk systems

A study of the transport coefficients of a system of elastic hard disks based on the use of Helfand-Einstein expressions is reported. The self-diffusion, the viscosity, and the heat conductivity are examined with averaging techniques especially appropriate for event-driven molecular dynamics algorithms with periodic boundary conditions. The density and size dependence of the results are analyzed, and comparison with the predictions from Enskogs theory is carried out. In particular, the behavior of the transport coefficients in the vicinity of the fluid-solid transition is investigated and a striking power law divergence of the viscosity with density is obtained in this region, while all other examined transport coefficients show a drop in that density range in relation to the Enskogs prediction. Finally, the deviations are related to shear band instabilities and the concept of dilatancy.

Energy partition and segregation for an intruder in a vibrated granular system under gravity.

The difference of temperatures between an impurity and the surrounding gas in an open vibrated granular system is studied. It is shown that, in spite of the high inhomogeneity of the state, the temperature ratio remains constant in the bulk of the system. The lack of energy equipartition is associated to the change of sign of the pressure diffusion coefficient for the impurity at certain values of the parameters of the system, leading to a segregation criterium. The theoretical predictions are consistent with previous experimental results, and also in agreement with molecular dynamics simulation results reported in this Letter.

STOCHASTIC RESONANCE IN A ONE-DIMENSIONAL ISING MODEL

Abstract The stochastic resonance phenomenon for a one-dimensional Ising model in an oscillating magnetic field is discussed. Within the linear field approximation, the amplitude of the induced magnetization presents a maximum as a function of the temperature. The behaviour of the phase shift is also studied.