Collisional invariants for the phonon Boltzmann equation

For the phonon Boltzmann equation with only pair collisions we characterize the set of all collisional invariants under some mild conditions on the dispersion relation

A volume-based hydrodynamic approach to sound wave propagation in a monatomic gas

We investigate sound wave propagation in a monatomic gas using a volume-based hydrodynamic model. In Physica A vol 387(24) (2008) pp6079-6094, a microscopic volume-based kinetic approach was proposed by analyzing molecular spatial distributions; this led to a set of hydrodynamic equations incorporating a mass-density diffusion component. Here we find that these new mass-density diffusive flux and volume terms mean that our hydrodynamic model, uniquely, reproduces sound wave phase speed and damping measurements with excellent agreement over the full range of Knudsen number. In the high Knudsen number (high frequency) regime, our volume-based model predictions agree with the plane standing waves observed in the experiments, which existing kinetic and continuum models have great difficulty in capturing. In that regime, our results indicate that the "sound waves" presumed in the experiments may be better thought of as "mass-density waves", rather than the pressure waves of the continuum regime.Comment: Revised to aid clarification (no changes to presented model); typos corrected, figures added, paper title change

From the Boltzmann equation to fluid mechanics on a manifold

We apply the Chapman-Enskog procedure to derive hydrodynamic equations on an arbitrary surface from the Boltzmann equation on the surface

Cauchy problem for the Boltzmann-BGK model near a global Maxwellian

In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform $L^2$-stability for nonlinear perturbations.Comment: 26 page

Quantum Kinetic Evolution of Marginal Observables

We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems to generally accepted approach on basis of kinetic equations.Comment: 18 page

Direct simulation for a homogenous gas

A probabilistic analysis of the direct simulation of a homogeneous gas is given. A hierarchy of equations similar to the BBGKY hierarchy for the reduced probability densities is derived. By invoking the molecular chaos assumption, an equation similar to the Boltzmann equation for the single particle probability density and the corresponding H-theorem is derived

Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation

We present a further theoretical extension to the kinetic theory based formulation of the lattice Boltzmann method of Shan et al (2006). In addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a new regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the non-equilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to the third order hydrodynamic moments. Numerical evidences demonstrate that the extended model overcomes some major defects existed in the conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number (Kn) can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsen's minimum and the asymptotic behavior of flow flux at large Kn

Fokker-Planck type equations for a simple gas and for a semi-relativistic Brownian motion from a relativistic kinetic theory

A covariant Fokker-Planck type equation for a simple gas and an equation for the Brownian motion are derived from a relativistic kinetic theory based on the Boltzmann equation. For the simple gas the dynamic friction four-vector and the diffusion tensor are identified and written in terms of integrals which take into account the collision processes. In the case of Brownian motion, the Brownian particles are considered as non-relativistic whereas the background gas behaves as a relativistic gas. A general expression for the semi-relativistic viscous friction coefficient is obtained and the particular case of constant differential cross-section is analyzed for which the non-relativistic and ultra relativistic limiting cases are calculated.Comment: To appear in PR