Einstein’s kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrödinger equation. The first step in this program is to verify the linear Boltzmann equation ...

We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was proved by Gallavotti [Phys. Rev. (2) 185, 308 (1969)]. The arguments presented here complete the analysis ...

In this paper, the lattice Boltzmann method for convectiondiffusion equation with source term is applied directly to solve some important nonlinear complex equations, including nonlinear Schrödinger (NLS) equation, coupled NLS equations, Klein-Gordon equation and coupled Klein-Gordon-Schrödinger equations, by using complex-valued distribution function and relaxation time. Detailed simulations o...

Ludwig Boltzmann had a hunch that irreversibility exhibited by a macroscopic system arises from the reversible dynamics of its microscopic constituents. He derived a nonlinear integro-differential equation-now called the Boltzmann equation-for the phase space density of the molecules of a dilute fluid. He showed that the Second law of thermodynamics emerges from Newton's equations of motion. Ho...

We determine the L p-L q-estimate of the linear equations of thermoelasticity for rhom-bic media in R 2. For this purpose we will transform the equations to an evolution equation showing that this equation has a unique solution with the help of the semigroup theory. In order to obtain the L p-L q-estimate, we will first apply the Fourier transformation to the evolution equation, thus determinin...

Lack of energy conservation in lattice Boltzmann models leads to unrealistically high values of the bulk viscosity. For this reason, the lattice Boltzmann method remains a computational tool rather than a model of a fluid. A novel lattice Boltzmann model with energy conservation is derived from Boltzmann's kinetic theory. Simulations demonstrate that the new lattice Boltzmann model is the valid...

This paper discusses the convergence of a new discrete-velocity model to the Boltzmann equation. First the consistency of the collision integral approximation is proved. Based on this we prove the convergence of solutions for a modiied model to DiPerna-Lions solutions of the Boltzmann equation. As a test numerical example, the solutions to the discrete problems are compared with the exact solut...

After considering the kinetical interaction principle (KIP) introduced in ref. Physica A 296, 405 (2001), we study in the Boltzmann picture, the evolution equation and the H-theorem for non extensive systems. The q-kinetics and the κ-kinetics are studied in detail starting from the most general non linear Boltzmann equation compatible with the KIP.

We consider the Fokker-Planck-Boltzmann equation consisting of the Boltzmann equation with an additional diffusion term in velocity space to simulate for instance the transport in thermal bath of binary elastic collisional particles, and investigate the (macroscopic) diffusion properties of solutions in large time.

Based on a recent L-L framework, we establish the acoustic limit of the Boltzmann equation for general collision kernels. The scaling of the fluctuations with respect to Knudsen number is optimal. Our approach is based on a new analysis of the compressible Euler limit of the Boltzmann equation, as well as refined estimates of Euler and acoustic solutions.