Stochastic models in kinetic theory

Abstract

The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent.

Low-noise Monte Carlo simulation of the variable hard sphere gas

“…Professor Cercignani has made contributions both to the mathematical theory ͑e.g., in stochastic models 39 ͒ and the physics of rarefied gases. Despite his strong mathematical inclination, he has made contributions that continue to have enormous impact in engineering applications: for example, using the relaxation-time approximation, he was the first to obtain results for slip flow coefficients; 40,41 he was also the first to provide a mathematical formulation and complete ͑numerical͒ solution of the pressure-driven flow ͑Knudsen minimum͒ problem.…”