A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

Abstract

In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixtures of gases.

Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting

“…Short range potentials were also discussed in [34]. A few articles deal with the derivation of kinetic models for higher order interactions [1,2], mixtures [3] and quantum particles [4,5,6]. The derivation of the Boltzmann equation is closely related to the derivation of the kinetic wave equation, but possibly more challenging, since dispersive equations can be thought of as an intermediary step between a quantum mechanical model with a large number of particles, and kinetic theory.…”

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

“…The derivation of Boltzmann equations from a system of classical interacting particles is an extraordinarily active research field, see [16,15,28,39,52,77,99,132,139,183,184,185,193] and references therein. The methods differ vastly from the quantum field theoretic approach developed in the work at hand.…”