Statistical-mechanical description of classical test-particle dynamics in the presence of an external force field: modelling noise and damping from first principles

TL;DR

This paper develops a rigorous kinetic-theoretical framework linking microscopic dynamics to macroscopic stochastic motion, addressing issues with traditional operators and providing explicit diffusion and drift coefficients in external force fields.

Contribution

It introduces a corrected kinetic operator for classical test-particle dynamics that preserves positivity and accurately models diffusion and friction phenomena.

Findings

Identified ill-defined nature of traditional kinetic operators

Proposed an alternative approach ensuring positivity of distribution functions

Derived explicit expressions for diffusion and drift coefficients

Abstract

Aiming to establish a rigorous link between macroscopic random motion (described e.g. by Langevin-type theories) and microscopic dynamics, we have undertaken a kinetic-theoretical study of the dynamics of a classical test-particle weakly coupled to a large heat-bath in thermal equilibrium. Both subsystems are subject to an external force field. From the (time-non-local) generalized master equation a Fokker-Planck-type equation follows as a "quasi-Markovian" approximation. The kinetic operator thus defined is shown to be ill-defined; in specific, it does not preserve the positivity of the test-particle distribution function $f(\mathbf{x}, \mathbf{v}; t)$. Adopting an alternative approach, previously proposed for quantum open systems, is proposed to lead to a correct kinetic operator, which yields all the expected properties. A set of explicit expressions for the diffusion and drift coefficients are obtained, allowing for modelling macroscopic diffusion and dynamical friction phenomena, in terms of an external field and intrinsic physical parameters.