Equilibrium stochastic dynamics of a Brownian particle in inhomogeneous space: derivation of an alternative model

TL;DR

This paper derives an alternative equilibrium stochastic model for Brownian particles in inhomogeneous spaces, extending Zwanzig's formulation, and discusses experimental implications and potential limitations of Boltzmann distribution.

Contribution

It introduces a new stochastic dynamics model for inhomogeneous systems, generalizing Zwanzig's approach and addressing limitations of traditional Boltzmann-based models.

Findings

Derived an alternative stochastic model for inhomogeneous space.

Proposed experimental tests to verify the new model.

Discussed potential limitations of Boltzmann distribution in such systems.

Abstract

An alternative equilibrium stochastic dynamics for a Brownian particle in inhomogeneous space is derived. Such a dynamics can model the motion of a complex molecule in its conformation space when in equilibrium with a uniform heat bath. The derivation is done by a simple generalization of the formulation due to Zwanzig for a Brownian particle in homogeneous heat bath. We show that if the system couples to different number of bath degrees of freedom at different conformations then the alternative model is derived. We discuss results of an experiment by Faucheux and Libchaber which probably has indicated possible limitation of the Boltzmann distribution as equilibrium distribution of a Brownian particle in inhomogeneous space and propose experimental verification of the present theory using similar methods.