Thermal equilibrium of a Brownian particle with coordinate dependent diffusion: comparison of Boltzmann and modified Boltzmann distributions with experimental results

TL;DR

This paper compares the traditional and modified Boltzmann distributions for a Brownian particle with coordinate-dependent diffusion, using experimental data to evaluate which better describes thermal equilibrium.

Contribution

It introduces a modified Boltzmann distribution derived from an Itô-process and validates it against experimental results without adjustable parameters.

Findings

Modified Boltzmann distribution aligns better with experimental data.

Supports the view of thermal equilibrium as an Itô-process for coordinate-dependent diffusion.

Experimental results favor the modified distribution over the classical one.

Abstract

In this paper we compare the Boltzmann distribution with a modified Boltzmann distribution, that results from an It\^o-process considering thermal equilibrium of a Brownian particle with coordinate dependent diffusion, in the light of an existing experiment. The experiment was reported in 1994 by Faucheux and Libchaber. The experiment made use of direct tracking of diffusion of Brownian particles near a wall. Results of this experiment allows us to compare the Boltzmann and the modified Boltzmann distribution without making use of any adjustable parameter. A comparison of these two distributions with the experimental results lends support to the consideration of thermodynamic equilibrium of a Brownian particle with coordinate-dependent diffusion to be an It\^o-process.