Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# An Ensemble Method of the Relationship between Diffusion Coefficient and Entropy in the Classical Brownian Motion

Version 1 : Received: 27 March 2019 / Approved: 28 March 2019 / Online: 28 March 2019 (13:46:29 CET)

How to cite: Liao, Y.; Gong, X. An Ensemble Method of the Relationship between Diffusion Coefficient and Entropy in the Classical Brownian Motion. Preprints 2019, 2019030268. https://doi.org/10.20944/preprints201903.0268.v1 Liao, Y.; Gong, X. An Ensemble Method of the Relationship between Diffusion Coefficient and Entropy in the Classical Brownian Motion. Preprints 2019, 2019030268. https://doi.org/10.20944/preprints201903.0268.v1

## Abstract

The diffusion coefficient--a measure of dissipation, and the entropy--a measure of fluctuation are found to be intimately correlated in many physical systems. Unlike the fluctuation dissipation theorem in linear response theory, the correlation is often strongly non-linear. To understand this complex dependence, we consider the classical Brownian diffusion in this work. Under certain rational assumption, i.e. in the bi-component fluid mixture, the mass of the Brownian particle $M$ is far greater than that of the bath molecule $m$, we can adopt the weakly couple limit. Only considering the first-order approximation of the mass ratio $m/M$, we obtain a linear motion equation in the reference frame of the observer as a Brownian particle. Based on this equivalent equation, we get the Hamiltonian at equilibrium. Finally, using canonical ensemble method, we define a new entropy that is similar to the Kolmogorov-Sinai entropy. Further, we present an analytic expression of the relationship between the diffusion coefficient $D$ and the entropy $S$ in the thermal equilibrium, that is to say, $D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}$, where $d$ is the dimension of the space, $k_B$ the Boltzmann constant, $\hbar$ the reduced Planck constant and $e$ the Euler number. This kind of scaling relation has been well-known and well-tested since the similar one for single component is firstly derived by Rosenfeld with the expansion of volume ratio.

## Keywords

diffusion coefficient；entropy；Brownian motion

## Subject

Physical Sciences, Thermodynamics

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