TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhysCore.4.2.015
TI  - A new derivation of the relationship between diffusion coefficient and entropy in classical Brownian motion by the ensemble method
PY  - 2021/06/02
UR  - https://scipost.org/SciPostPhysCore.4.2.015
JF  - SciPost Physics Core
JA  - SciPost Phys. Core
VL  - 4
IS  - 2
SP  - 015
A1  - Liao, Yi
AU  - Gong, Xiao-Bo
AB  - 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.
ER  -

@Article{10.21468/SciPostPhysCore.4.2.015,
	title={{A new derivation of the relationship between diffusion coefficient and entropy in classical Brownian motion by the ensemble method}},
	author={Yi Liao and Xiao-Bo Gong},
	journal={SciPost Phys. Core},
	volume={4},
	pages={015},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhysCore.4.2.015},
	url={https://scipost.org/10.21468/SciPostPhysCore.4.2.015},
}