A new derivation of the relationship between diffusion coefficient and entropy in classical Brownian motion by the ensemble method

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 MM is far greater than that of the bath molecule mm, we can adopt the weakly couple limit. Only considering the first-order approximation of the mass ratio m/Mm/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 DD and the entropy SS in the thermal equilibrium, that is to say, D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}D=ℏeMexp[S/(kBd)], where dd is the dimension of the space, k_BkB the Boltzmann constant, h the reduced Planck constant and ee 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.