2017/07/07 by Yi Liao, Liao, Yi, Xiao-Bo Gong +1
Physics and Astronomy · Economics, Econometrics and Finance · #Advanced Thermodynamics and Statistical Mechanics #Statistical Mechanics and Entropy #Complex Systems and Time Series Analysis
paper · pdf · doi:10.48550/arxiv.1707.02068
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 =(ℏ)/(eM) exp[S/(kBd)], where d is the dimension of the space, kB the Boltzmann constant, ℏ 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.