Diffusion Of Brownian Particles In One Dimensional Periodic Potential

Brownian motion is the basic subject in non-equilibrium statistical physics,many problems in the nature can be modeled as a Brownian motion.Langevin equation and Fokker-Planck equation provide a theoretical basis for Brownian motion.In this thesis,the diffusion behavior of Brownian particles in a periodic potential driven by an internal white noise are studied.The physical mechanisms of the results are explored.In the first part,we study the diffusion behavior of a Brownian particle in the periodic potential with a basic cell composed of a parabolic potential barrier linked smoothly with a harmonic potential well in moderate friction region,and calculate the diffusion coefficient.The diffusion of Brownian particles driven by internal white noise is studied analytically.A simplified model is proposed to calculate the diffusion coefficient of Brownian particles.When a particle passes through the joint point of harmonic potential and parabolic potential,a time coarse-graining scheme is used to obtain a simple analytical expression of the probability distribution.The escape rate and long jump probability can be obtained by combining the analytical expression with the random walk model,and the diffusion coefficient is obtained.The theoretical results of diffusion coefficient are confirmed by the numerical simulation results.In the second part,the diffusion behavior of Brownian particles in moderate to large friction region is studied by using the proposed reactive flow theory with finite barrier height,the periodic potential is a cosine potential.By moving the starting point of particle to the inside of the potential well,the effective barrier frequency is obtained by using the equivalent parabolic potential barrier of the parabolic potential,and the analytical expression of the diffusion coefficient is obtained.The analytical result can be used as the zero-order approximation of the diffusion coefficient in moderate to large friction region.Then the Kramers function is improved by perturbation theory,and the first order correction of escape rate and diffusion coefficient is calculated by the method of the flux over population expression.An improvement of this function by means of a perturbation theory allows one to calculate corrections to the rate by means of the flux over population expression.The analytical results of diffusion coefficient are in good agreement with the numerical simulation results.This method is simple and high precision.