Diffusion Of Brownian Particles In A Periodic Potential

The diffusion of Brownian particles in a tilted periodic potential can be used to describe many physical situations.In this paper,Langevin simulation is used to reveal the dimensional effect of diffusion in a tilted periodic potential.Diffusion behavior of Brownian particles driven by white noise in one-dimensional inclined periodic potential is studied by model soving the Fokker-Planck equation and the reaction flow theory with finite barrier height correction.In the first part,we study the diffusion motion of Brownian particles driven by white noise in one-dimensional and two-dimensional tilted periodic potentials.Applying bias force in the x direction,using the second-order Runge-Kutta algorithm to simulate the Langevin equation for two-dimensional inclined periodic potential,the dimensional effect is revealed.Applying a bias force in the x-direction is conducive to the formation of the running state in the y-direction and maintaining the locked state in the x-direction.The coexistence of the two states leads to rapid diffusion.Thus,diff usion in the one direction can be controlled by applying a bias force in the vertical direction.A comparison of diffusion behavior in one-dimensional and two-dimensional inclined potential reveals: in one-dimensional case,the diffusion coefficient reaches maximum when the force is less than the critical tilt value,but it reaches maximum when the force is greater than the critical tilt value in the two-dimensional tilted periodic potential.In two-dimensional case,due to the diffusion in y direction and the coexistence of lock state and run state,the lock state still exists when the force is greater than the critical tilt value.In the second part,the diffusion of Brownian particles driven by white noise in one-dimensional tilted periodic potential is studied analytically.The coordinate is integrated on both sides of the Fokker-Planck equation in the phase space.The Fokker-Planck equation for the probability density of velocity is obtained.The coordinate variance is the core of the diffusion problem of Brownian particles in tilted periodic potential.By solving the reduced Fokker-Planck equation,the analytical expression of the variance is obtained.Considering the finite barrier height correction,the starting point of particle motion is moved into the potential well,the potential in the barrier area is equivalent to a parabolic potential,and the effective frequency of the barrier is obtained by potential approaching method.The analytical expression of the coordinate variance in one dimensional tilted periodic potential is obtained by using the improved reactive flux theory.The theoretical results match the simulation results well within a certain range of parameters.The escape rate in the theory is obtained analytically by using a weak noise approximation.The theoretical results are in good agreement with the simulation results.