Nonergodicity of Brownian motion in a periodic potential

Nonergodicity in Brownian dynamics can be divided into two classes by adding a periodic potential in a force-free ballistic diffusive system. Class-I is the system in which the Laplace transform of the damping kernel is equal to zero at low frequency. When the temperature is much higher than the barrier height, the kinetic part of the mean energy depends on the initial distribution of the velocity; with the temperature decreasing, the ergodicity is recovered. Thinking the stable velocity variance of class-I as an internal noise to drive a force-free Brownian particle, the Laplace transform of the damping kernel is infinite at zero frequency. It is found that the diffusion coefficient approaches vanishing with the temperature increasing, which exhibits the characteristic of classical locality. The asymptotic mean-square coordinates of the class-H depends on its initial coordinates and the ergodicity cannot be ensured through introducing a potential.