Extension of generalized Kramers theory in the spatial diffusion regime: Diffusion of Brownian particles in periodic potential

Extension of generalized Kramers theory in the spatial diffusion regime to extensive noise parameters is illustrated through the internal Ornstein–Uhlenbeck noise-induced diffusion of a particle in one-dimensional periodic potential. In the current over population method, the Fokker–Planck equation does not have a non-equilibrium stationary-state solution for a large scope of Ornstein–Uhlenbeck noise parameters. Instead, an effective non-equilibrium stationary-state transition probability density can be obtained and can serve as the long-term mean transition probability density for the Langevin simulation of the escape rate. The analytical results of the diffusion coefficient in the spatial diffusion regime are in good agreement with the Langevin simulation results at a low temperature. This theoretical approach can be applied to Gaussian noise processes in potential barrier crossing problems, including the processes in the presence of external noises.