Ergodic properties of heterogeneous diffusion processes in a potential well.

Abstract

Heterogeneous diffusion processes can be well described by an overdamped Langevin equation with space-dependent diffusivity D(x). We investigate the ergodic and nonergodic behavior of these processes in an arbitrary potential well U(x) in terms of the observable-occupation time. Since our main concern is the large-x behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms D(x) = D0|x|α and U(x) = U0|x|β for simplicity. Based on the competition roles played by D(x) and U(x), three different cases, β > α, β = α, and β < α, are discussed. The system is ergodic for the first case β > α, where the time average agrees with the ensemble average, both determined by the steady solution for long times. By contrast, the system is nonergodic for β < α, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case β = α, the ergodic property, depending on the prefactors D0 and U0, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases is also evaluated from Monte Carlo simulations.