Ergodic properties of heterogeneous diffusion processes in a potential well.

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) = D|x| and U(x) = U|x| 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 D and U, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases is also evaluated from Monte Carlo simulations.