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 non-ergodic 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 $D(x)=D_0|x|^\alpha$ and $U(x)=U_0|x|^\beta$ for simplicity. Based on the competition roles played by $D(x)$ and $U(x)$, three different cases, $\beta>\alpha$, $\beta=\alpha$, and $\beta<\alpha$, are discussed. The system is ergodic for the first case $\beta>\alpha$, where the time average agrees with the ensemble average, being both determined by the steady solution for long times. In contrast, the system is non-ergodic for $\beta<\alpha$, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case $\beta=\alpha$, the ergodic property, depending on the prefactors $D_0$ and $U_0$, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases are also evaluated from Monte Carlo simulations.