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 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 $D(x)=D_0|x|^α$ and $U(x)=U_0|x|^β$ 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, being both determined by the steady solution for long times. In contrast, the system is non-ergodic 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 $D_0$ 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.