Ergodic properties of occupation times in heterogeneous media

TL;DR

This work addresses ergodicity in Brownian motion within spatially heterogeneous media by focusing on occupation-time statistics. Using the backward Feynman–Kac framework, the authors derive exact and asymptotic results for half-occupation and interval-occupation times under two explicit diffusion profiles: piecewise-constant and power-law. They reveal nonergodic behavior and explicit EB parameters for the piecewise case, and an α-dependent ergodicity transition in the power-law case, supported by numerical simulations. The results establish a robust, general method to quantify ergodicity in heterogeneous diffusion and highlight distinct limiting distributions (asymmetric arcsine, Lamperti, Mittag–Leffler) that govern occupation-time fluctuations. This provides a complementary perspective to MSD-based analyses and offers practical insights for transport in complex media.

Abstract

We investigate the ergodic properties of Brownian motion in heterogeneous media through the statistics of occupation times. Using the Feynman-Kac formalism, we derive analytical expressions for the distributions, moments, and ergodicity breaking parameters of occupation times in two models with spatially varying diffusion coefficient: a piecewise-constant profile and a power-law profile. In the piecewise model, the half occupation time and the occupation time within an interval follow asymmetric arcsine and half-Gaussian distributions, respectively, indicating non-ergodic behavior. For the power-law case, the corresponding distributions are the Lamperti and Mittag-Leffler. In both models, we identify a transition from non-ergodic to ergodic dynamics as the exponent vary. Numerical simulations fully corroborate the analytical results, demonstrating the effectiveness of the Feynman-Kac approach for quantifying ergodicity in heterogeneous diffusion processes.

Figures (11)

• Figure 1: Limiting distribution of the half-occupation time for the piecewise heterogeneity. The different symbols are for trajectories with $t=10^4,10^5,$ and $10^6$. The black solid line corresponds to Eq. \ref{['ld1']}. $D_{+}=0.5,1.0$, and $2.0$ in panels (a), (b), and (c), respectively. In all panels $D_{-}=1.0$, $x_0=0$, simulation timestep $dt=0.1$, and $N=10^4$ trajectories.
• Figure 2: (a) $\langle T^{+}(t)\rangle$, (b) $\langle T^{+}(t)^2\rangle$, and (c) $\textrm{EB}_+$ of the half-occupation time for the piecewise heterogeneity. In panels (a), and (b) the solid lines are computed with Eq. \ref{['t1mas']}, $D_{+}=0.5,1.0$, and $2.0$; In panel (c) the solid line is Eq. \ref{['EBmas2']}. In all panels $D_{-}=1.0$, $x_0=0$, simulation timestep $dt=0.1$, and $N=10^4$ trajectories.
• Figure 3: PDF of the time averaged half-occupation time for the piecewise heterogeneity. The different symbols are for trajectories with $t=10^4,10^5,$ and $10^6$. $D_{+}=0.5,1.0$, and $2.0$ in panels (a), (b), and (c), respectively. The solid line is computed wiht Eq. \ref{['Peta1']} In all panels $D_{-}=1.0$, $x_0=0$, simulation timestep $dt=0.1$, and $N=10^4$ trajectories.
• Figure 4: Limiting distribution of the half-occupation time for the power-law heterogeneity. The different symbols are for trajectories with $t=10^4,10^5,$ and $10^6$. In panels (a), and (b) the black solid line corresponds to Eq. \ref{['lamperti']}; In panel (c) corresponds to $e^{-0.5u}$, and to $\delta(t-0.5)$ in its inset. $\alpha=-0.05,0.5$, and $1.5$ in panels (a), (b), and (c), respectively. If $\alpha>0$$\epsilon=10^{-10}$, and $\epsilon=10^{-3}$ if $\alpha<0$. In all panels $D_{0}=1.0$, $x_0=0$, simulation timestep $dt=0.1$, and $N=10^4$ trajectories.
• Figure 5: (a) $\langle T^{+}(t)\rangle$, (b) $\langle T^{+}(t)^2\rangle$, and (c) $\textrm{EB}_+$ of the half-occupation time for the power-law heterogeneity. In panels (a), and (b) the solid lines are computed with Eq. \ref{['tm1']} and $\alpha=-0.05,0.5$, and $1.5$; In panel (c) the solid line is Eq. \ref{['ebmas']}. If $\alpha>0$$\epsilon=10^{-10}$, and $\epsilon=10^{-3}$ if $\alpha<0$. In all panels $D_{0}=1.0$, $x_0=0$, simulation timestep $dt=0.1$, and $N=10^4$ trajectories.