A fractional Feynman-Kac equation for weak ergodicity breaking

TL;DR

This paper derives fractional Feynman-Kac equations for continuous-time random walks with power-law waiting times, revealing how weak ergodicity breaking affects long-time time-averages of observables in anomalous diffusion.

Contribution

It introduces forward and backward fractional Feynman-Kac equations for CTRW in binding potentials, extending the analysis of time-averages and ergodicity breaking in anomalous diffusion.

Findings

Derived probability density functions for time-averages at long times.

Showed that time-averages are random variables for <<1, but equal ensemble averages at =1.

Analyzed convergence of fluctuations to asymptotic values.

Abstract

Continuous-time random walk (CTRW) is a model of anomalous sub-diffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, $\psi(\tau) \tau^{-(1+\alpha)}$, leads to sub-diffusion ($<x^2>~t^{\alpha}$) for 0<\alpha<1. In closed systems, the long stagnation periods cause time-averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time-average of a general observable $\bar{U} = \int_0^t U[x(\tau)]d\tau / t$ is a functional of the path and is described by the well known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time-averages: the fraction of time spent by a particle in half box, and the time-average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time-averages for $t \rightarrow \infty$ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long-times, except for \alpha=1 when they are identical to their ensemble averages. Using the fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.