A fractional Feynman-Kac equation for weak ergodicity breaking

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<α<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 $t\to \infty$ 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 α=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.

14 pages, 4 figures

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