Fractional Feynman-Kac equation for weak ergodicity breaking

Shai Carmi*, Eli Barkai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

53 Scopus citations


The continuous-time random walk (CTRW) is a model of anomalous subdiffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, ψ(τ)∼τ -(1 +α ), leads to subdiffusion (ψ∼tα) 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 Ū(t)=1t U[x(τ)]dτ 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→ ∞ 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 our fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.

Original languageAmerican English
Article number061104
JournalPhysical Review E
Issue number6
StatePublished - 5 Dec 2011
Externally publishedYes


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