TY - JOUR

T1 - Fractional Feynman-Kac equation for weak ergodicity breaking

AU - Carmi, Shai

AU - Barkai, Eli

PY - 2011/12/5

Y1 - 2011/12/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84555189320&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.84.061104

DO - 10.1103/PhysRevE.84.061104

M3 - Article

AN - SCOPUS:84555189320

SN - 1539-3755

VL - 84

JO - Physical Review E

JF - Physical Review E

IS - 6

M1 - 061104

ER -