Weak Ergodicity Breaking in the Continuous-Time Random Walk

The continuous-time random walk (CTRW) model exhibits a nonergodic phase when the average waiting time diverges. Using an analytical approach for the nonbiased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the nonergodic properties of the random walk which show strong deviations from Boltzmann-Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann-Gibbs theory, while in the nonergodic phase yields a generalized nonergodic statistical law.