Abstract
The continuous-time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory in this case? Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann-Gibbs theory 1. 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 non-ergodic phase yields a generalized non-ergodic statistical law. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle in a finite region of space approaches U- or W-shaped distributions related to the arcsine law. When conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. The relation of our work to single-molecule experiments is briefly discussed.\\
1) G. Bel and E. Barkai, Phys. Rev. Lett., 94, 240602 (2005), Phys. Rev. E, 73, 016125 (2006).
Users
Please
log in to take part in the discussion (add own reviews or comments).