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Center for Nonlinear Studies |
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 nonergodic properties of the random walk which show strong deviations from Boltzmann-Gibbs theory [1,2]. 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). [2] G. Bel and E. Barkai, Phys. Rev. E, 73, 016125 (2006).