TY - JOUR

T1 - Random walk to a nonergodic equilibrium concept

AU - Bel, G.

AU - Barkai, E.

PY - 2006/1

Y1 - 2006/1

N2 - Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic 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. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a δ function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

AB - Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic 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. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a δ function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

UR - http://www.scopus.com/inward/record.url?scp=32844465286&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.73.016125

DO - 10.1103/PhysRevE.73.016125

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C2 - 16486234

AN - SCOPUS:32844465286

SN - 1539-3755

VL - 73

JO - Physical Review E

JF - Physical Review E

IS - 1

M1 - 016125

ER -