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 manuscript a
non-ergodic equilibrium concept is investigated, for a continuous time random
walk model in a potential field. In particular we show that in the non-ergodic
phase the distribution of the occupation time of the particle on a given
lattice point, approaches U or W shaped distributions related to the arcsin
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 non-ergodic dynamics and canonical
statistical mechanics. In the ergodic phase the distribution function of the
occupation times approaches a delta function centered on the value predicted
based on standard Boltzmann-Gibbs statistics. Relation of our work with single
molecule experiments is briefly discussed.Comment: 14 pages, 6 figure
