Occupation Time Statistics in the Quenched Trap Model

We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width $T_g$, we find in thermal equilibrium a transition between Boltzmann statistics when $T>T_g$ to Lamperti statistics when $T<T_g$. We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.

Figure 1

The PDF of the occupation fraction for the deterministic field $U(x)=\mathcal{F}x$. (a) For $T=3T_g$ we find a delta function centered on the value given by Boltzmann’s statistics. When $T/T_g=0.7$ [panel (b)] the nontrivial distribution of the occupation fraction has three peaks while for $T/T_g=0.3$ [panel (c)] the distribution is bimodal. The $+$ symbols are simulations and the curve is the theoretical prediction [Eq. (15)] without fitting. We used $\mathcal{F}=1$, $T_g=1$, $a={10}^{-5}$ the observation domain $0<x<1$, system size 20, and $3\times {10}^4$ disordered systems.

Figure 2

The averaged occupation fraction versus $T/T_g$. When $T>T_g$, $⟨\overline{p}⟩=P_B(T)$, namely, the usual Boltzmann theory applies, while for $T<T_g$, $⟨\overline{p}⟩=P_B(T_g)$, which is independent of the temperature $T$. The lines are theoretical predictions [Eq. (17)] and the diamonds are simulation results with no fitting.