Random walk to a nonergodic equilibrium concept

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 $\delta$ function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

Figure 1

(Color online) Boltzmann’s equilibrium for an ensemble of CTRW particles in a double-well potential field, and fixed temperature. In simulations (crosses and circles) the CTRW with $\alpha =0.5$ and 0.7 is considered. The figure illustrates that for an ensemble of particles, standard equilibrium is obtained; ergodicity breaking is found only when long time averages of single particle trajectories are analyzed. The scaled potential (dashed curve) is the double-well potential field, and the solid curve is the Boltzmann equilibrium distribution. To construct the histogram we measured the position of $N={10}^6$ particles, after each is evolved for a time $t={10}^8$. The temperature is $T=0.3$. Note that all quantities are dimensionless.

Figure 2

(Color online) The PDF of occupation times ${\overline{p}}_X=t_X∕t$ where $t_X$ is the total time spent on a finite domain, and $t$ is the measurement time. Here we used $\alpha =0.5$. The domains we considered are $x>0.6$, $x>0$, and $x>-0.6$, corresponding to (a), (b), and (c) in the figure. For an ergodic process satisfying detailed balance, the PDF $f\left({\overline{p}}_X\right)$ would be a $\delta$ function centered around the value predicted by Boltzmann which is given by the arrows. In a given numerical experiment, it is unlikely to obtain the value of ${\overline{p}}_X$ predicted by Boltzmann, though Boltzmann statistics does yield the average of ${\overline{p}}_X$ over many particles. To construct histograms we used ${10}^6$ trajectories, measurement time $t={10}^8$, and temperature $T=0.3$. The solid curves correspond to the analytical formula Eqs. (7, 8) used without any fitting parameters, and the pluses show the simulation results.

Figure 3

(Color online) Same as Fig. 2 but now $\alpha =0.7$. In this case the probability of obtaining the ergodic value (arrow) in a single measurement is higher than for the case $\alpha =0.5$ (see Fig. 2). Note that the larger the region we observe the more the PDF is tilted toward 1 since it is more probable that the particle is within the region of observation.

Figure 4

(Color online) The PDF of the fraction of occupation time in the disconnected regions $x<-0.6$ and $x>0.6$, for $\alpha =0.75$ and temperature $T=0.3$. The theoretical line (solid curve) is determined by summing the equilibrium probabilities of occupying all the cells within the regions, and substituting it into the constant ${\mathcal{R}}_X$. The good agreement between the theory and the numerical simulation (pluses) illustrates that our theory is not limited to connected regions.

Figure 5

(Color online) The PDF of the fraction of occupation time of the regions near the potential minima, $-0.9<x<-0.5$ and $0.5<x<0.9$, for $\alpha =0.8$ when temperature $T$ is varied. When the temperature is such that the probability of occupying the minima regions is equal to $1∕2$ the PDF is symmetric; for our parameters this temperature is $T_s=0.4$. For $T⪡T_s$ the particle tends to be located all the time in the vicinity of the minima; hence for $T=0.1<T_s$ (dot-dashed curve) the PDF has more weight on values ${\overline{p}}_X>1∕2$. For $T⪢T_s$ the particle is hardly found in the minima regions; hence for $T=5>T_s$ (dotted curve) the PDF has more weight on values ${\overline{p}}_X<1∕2$.

Figure 6

(Color online) A schematic diagram of the two-state process.