Conditional Ergodicity and Universal Fluctuations in Weak Ergodicity Breaking

Abstract

Time averages extracted from single-particle trajectories in complex media often vary strongly from one trajectory to another, even for long measurement times. Such persistent trajectory-to trajectory scatter is commonly observed in anomalous diffusion and signals weak ergodicity breaking driven by scale-free trapping. Here we identify conditional ergodicity: conditioning on a natural internal clock restores self-averaging of time-averaged observables. Combining conditional ergodicity with the stochastic mapping between the internal clock and physical time implies a universal law: once rescaled by their mean, time-averaged transport coefficients in systems exhibiting weak ergodicity breaking follow the Mittag-Leffler distribution. We demonstrate this universality across multiple models of disordered media displaying anomalous diffusion.

Figures (5)

• Figure 1: The scatter of TA observables across different models. The spread between trajectories indicates non-ergodic behavior for the models discussed: (a) TA-MSD in CTRW ($t=10^8$), (b) TA-displacement, $\overline{\delta(\Delta,t)} \equiv (t-\Delta)^{-1}\int_{0}^{t-\Delta} [x(t'+\Delta)-x(t')]\,dt'$, in transient QTM ($F = 0.1, \, t=10^{14}$), (c) TA-MSD in Comb model ($t=10^5$), (d) absolute TA displacement, $|\overline{\delta(\Delta,t)}|$, in Barrier model ($t=10^8$). For each model, 10 examples of the TA are given, color-coded. The thick black line corresponds to the numerical average over 1000 simulated trajectories and the red dashed line in (a) and (b) are the known analytical solution - $\langle \overline{\delta^2} \rangle \sim \Delta / [A t^{1-\alpha} \Gamma(1 + \alpha)]$ for the symmetrical CTRW in $1d$he2008randomlubelski2008nonergodicity and $\langle \overline{\delta} \rangle \sim \Delta / [A \Lambda t^{1-\alpha} \Gamma(1 + \alpha)]$ (see Eq. (S.11) in the SM SM) for the transient QTM in $2d$shafir2022case. All models exhibit the same trajectory-to-trajectory scattering of the TA observable, as captured by Eq. \ref{['eq:MittagLefflerDistribution']}.
• Figure 2: Conditional ergodicity. In each model—(a) CTRW, (b) QTM, and (c) comb model —two independent TA-MSD trajectories are plotted as functions of the physical time $t$. Both trajectories are conditioned to reach the same internal clock value $\mathcal{S}$ at the end of the measurement. In all cases, the two TAs overlap well before the final time, illustrating conditional ergodicity [Eq. \ref{['eq:conditional_ergodicity']}].
• Figure 3: The universal distribution across all models. The PDF $\phi_\alpha (\xi)$ [in Eq. \ref{['eq:MittagLefflerDistribution']}] capturing the scatter of the rescaled TA observable $\xi=\overline{\mathcal{O} }/\langle \overline{\mathcal{O}}\rangle$ across different models and under varying geometric constraints and external forces (see SM SM). Once $\alpha$ associated with the trapping times is fixed, all models collapse to the same Mittag-Leffler form without any fitting. Panel (a): $\alpha=0.285$; the scatter of $\overline{\mathcal{O}}=|\overline{\delta}|$ in the Barrier model, $\overline{\mathcal{O}} = \overline{\delta}$ in biased $1d$ CTRW and transient $2d$ QTM, and of $\overline{\mathcal{O}} = \overline{\delta^2}$ in the symmetrical $2d$ QTM . Panel (b): $\alpha=0.6$; the scatter of $\overline{\mathcal{O}} = \overline{\delta^2}$ in symmetrical $1d$ CTRW, $\overline{\mathcal{O}} = \overline{\delta}$ in biased QTM under geometrical constraints of a channel for two cases of width $W=10,20$ and of $\overline{\mathcal{O}} = \overline{\delta^2}$ in the Comb model.
• Figure 4: Simulation results of $\overline{\delta^2}$ for the Barrier model with $\alpha=0.5$, $T=10^8$ showing ergodic behavior. Upper plot: The ensemble average of the TA-MSD $\langle \overline{\delta^2} \rangle$ (solid black line) is computed over $1000$ trajectories. Ten individual realizations of $\overline{\delta^2}$ are also shown, color-coded. Lower plot: The corresponding EA-MSD. The fitting exponent $\beta$ in both plots agrees very well with the expected theoretical value bouchaud1990anomalousalexander1981excitation, $\beta \approx 2 \alpha_{\text{eff}} = 2 \alpha / (1+\alpha) = 2/3$.
• Figure 5: Simulation results of $\overline{\delta}$ for the Barrier model with $\alpha=0.8$, $T=10^8$ showing 20 independent examples color coded.