Genuine and spurious (non-)ergodicity in single particle tracking

Abstract

In single-particle tracking experiments measuring anomalous diffusion dynamics, understanding ergodicity is crucial, as it ensures that the time average of an observable matches the ensemble average, and can thus be fitted with known ensemble-averaged observables. A commonly used criterion for assessing the ergodicity of a stochastic process is based on the comparison of the mean-squared displacement (MSD) with the time-averaged MSD (TAMSD). This approach has been widely applied and proves effective in cases of weak ergodicity breaking across various systems in both theoretical and experimental studies. However, there is relatively little discussion regarding the theoretical justification and limitations of this definition. Here, we demonstrate that this widely accepted criterion to some extent contradicts the classical definition of ergodicity as well as physical intuition, leading to spurious (non-)ergodicity results when applied to several well-known stochastic models. To address this limitation, we propose using the mean-squared increment (MSI) instead of the MSD for comparison of ensemble- and time-averaged observables. Several well-established examples demonstrate that our MSI-TAMSD criterion not only effectively reveals weak ergodicity breaking, equivalent to the MSD-TAMSD approach, but also provides a more accurate characterization of the genuine (non-)ergodicity of systems where the MSD-TAMSD method fails. Additionally, for systems exhibiting "ultraweak" ergodicity breaking, the MSI can reveal the asymptotic stationarity and ergodic nature of the process' increments. Our findings emphasize the important role of the MSI observable for SPT experiments and anomalous diffusion studies.

Figures (8)

• Figure 1: Schematic for the ergodicity of an observable $\mathcal{O}$ of a stochastic process $x(t)$ with stationary PDF $P_{\mathrm{st}}(x)$. The ensemble average $\left<\mathcal{O}\right>=\int_{-\infty}^\infty\mathcal{O} (x)P_{\mathrm{st}}(x)dx$ and the time average $\overline{\mathcal{O}}=\frac{1 }{T}\int_0^T{\mathcal{O}(x(t'))}dt'$ of one single trajectory are highlighted by magenta and green colors respectively. The process $x(t)$ is ergodic with respect to $\mathcal{O}$ if $\lim_{T\to\infty}\overline{\mathcal{O}}=\left< \mathcal{O}\right>$.
• Figure 2: Simulation results for Fractional OUP (\ref{['eq-foup']}) with initial condition $x_0=0$: MSD (black circles), MSI (green rectangles) and TAMSDs of 50 individual trajectories (magenta curves) for (a) normal diffusion with $H=0.5$, (b) subdiffusion with $H=0.2$ and (c) and superdiffusion with $H=0.8$. When $H=0.5$, the fractional OUP reduces to the normal OUP (\ref{['eq-ou']}) with initial condition $x_0=0$. The EA-TAMSD is represented by solid blue curves. The variation (scatter) of the amplitudes between different TAMSDs of individual trajectories is negligible, indicating that individual trajectories are completely reproducible. Parameters: time step $dt=0.01$, measurement time $T=500$, starting time for the MSI $t=10$, initial condition $x_0=0$, noise strength $\sigma_{2H}=1$ and relaxation rate $\lambda=1$.
• Figure 3: Simulation results for FBM under Poissonian resetting with the rate $r$ to the initial position $x(0)=0$, showing MSD (black circles), MSI (green rectangles) and TAMSDs of individual trajectories (magenta curves) with (a) $H=0.5$, (b) $H=0.2$ and (c) $H=0.8$. The EA-TAMSD is represented by solid blue curves. Other parameters: time step $dt=0.01$, measurement time $T=500$, starting time of the MSI $t=10$, and resetting rate $r=1$.
• Figure 4: Simulation results for RL-FBM (\ref{['eq-rlfbm']}) showing MSD (black circles), MSI (green rectangles) and TAMSDs of individual trajectories (magenta curves) with $\alpha=1.4$. The EA-TAMSD is represented by solid blue curves. When $1/2< \alpha<3/2$, the EA-TAMSD of RL-FBM (\ref{['eq-rlfbm']}) is identical to the MSI rather than the MSD, leading to a spurious nonergodicity. Other parameters: time step $dt=0.01$, measurement time $T=500$, starting time of MSI $t=10$. The algorithm of numerical simulations is presented in App. C of wei2025.
• Figure 5: Simulation results for Lévy walks showing the MSD (black circles), MSI (green rectangles), and TAMSDs of 50 individual trajectories (magenta curves) for (a) exponential waiting time PDF (\ref{['expwtd']}) with $\lambda=1$ and a power-law waiting time PDF (\ref{['plwtd']}) with (b) $\beta=0.7$, and (c) $\beta=1.2$. The EA-TAMSD is represented by solid blue curves. The variation (scatter) of the amplitudes between different TAMSDs of individual trajectories is negligible, indicating that individual trajectories are completely reproducible. Other parameters: time step $d t=0.01$, measurement time $T= 500$, starting time of MSI $t=100$, and $t_0=0.01$. It is important to note that discrepancies in the MSD arise as the power-law density cannot be accurately generated in the short waiting time regime using the inverse transform sampling method.