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Genuine and Spurious (Non-)Ergodicity in Single Particle Tracking

Wei Wang1,*, Qing Wei2,3,*, Igor M. Sokolov4,†, Ralf Metzler1,5,‡, and Aleksei Chechkin5,6,7,8,§

  • *These authors contributed equally to this work.
  • †Contact author: igor.sokolov@physik.hu-berlin.de
  • ‡Contact author: rmetzler@uni-potsdam.de
  • §Contact author: achechkin@mpi-halle.mpg.de
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Phys. Rev. X 16, 031071 – Published 17 September, 2026

DOI: https://doi.org/10.1103/m3jj-6sqz

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. For processes with stationary increments, the results for MSI and MSD coincide. However, if the increments of the process reach stationarity at long times only, they differ, and the MSI-TAMSD comparison still provides the test for ergodicity. Several well-established examples demonstrate, through both trajectory simulations and theoretical analysis, that our MSI-TAMSD criterion 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’s increments. Our findings emphasize the important role of the MSI observable for SPT experiments and anomalous diffusion studies.

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Popular Summary

Ergodicity is a fundamental concept of statistical physics: The long-time average of a physical observable converges to its ensemble average. This notion is essential for interpreting time series from single-particle tracking in bio- and soft-matter systems as well as financial market data. Typically, ergodicity is assessed by comparing the ensemble-averaged mean-squared displacement (MSD) with the time-averaged MSD (TAMSD) along individual trajectories. However, the (non-)equivalence of these two quantities can be ambiguous and may not reliably distinguish different stochastic dynamics. Here, we introduce the mean-squared increment (MSI) as a new statistic and compare it with the TAMSD. This comparison provides an operational test of ergodicity for the squared-increment observable. More broadly, the joint behavior of the MSI and TAMSD provides an additional tool, which is useful in discriminating different classes of stochastic processes.

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Phys. Rev. X 16, 031071– Published 17 September, 2026
  • Received 12 December 2025
  • Revised 24 June 2026
  • Accepted 17 August 2026
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