The standard test for ergodicity in single-particle tracking compares two numbers: the mean-squared displacement (MSD, averaged over many particles) and the time-averaged MSD (TAMSD, averaged over one particle's trajectory). If they converge, the system is ergodic. If they don't, it isn't.
Wang, Wei, Sokolov, Metzler, and Chechkin (arXiv:2603.22989) show that this test gives wrong answers in established stochastic models. Systems that are ergodic by the classical definition — where time averages converge to ensemble averages — can fail the MSD-TAMSD comparison. Conversely, systems with genuine ergodicity breaking can pass it.
The problem is the observable. MSD measures cumulative displacement from the origin, which grows without bound in diffusive systems. It's non-stationary by construction. Comparing the time average of a non-stationary quantity to its ensemble average asks a question that doesn't have a clean answer — the comparison conflates the growth of the observable with the ergodic properties of the underlying process.
The fix is to use mean-squared increments (MSI) instead — the displacement over a fixed lag, not from the origin. Increments are stationary for the processes where MSD isn't. The MSI diagnostic correctly identifies weak ergodicity breaking and reveals when increments are themselves ergodic even though the full trajectory isn't.
What's structural isn't the fix. It's the error. For decades, the MSD-TAMSD comparison has been the default tool for assessing ergodicity in experiments on anomalous diffusion — in cell biology, polymer physics, granular media. The results generated by this tool shaped conclusions about whether these systems are ergodic. Some of those conclusions are wrong. Not because the data were wrong, or because the models were wrong, but because the diagnostic itself created the apparent pathology. The tool measured its own inadequacy and attributed it to the system.