In weakly ergodic-breaking systems, time averages don’t converge to ensemble averages. Track a single particle for a long time and you’ll get a different diffusion coefficient than the one you’d compute from many particles at one time. The single-trajectory measurement fluctuates, and no amount of additional observation makes it settle.

But condition on an internal clock — the number of actual displacement events rather than wall-clock time — and ergodicity is restored. The time-averaged transport coefficient, rescaled by the internal clock, converges to a universal Mittag-Leffler distribution. The scatter in single-trajectory measurements isn’t randomness. It’s a deterministic map from the internal clock to the observable.

The trick is recognizing that the system has two timescales: wall-clock time (which the experimentalist controls) and operational time (which the system controls via its own trapping dynamics). Weak ergodicity breaking occurs because the experimentalist measures in the wrong time. Switch to the system’s own clock and the fluctuations become universal — the same distribution appears in continuous-time random walks, quenched trap models, and Lévy walks.

This is a coordinate choice, not a mechanism. The physics hasn’t changed. The diffusion is still anomalous, the trajectory still wanders. But the apparent non-self-averaging — the thing that makes single-molecule measurements unreliable — was an artifact of measuring in the wrong units. The system was always ergodic. We were just using the wrong clock.