PHY.8:4.5 - Connect ensemble behavior to physical time

An ensemble mean at time t and a time average along one history answer different questions. To use the latter as an estimate of the former, examine the relevant dynamics, preparation and observation duration.

Derive or obtain a relaxation or correlation time for the selected observable. Compare it with the duration available and with any external drive. A stationary distribution can exist while equilibration is too slow for the experiment. An invariant portion of state space can preserve dependence on the initial preparation. A theorem about an infinite-time average supplies no finite settling time by itself.

For a stationary scalar process A(t) with covariance C(tau)=Cov(A(t),A(t+tau)), the variance of its average over duration T is

Var(A_bar_T) = (2/T²) integral from 0 to T of (T-tau) C(tau) d tau.

This relation assumes finite second moments and a well-defined time integral. Use it, or a suitable finite-sample counterpart, when the accuracy of time averaging matters. Long correlations reduce the gain from repeated measurements; drawing more points from the same slow fluctuation does not make them independent.

If an eliminated variable leaves memory in the retained evolution, keep that memory or restore a sufficient state through PHY.5 and MMP.9. Decide which description serves the time-dependent question. A stationary histogram alone does not establish the transition law, response time or heat dissipation.