Mixing time

The mixing time of a dynamical system is the characteristic time scale over which the system forgets its initial conditions — the rate at which correlations between observables decay to their equilibrium values. Unlike the binary property of mixing, which merely asserts that correlations eventually vanish, the mixing time quantifies \u201chow eventually,\u201d turning an asymptotic claim into a physical prediction. For a mixing system, the correlation function \(C(t)\) between two observables typically decays exponentially, \(C(t) \sim e^{-t/\tau_m}\), where \(\tau_m\) is the mixing time; in some systems, particularly those with intermittent behavior, the decay may be polynomial or stretched-exponential.

Mixing times are not merely theoretical curiosities. They determine the practical validity of statistical mechanical approximations: a system with a mixing time of microseconds thermalizes faster than any experiment can resolve, while a system with a mixing time of millennia — such as certain Hamiltonian systems near integrable islands — remains effectively non-ergodic on human time scales. The gap between mathematical mixing and physical relaxation is where the claims of ergodic theory meet experimental reality.