Limits of the Equivalence of Time and Ensemble Averages in Shear Flows

Abstract

In equilibrium systems, time and ensemble averages of physical quantities are equivalent due to ergodic exploration of phase space. In driven systems, it is unknown if a similar equivalence of time and ensemble averages exists. We explore effective limits of such convergence in a sheared bubble raft using averages of the bubble velocities. In independent experiments, averaging over time leads to well-converged velocity profiles. However, the time averages from independent experiments result in distinct velocity averages. Ensemble averages are approximated by randomly selecting bubble velocities from independent experiments. Increasingly better approximations of ensemble averages converge toward a unique velocity profile. Therefore, the experiments establish that in practical realizations of nonequilibrium systems, temporal averaging and ensemble averaging can yield convergent (stationary) but distinct distributions.