Limits of the equivalence of time and ensemble averages in shear flows

TL;DR

This paper investigates whether time and ensemble averages of bubble velocities in a sheared bubble raft are equivalent, revealing that in driven non-equilibrium systems they can converge to different distributions despite stationarity.

Contribution

It demonstrates experimentally that in non-equilibrium shear flows, time and ensemble averages can produce distinct but stationary velocity distributions.

Findings

Time averages in single experiments converge to stable velocity profiles.

Ensemble averages from multiple experiments differ from time averages.

Increasing ensemble size improves approximation of true ensemble averages.

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 non-equilibrium systems, temporal averaging and ensemble averaging can yield convergent (stationary) but distinct distributions.