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

Yuhong Wang·Kapilanjan Krishan·Michael Dennin
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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.

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Yuhong Wang, Kapilanjan Krishan, Michael Dennin. 2007-06-11. Limits of the equivalence of time and ensemble averages in shear flows. https://doi.org/10.1103/physrevlett.98.220602

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