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

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.

Figure 1

The probability distribution of velocities is indicated for ensemble averaging (open circles) and time averaging (closed squares). The solid lines are Lorenztian fits to the data, red curves for time-averaged data and blue for ensemble averaged data. (a) Results for a position at the center of the trough. For the ensemble (time) average the center of the Lorentzian fits are at $0.01\pm 0.01$ ($-0.01\pm 0.01$) and the widths are $0.68\pm 0.02$ ($0.72\pm 0.02$), respectively. (b) For a position $0.25d$ from the center. For the ensemble (time) average the center of the Lorentzian fits are at $0.41\pm 0.01$ ($0.36\pm 0.01$) and the widths are $0.47\pm 0.03$ ($0.52\pm 0.02$), respectively.

Figure 2

(a) Deviation of time-average velocity profiles (normalized by the wall speed $v_w$) from the mean profile ($v_m$) generated by ten members of the time set. (b) Deviation of ten different ensemble averages of the velocities from the mean profile. Plotting the deviation highlights the differences between the different time-average profiles. The average standard deviation for time-average profiles in (a) is 0.081; in contrast, the ensemble averages shown in (b) are relatively indistinguishable with a standard deviation of 0.017. The $y$ position is scaled by the bubble diameter $D$.

Figure 3

This is the decay in the correlation of the mean velocity $0.25d$ away from the trough center. The total strain in each experiment is 5, and we only show the autocorrelation to a unit strain to make it visually easier to read. The remainder of the curve is qualitatively unchanged at higher strains. The inset shows a close-up at short strain along with an exponential fit (correlation $\propto \exp (-\gamma /0.01)$.

Figure 4

The mean values of the velocity $0.25d$ from the center as a function of the number of bubbles used to compute the average for (a) time averaging and (b) ensemble averaging. For illustration purposes, 10 of the 30 runs are shown. The deviation in the final values (indicated by the arrows and $\delta$) is larger for time averages than ensemble averages. When we use the complete set of 30 runs, we obtain $\delta =0.258\mathrm{mm}/\mathrm{s}$ for time averages and $\delta =0.052\mathrm{mm}/\mathrm{s}$ for ensemble averages.

Figure 5

Plot of the average deviation $⟨\delta (M)⟩$ between different ensemble averages as a function of $M$, the number of runs used to generate the ensemble set. As the number of independent runs used to generate the ensemble increases, the deviation between computed averages decreases.