Far-from-equilibrium distribution from near-steady-state work fluctuations

A long-standing goal of nonequilibrium statistical mechanics has been to extend the conceptual power of the Boltzmann distribution to driven systems. We report some new progress towards this goal. Instead of writing the nonequilibrium steady-state distribution in terms of perturbations around thermal equilibrium, we start from the linearized driven dynamics of observables about their stable fixed point, and expand in the strength of the nonlinearities encountered during typical fluctuations away from the fixed point. The first terms in this expansion retain the simplicity of known expansions about equilibrium, but can correctly describe the statistics of a certain class of systems even under strong driving. We illustrate this approach by comparison with a numerical simulation of a sheared Brownian colloid, where we find that the first two terms in our expansion are sufficient to account for the shear thinning behavior at high shear rates.

Figure 1

Shear cell with periodic boundary conditions along flow direction. The reflecting walls on the top and bottom are separated by a distance $d$. The top wall moves at constant speed $v$ in the $x$ direction, while the bottom wall is fixed, causing a linear gradient $\dot{\gamma }=v/d$ in the solvent flow velocity along the $y$ direction.

Figure 2

(a) ${\sigma }_{xy}^I\left(t\right)$ averaged over all trajectory segments of length $\mathrm{\Delta }t=0.5$ that end at a specified value, from a simulation run with the same parameters as Fig. 4, and Pe = 19. Two arbitrarily chosen ending state restrictions are indicated by dotted lines, with the corresponding average trajectories shown in the same colors. (b) Measured probability distributions of work done over all the trajectories that go into the averages of panel (a), shaded in the same colors. (c) A portion of the raw ${\sigma }_{xy}^I$ time series from which the other two panels were generated. The ending state restrictions of panel (a) are indicated with dotted lines of the same color, and two typical trajectory segments ending at these values are shaded in these colors. The shaded areas are proportional to the interaction-dependent part of the work, according to Eq. (51).

Figure 3

Using the method illustrated in Fig. 2, we compute ${\langle W\rangle }_{\mathrm{\Delta }t}\left({\sigma }_{xy}^I\right)$ and $(\beta /2){\langle W^2\rangle }_{\mathrm{\Delta }t}^c\left({\sigma }_{xy}^I\right)$ for a range of values of ${\sigma }_{xy}^I$, and plot this data for Pe values 4, 10, and 19, increasing from bottom to top. Also plotted is the free energy $F$ extracted from the Pe = 0 simulation run. Traces have been vertically shifted for easier comparison of slopes.

Figure 4

Top: Location ${\sigma }_{xy}^{I*}$ of peak of probability distribution, computed via Eq. (46), compared with directly measured most frequent values in the simulation [28]. Also plotted is the prediction using the mean work alone via Eq. (23), without the $O\left(\tilde{\mu }\right)$ variance term, as well as the linear response prediction ${\sigma }_{xy}^{I*,\mathrm{LR}}$. Bottom: Relative size of departure from linear-response prediction.

Figure 5

Full probability distribution from Eq. (46), compared with shifted equilibrium distribution and with distribution directly sampled from simulation at Pe = 4, Pe = 10, and Pe = 19.