Statistical mechanics far from equilibrium: Prediction and test for a sheared system

We report the application of a far-from-equilibrium statistical-mechanical theory to a nontrivial system with Newtonian interactions in continuous boundary-driven flow. By numerically time stepping the force-balance equations of a one-dimensional model fluid we measure occupancies and transition rates in simulation. The high-shear-rate simulation data reproduce the predicted invariant quantities, thus supporting the theory that a class of nonequilibrium steady states of matter, namely, sheared complex fluids, is amenable to statistical treatment from first principles.

Figure 1

(Color online) (a) The one-dimensional rotor model of length $L$. Each of the $L$ rotors is characterized by its angle $\theta$ and angular velocity $\dot{\theta }$. (b) The potential of interaction between neighbors, $U\left(\Delta \theta \right)=-\text{cos}\Delta \theta -\text{cos}4\Delta \theta$, a symmetric periodic function of their angular difference $\Delta \theta$, with four wells, labeled $a,b,c,d$.

Figure 2

(Color online) (a) Test of the prediction ${\omega }_{da}+{\omega }_{dc}={\omega }_{ba}+{\omega }_{bc}\forall \dot{\gamma }$ with noise strength $\sigma =10$. The left-hand ordinate measures rates, using the same units as the abscissa, while the right-hand ordinate measures the dimensionless ratio. (b) Test of the predicted relationship ${\omega }_{ab}{\omega }_{ba}={\omega }_{ad}{\omega }_{da}\forall \dot{\gamma }$, with $\sigma =20$. At this higher noise strength, higher shear rates are numerically accessible because of the greater number of observed backward transitions. (c) Test of the prediction ${\omega }_{cd}{\omega }_{dc}={\omega }_{cb}{\omega }_{bc}\forall \dot{\gamma }$, for the same parameters as in (a). (d) Contours of the measured ratio $\left({\omega }_{ba}+{\omega }_{bc}\right)/\left({\omega }_{da}+{\omega }_{dc}\right)$, predicted to be unity across the whole parameter space of shear rate $\dot{\gamma }$ and noise strength $\sigma$. Data acquisition time limited simulations in the bottom left-hand corner. See supplementary material for more data [24].