Nonequilibrium Langevin dynamics: A demonstration study of shear flow fluctuations in a simple fluid

The present paper is based on a recent success of the second-order stochastic fluctuation theory in describing time autocorrelations of equilibrium and nonequilibrium physical systems. In particular, it was shown to yield values of the related deterministic parameters of the Langevin equation for a Couette flow in a microscopic molecular dynamics model of a simple fluid. In this paper we find all the remaining constants of the stochastic dynamics, which then is simulated numerically and compared directly with the original physical system. By using these data, we study in detail the accuracy and precision of a second-order Langevin model for nonequilibrium physical systems theoretically and computationally. We find an intriguing relation between an applied external force and cumulants of the resulting flow fluctuations. This is characterized by a linear dependence of an athermal cumulant ratio, an apposite quantity introduced here. In addition, we discuss how the order of a given Langevin dynamics can be raised systematically by introducing colored noise.

Figure 1

Plot of the athermal cumulant ratio $k\left(P_{yx}\right)$ as a function of the shear rate $\gamma$ for our MD simulations. The error bars are given by one standard deviation. Values of $k\left(P_{yx}\right)$, which by magnitude are less than their respective uncertainties, are disregarded and not plotted in the figure. The data from the linear shear regime ($•$) can be fit to a line $k\left(P_{yx}\right)=\mathrm{const}\gamma$. The points, which extend into the region of shear thinning ($\gamma >2.0\mathrm{r}.\mathrm{u}.$) and, therefore, are excluded from the fitting procedure, still conform to the observed linear trend.

Figure 2

Qualitative comparison of sample trajectories for $\alpha \left(t\right)=P_{yx}$ in our MD and LD simulations: (a) the MD simulation vs the LD model with simple ex and (b) the MD simulation vs the LD model with exponential sh.