Nonequilibrium Maxwell-Demon NEMD formalism for transport: I. Extrapolating shear viscosity to the hydrodynamic limit

Abstract

The Maxwell-Demon nonequilibrium molecular dynamics method for measuring the shear viscosity divides the fluid simulation cell into two regions of width w in the $x$-direction, with the particles allowed to move freely between the two sides. The Demon maintains equal and opposite regional average particle velocities in the $y$-direction by applying an acceleration needed to sustain the nonequilibrium steady state (NESS), which includes both total force balance and an instantaneous correction for diffusion of particles across boundaries. We show that the driven velocity profile in x is a natural hydrodynamic response and not imposed by the constraint. For the present shear geometry, the measured NESS profile in Eulerian slabs is represented accurately by a piecewise parabolic form, reminiscent of planar Poiseuille flow. The parabolic profile gives an estimate of the kinematic viscosity from work done on the shearing fluid, as well as an entropy production estimate, derived from heat removal by the Nosé-Hoover thermostat that keeps each regional average temperature constant. We show that the mechanical work supplied by the Demon equals the heat removed by the thermostat. Once NESS driving is removed, the parabolic velocity profile relaxes exponentially in short order to sinusoidal, the natural transverse momentum-diffusion eigenmode. Our Maxwell-Demon shear method provides a direct NEMD formalism for obtaining shear viscosity from momentum diffusion, work, and entropy balances. Our results in three dimensions for increasing system size $N$ (the number of particles) demonstrate that shear viscosity approaches an asymptote (the so-called hydrodynamic limit) as $N^{-1/3}$.