Nonequilibrium free energy, coarse-graining, and the Liouville equation

The Helmholtz free energy is computed for an ensemble of initial conditions for a one-dimensional particle falling down a staircase potential, while in contact with a thermal reservoir. Initial conditions are chosen from the equilibrium canonical ensemble, with the gravitational field applied either as a step function (steady field) or a δ function (pulsed perturbation). The first case leads to a fractal steady-state distribution, while the second case leads to relaxation of a perturbed distribution back toward equilibrium. Coarse-graining is applied to the computation of the non- equilibrium entropy, with finer resolution in phase space accompanied by an increase in the number of trajectories. The limiting fine-grained (continuum) prediction of the Liouville equation is shown to be consistent with the numerical simulations for the steady state, but with incredibly slow (logarithmic) divergence appropriate to a lower-dimensional fractal distribution. On the other hand, simulations of the relaxation process show little or no sign of converging to the prediction obtained from the Liouville equation. Irreversible phase-space mixing of trajectories appears to be a necessary modification to the Liouville equation, if one wants to make predictions of numerical simulations in nonequilibrium statistical mechanics.