Sensitivity of nonequilibrium relaxation to interaction potentials: Timescales of response from Boltzmann's H function

We investigate, by simulations and analytic theory, the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using Boltzmann's $H$ function $H(t)$. We evaluate $H(t)$ for three different intermolecular potentials in all three dimensions and find that the well-known $H$ theorem is valid and that the $H$ function exhibits rather strong sensitivity to all these factors. The relaxation of $H(t)$ is long in one dimension, but short in three dimensions, longer for the Lennard-Jones potential than for the hard spheres. The origin of the ultraslow approach to the equilibrium of $H(t)$ in one-dimensional systems is discussed. Importantly, we obtain a closed-form analytic expression for $H(t)$ using the solution of the Fokker-Planck equation for velocity space probability distribution and compare its predictions with the simulation results. Interestingly, $H(t)$ is found to exhibit a linear response when vastly different initial nonequilibrium conditions are employed. The microscopic origin of this linear response is discussed. The oft-quoted relation of $H$ function with Clausius's entropy theorem is discussed.