The Boltzmann Entropy for Dense Fluids Not in Local Equilibrium

TL;DR

This study uses computer simulations to analyze the evolution of Boltzmann entropy in dense fluids not in local equilibrium, showing monotonic increase over time and discussing implications for isolated Hamiltonian systems.

Contribution

It demonstrates that Boltzmann entropy increases monotonically in dense fluids out of local equilibrium and discusses its general behavior in isolated Hamiltonian systems.

Findings

Boltzmann entropy is monotone increasing in simulations.

Kinetic part of entropy can decrease while total increases.

Monotonicity likely holds for typical initial microstates in isolated systems.

Abstract

We investigate, via computer simulations, the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables $M$ describing the system are the (empirical) particle density $f=\{f(\un{x},\un{v})\}$ and the total energy $E$. We find that $S(f_t,E)$ is monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of $S(M_t) = S(M_{X_t})$ should hold generally for ``typical'' (the overwhelming majority of) initial microstates (phase-points) $X_0$ belonging to the initial macrostate $M_0$, satisfying $M_{X_0} = M_0$. This is a direct consequence of Liouville's theorem when $M_t$ evolves autonomously.