Nonequilibrium Entropy

TL;DR

This paper introduces a general formulation of nonequilibrium entropy for isolated systems, demonstrating its consistency with thermodynamic entropy and exploring its behavior in specific models like ideal gases and lattice systems.

Contribution

It provides a first-principles-based statistical definition of nonequilibrium entropy that aligns with thermodynamic entropy and clarifies its relation to classical models.

Findings

Statistical entropy is less than equilibrium entropy during free expansion.

In lattice models, entropy relates to the Tonks gas entropy as lattice spacing approaches zero.

Standard Boltzmann H-theorem quantity does not directly represent the statistical entropy.

Abstract

We consider an isolated system in an arbitrary state and provide a general formulation using first principles for an additive and non-negative statistical quantity that is shown to reproduce the equilibrium thermodynamic entropy of the isolated system. We further show that the statistical quantity represents the nonequilibrium thermodynamic entropy when the latter is a state function of nonequilibrium state variables; see text. We consider an isolated 1-d ideal gas and determine its non-equilibrium statistical entropy as a function of the box size as the gas expands freely isoenergetically, and compare it with the equilibrium thermodynamic entropy S_{0eq}. We find that the statistical entropy is less than S_{0eq} in accordance with the second law, as expected. To understand how the statistical entropy is different from thermodynamic entropy of classical continuum models that is known to become negative under certain conditions, we calculate it for a 1-d lattice model and discover that it can be related to the thermodynamic entropy of the continuum 1-d Tonks gas by taking the lattice spacing {\delta} go to zero, but only if the latter is state-independent. We discuss the semi-classical approximation of our entropy and show that the standard quantity S_{f}(t) in the Boltzmann's H-theorem does not directly correspond to the statistical entropy.