Generalization of Gibbs Entropy and Thermodynamic Relation

TL;DR

The work develops a generalized non-equilibrium entropy framework that extends Gibbs–Einstein concepts to arbitrary non-equilibrium thermodynamic processes by employing a coarse-grained phase-space density $\tilde{\rho}(x,t)$ and a microscopic entropy involving a transition kernel $\phi_{\Delta t}$. It proves an entropy theorem showing that the macroscopic entropy $\bar{S}(t)$ increases for non-equilibrium states while reducing to the classical Gibbs entropy at perfect resolution, and it derives a generalized thermodynamic relation with multiple effective temperatures $T_n$ tied to energy moments $\langle E^n\rangle$. The paper then connects these results to a fluctuation theorem in the spirit of Evans, demonstrating that time-reversed trajectories have a well-defined probability weight ratio governed by an entropy increment $\delta s$, under macroscopic reversibility conditions. Overall, the framework links fast chaotic microdynamics, coarse-graining, and environmental coupling to irreversibility and fluctuations in non-equilibrium thermodynamics, offering a principled basis for generalized entropy, thermodynamics, and fluctuations.

Abstract

In this paper, we extend Gibbs's approach of quasi-equilibrium thermodynamic processes, and calculate the microscopic expression of entropy for general non-equilibrium thermodynamic processes. Also, we analyze the formal structure of thermodynamic relation in non-equilibrium thermodynamic processes.