Foundations of a Finite Non-Equilibrium Statistical Thermodynamics: Extrinsic Quantities

TL;DR

This paper develops a framework for finite non-equilibrium statistical thermodynamics, addressing foundational questions, redefining extrinsic variables, and resolving classical paradoxes while connecting to equilibrium limits.

Contribution

It introduces definitions for extrinsic variables in non-equilibrium thermodynamics consistent with equilibrium results and modifies the Gibbs entropy to incorporate entropy creation.

Findings

Extrinsic variables are expressed as functions of time and space.

Resolutions are provided for classical paradoxes like Gibbs' and Maxwell's demon.

The framework aligns with expected equilibrium values in classical contexts.

Abstract

Statistical thermodynamics is valuable as a conceptual structure that shapes our thinking about equilibrium thermodynamic states. A cloud of unresolved questions surrounding the foundations of the theory could lead an impartial observer to conclude that statistical thermodynamics is in a state of crisis though. Indeed, the discussion about the microscopic origins of irreversibility has continued in the scientific community for more than a hundred years. This paper considers these questions while beginning to develop a statistical thermodynamics for finite non-equilibrium systems. Definitions are proposed for all of the extrinsic variables of the fundamental thermodynamic relation that are consistent with existing results in the equilibrium thermodynamic limit. The probability density function on the phase space is interpreted as a subjective uncertainty about the microstate, and the Gibbs entropy formula is modified to allow for entropy creation without introducing additional physics or modifying the phase space dynamics. Resolutions are proposed to the mixing paradox, Gibbs' paradox, Loschmidt's paradox, and Maxwell's demon thought experiment. Finally, the extrinsic variables of the fundamental thermodynamic relation are evaluated as functions of time and space for a diffusing ideal gas, and the initial and final values are shown to coincide with the expected equilibrium values when interpreted in a classical context.