Mathematical Representation of Clausius' and Kelvin's Statements of the Second Law and Irreversibility

TL;DR

This paper introduces a stochastic mathematical framework for Clausius' and Kelvin's Second Law statements, linking entropy production with potential energy decrease in lifted Markov systems, providing a dynamic foundation for thermodynamics.

Contribution

It offers a rigorous mathematical representation of the Second Law using Markov processes and establishes a connection between entropy production and potential energy in lifted systems.

Findings

Entropy production equals potential energy decrease in the long-time limit.

Lifted processes have detailed balance and a natural potential function.

Provides a dynamic, modernized statement of the Second Law.

Abstract

We provide a stochastic mathematical representation for Clausius' and Kelvin-Planck's statements of the Second Law of Thermodynamics in terms of the entropy productions of a finite, compact driven Markov system and its lift. A surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n-dimensional torus T^n. The corresponding lifted processes have detailed balance thus a natural potential function but no stationary probability. We show that in the long-time limit the entropy production of the finite driven system precisely equals the potential energy decrease in the lifted system. This theorem provides a dynamic foundation for the two equivalent statements of Second Law of Thermodynamics, a la Kelvin's and Clausius'. It suggests a modernized, combined statement: "A mesoscopic engine that works in completing irreversible internal cycles statistically has necessarily an external effect that lowering a weight accompanied by passing heat from a warmer to a colder body."