Mathematical Formalism for Isothermal Linear Irreversibility

TL;DR

This paper establishes a rigorous mathematical framework linking symmetry, reversibility, and entropy production in linear stochastic systems, extending classical thermodynamics principles to complex molecular processes.

Contribution

It provides necessary and sufficient conditions for reversibility in linear stochastic differential equations and generalizes Einstein's fluctuation-dissipation relation.

Findings

Reversibility is characterized by symmetry and zero entropy production.

Existence of irreversible stationary processes is demonstrated.

A criterion to distinguish between stationary and sweeping behaviors is proposed.

Abstract

We prove the equivalence among symmetricity, time reversibility, and zero entropy production of the stationary solutions of linear stochastic differential equations. A sufficient and necessary reversibility condition expressed in terms of the coefficients of the equations is given. The existence of a linear stationary irreversible process is established. Concerning reversibility, we show that there is a contradistinction between any 1-dimensional stationary Gaussian process and stationary Gaussian process of dimension $n>1$. A concrete criterion for differentiating stationarity and sweeping behavior is also obtained. The mathematical result is a natural generalization of Einstein's fluctuation-dissipation relation, and provides a rigorous basis for the isothermal irreversibility in a linear regime which is the basis for applying Onsager's theory to macromolecules in aqueous solution.