Derivation of the Onsager Principle from Large Deviation Theory

TL;DR

This paper derives the Onsager linear relations from large deviation theory by analyzing short-time fluctuations near equilibrium, establishing a connection between rate functions and thermodynamic entropy, and deriving the Green-Kubo formula.

Contribution

It introduces a novel derivation of Onsager relations using large deviation theory and short-time asymptotics, providing new insights into macroscopic fluctuation behavior.

Findings

Derived Onsager relations from large deviation principles.

Established a Green-Kubo formula for the Onsager matrix.

Illustrated the theory with an ideal Knundsen gas example.

Abstract

The Onsager linear relations between macroscopic flows and thermodynamics forces are derived from the point of view of large deviation theory. For a given set of macroscopic variables, we consider the short-time evolution of near-equilibrium fluctuations, represented as the limit of finite-size conditional expectations. The resulting asymptotic conditional expectation is taken to represent the typical macrostate of the system and is used in place of the usual time-averaged macrostate of traditional approaches. By expanding in the short-time, near-equilibrium limit and equating the large deviation rate function with the thermodynamic entropy, a linear relation is obtained between the time rate of change of the macrostate and the conjugate initial macrostate. A Green-Kubo formula for the Onsager matrix is derived and shown to be positive semi-definite, while the Onsager reciprocity relations readily follow from time reversal invariance. Although the initial tendency of a macroscopic variable is to evolve towards equilibrium, we find that this evolution need not be monotonic. The example of an ideal Knundsen gas is considered as an illustration.