Thermodynamic curvature measures interactions

TL;DR

This paper reviews the extension of thermodynamic fluctuation theory beyond Gaussian approximation, introducing a Riemannian curvature scalar that relates to particle interactions and correlation length.

Contribution

It introduces the thermodynamic Riemannian curvature scalar as a new invariant that captures interaction effects beyond Gaussian fluctuations.

Findings

|R| correlates with the correlation length

Sign of R indicates attractive or repulsive interactions

Generalizes fluctuation theory beyond Gaussian approximation

Abstract

Thermodynamic fluctuation theory originated with Einstein who inverted the relation $S=k_B\ln\Omega$ to express the number of states in terms of entropy: $\Omega= \exp(S/k_B)$. The theory's Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $|R|$ is related to the correlation length and suggest that the sign of $R$ corresponds to whether the interparticle interactions are effectively attractive or repulsive.