Thermodynamic curvature measures interactions

Abstract

Thermodynamic fluctuation theory originated with Einstein who inverted the relation S = k B ln Ω to express the number of states in terms of entropy: Ω = 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.

Thermodynamic R-diagrams reveal solid-like fluid states

“…Starting in the narrow slab region, and lowering T at constant p into the supercooled liquid state, leads to negative and decreasing R. This would be expected if there were a second critical point, since all of the known fluid critical point models tabulated in [10] have R decreasing to negative infinity at the critical point. The LLCP is expected to be in the same universality class as the liquid-vapor critical point [32].…”

“…In this paper, we key on the sign of R, which appears to be connected with the character of interparticle interactions: R > 0 if repulsive interactions dominate, as in solids, and R < 0 if attractive interactions dominate, as near the critical point [10]. The first indication of a repulsive/attractive sign interpretation for R came from the quantum Fermi/Bose gasses, with uniformly positive/negative R values (in the R sign convention used here 1 ) [13,14].…”

“…Our thesis here and elsewhere is that attractive interactions correspond locally to the 2-sphere, and repulsive interactions correspond locally to the pseudo 2-sphere. R has been physically interpreted with an extension of Einstein's thermodynamic fluctuation theory (see [10] for a recent review). This extended theory includes covariance, conservation, and consistency, and clearly marks |R| as a thermodynamic measure of the mesoscopic size scale where the character of fluctuations changes qualitatively due to fluctuating structures resulting from microscopic interactions.…”

“…2 The calculation of R R follows from the Helmholtz free energy per volume f = f (T, ρ), with T the temperature, and ρ the number of particles per volume [10]. Let s = −f ,T be the entropy per volume, and µ = f ,ρ the chemical potential.…”

Gaussian and non-Gaussian fluctuations in pure classical fluids

“…4 Ruppeiner and others have indicated that there are significant problems with (energy) conservation and non-covariant behavior of entropy expansions beyond the Gaussian approximation. 4,11,12 Nevertheless, there is interest in the properties of the resulting non-Gaussian distributions. 4,[12][13][14] The third and higher moments of these distributions are finite and have been developed by Greene and Callen.…”