Information Geometry, One, Two, Three (and Four)

TL;DR

This paper explores how information geometry, specifically the scalar curvature of the Fisher-Rao metric, can characterize phase transitions across various statistical models and even in black hole solutions.

Contribution

It extends previous work by analyzing the scalar curvature in additional models like the 1D Potts, 2D Ising with quantum gravity, and 3D spherical models, highlighting the geometric approach to phase transitions.

Findings

Scalar curvature diverges at phase transition points.

Extended analysis to new models confirms geometric indicators of criticality.

Connections made between statistical models and black hole solutions.

Abstract

Although the notion of entropy lies at the core of statistical mechanics, it is not often used in statistical mechanical models to characterize phase transitions, a role more usually played by quantities such as various order parameters, specific heats or suscept ibilities. The relative entropy induces a metric, the so-called information or Fisher-Rao m etric, on the space of parameters and the geometrical invariants of this metric carry information about the phase structure of the model. In various models the scalar curvature, ${\cal R}$, of the information metric has been found to diverge at the phase transition point and a plausible scaling relation postulated. For spin models the necessity of calculating in non-zero field has limited analytic consideration to one-dimensional, mean-field and Bethe lattice Ising models. We report on previous papers in which we extended the list somewhat in the current note by considering the {\it one}-dime nsional Potts model, the {\it two}-dimensional Ising model coupled to two-dimensional quantum gravity and the {\it three}-dimensional spherical model. We note that similar ideas have been ap plied to elucidate possible critical behaviour in families of black hole solutions in {\it four} space-time dimensions.