Generalized Ehrenfest's Equations and phase transition in Black Holes

TL;DR

This paper extends Ehrenfest's equations to complex thermodynamic systems with multiple work terms, specifically applied to black holes, providing a generalized framework for analyzing second order phase transitions.

Contribution

It introduces a generalized set of Ehrenfest's equations for systems with multiple degrees of freedom, including black holes, and derives the number of equations needed at phase transition points.

Findings

Nine equations for black holes with two work terms

General formula for systems with N degrees of freedom: N(N+1)^2/2 equations

Framework applicable to analyzing second order phase transitions in complex systems

Abstract

We generalize Ehrenfest's equations to systems having two work terms, i.e. systems with three degrees of freedom. For black holes with two work terms we obtain nine equations instead of two to be satisfied at the critical point of a second order phase transition. We finally generalize this method to a system with an arbitrary number of degrees of freedom and found there is $\frac{N(N+1)^{2}}{2}$ equations to be satisfied at the point of a second order phase transition where $N$ is number of work terms in the first law of thermodynamics.