[Paper Review] Generalized Maxwell equal area law and black holes in complex free energy

Zhen-Ming Xu, Yushan Wang+2 more|arXiv (Cornell University)|May 10, 2023

Black Holes and Theoretical Physics38 references4 citations

TL;DR

This paper introduces a generalized Maxwell equal area law that extends thermodynamic phase transition analysis to complex free energy landscapes, enabling the identification of black hole phase transitions through analytic continuation and winding number topology. It establishes that the local maximum winding number in the complex plane corresponds to second-order phase transitions (e.g., Hawking-Page), while higher winding numbers signal first-order transitions, with Riemann surface foliations providing a geometric counterpart to phase behavior in black holes.

ABSTRACT

Maxwell equal area law is an important and traditional analytical tool in thermodynamic phase transition, especially in the calculation of gas-liquid phase transition, which reconciles the theoretical calculation with the experimental results. Undoubtedly, its importance is also self-evident for the black hole thermodynamic system. In this study, we construct a generalized Maxwell equal area law, which allows different states of thermodynamic systems to be within the generalized free energy. The black hole thermodynamic characteristics are spontaneously emerged in the free energy landscape. Furthermore, by analytic continuation, we utilize the properties of analytical functions to investigate some universal characteristics of thermodynamic phase transitions in black holes, and preliminarily establish the counterpart of thermodynamic phase transitions in the complex domain.

Motivation & Objective

• To extend the classical Maxwell equal area law to a generalized framework applicable to thermodynamic systems with complex free energy.
• To investigate black hole thermodynamic phase transitions using complex analysis and analytic continuation of free energy functions.
• To identify universal topological invariants—specifically winding numbers—in the complex plane that characterize different types of black hole phase transitions.
• To establish a geometric and analytical correspondence between phase transition types and the structure of Riemann surfaces in the complex domain.
• To unify the description of diverse black hole systems (Reissner-Nordström, Schwarzschild-AdS, charged AdS, Gauss-Bonnet) under a single topological framework.

Proposed method

• Construct a generalized free energy function ψ(z) by analytic continuation of thermodynamic potentials into the complex plane, parameterized by the event horizon radius z.
• Define the local maximum winding number W_local-max as the maximum number of times the complex function ψ(z) winds around the origin in the complex plane, indicating the number of real positive zeros.
• Use the Argument Principle to compute the winding number W_global = Z − P, where Z is the number of zeros and P the number of poles of ψ(z) in C\{0}.
• Relate the winding number to phase transition order: W_local-max ≥ 2 implies second-order transition, W_local-max ≥ 3 implies first-order transition.
• Analyze specific black hole systems (Reissner-Nordström, Schwarzschild-AdS, charged AdS, Gauss-Bonnet) by deriving their respective ψ(z) functions and computing their winding numbers.
• Establish a geometric interpretation: a second-order transition corresponds to a Riemann surface with two foliations, while higher winding numbers correspond to multiple foliations.

Experimental results

Research questions

• RQ1 How can the classical Maxwell equal area law be generalized to describe thermodynamic phase transitions in complex free energy landscapes?
• RQ2 What topological invariant in the complex plane corresponds to the occurrence of first- and second-order phase transitions in black hole systems?
• RQ3 How do the winding numbers of the complex free energy function relate to the number and type of thermodynamic phases in black holes?
• RQ4 Can analytic continuation of thermodynamic potentials reveal universal patterns in black hole phase transitions across different black hole models?
• RQ5 What is the geometric interpretation of phase transitions in terms of Riemann surface foliations in the complex domain?

Key findings

• The local maximum winding number W_local-max = 2 corresponds to second-order phase transitions in Reissner-Nordström and Schwarzschild-AdS black holes, consistent with divergent heat capacity.
• For the charged AdS black hole, W_local-max = 3 indicates the presence of both first-order and second-order phase transitions, matching its van der Waals-type behavior.
• The charged Gauss-Bonnet black hole in six dimensions exhibits W_local-max = 5, decomposed as 3 + 2, indicating one first-order and two second-order phase transitions, consistent with a triple point.
• The global winding number W_global is consistently 1 across all analyzed systems, reflecting the topological stability of the system's analytic structure.
• The complex free energy function ψ(z) for each black hole model is derived explicitly, with zeros corresponding to critical points in the phase diagram.
• The geometric counterpart of phase transitions is identified as Riemann surfaces with foliations: two for second-order, three for first-order, and five for the triple-point system.