Canonical Quantum Statistics of Schwarzschild Black Holes and Ising Droplet Nucleation

TL;DR

This paper links the quantum thermodynamics of Schwarzschild black holes to droplet nucleation theory and the Ising model, providing new insights into black hole entropy and temperature through exact partition function analysis.

Contribution

It interprets the imaginary part of the black hole partition function within droplet nucleation theory and relates it to the Ising droplet model for phase transitions.

Findings

Partition function matches that of the 2D Ising droplet model.

Exact complex free energy for negative magnetic fields is derived.

Black hole thermodynamics can be studied via the Ising model analogy.

Abstract

Recently is was shown that the imaginary part of the canonical partition function of Schwarzschild black holes with an energy spectrum E_n = \sigma \sqrt{n} E_P, n= 1,2, ..., has properties which - naively interpreted - leads to the expected unusual thermodynamical properties of such black holes (Hawking temperature, Bekenstein-Hawking entropy etc). The present paper interprets the same imaginary part in the framework of droplet nucleation theory in which the rate of transition from a metastable state to a stable one is proportional to the imaginary part of the canonical partition function. The conclusions concerning the emerging thermodynamics of black holes are essentially the same as before. The partition function for black holes with the above spectrum was calculated exactly recently. It is the same as that of the primitive Ising droplet model for nucleation in 1st-order phase transitions in 2 dimensions. Thus one might learn about the quantum statistics of black holes by studying that Ising model, the exact complex free energy of which is presented here for negative magnetic fields, too.