Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions

TL;DR

This paper explores the statistical mechanics of self-gravitating fermions in D-dimensional space, revealing that for dimensions four and higher, quantum mechanics cannot prevent gravitational collapse, highlighting the special nature of our three-dimensional universe.

Contribution

It provides a detailed analysis of phase transitions and stability in self-gravitating fermion systems across different dimensions, emphasizing the unique stability of three-dimensional space.

Findings

For D≥4, a critical temperature and energy prevent equilibrium states.

Quantum mechanics cannot stabilize matter against collapse in D≥4.

The three-dimensional space is unique in allowing stable matter due to quantum effects.

Abstract

We discuss the statistical mechanics of a system of self-gravitating fermions in a space of dimension $D$. We plot the caloric curves of the self-gravitating Fermi gas giving the temperature as a function of energy and investigate the nature of phase transitions as a function of the dimension of space. We consider stable states (global entropy maxima) as well as metastable states (local entropy maxima). We show that for $D\ge 4$, there exists a critical temperature (for sufficiently large systems) and a critical energy below which the system cannot be found in statistical equilibrium. Therefore, for $D\ge 4$, quantum mechanics cannot stabilize matter against gravitational collapse. This is similar to a result found by Ehrenfest (1917) at the atomic level for Coulombian forces. This makes the dimension D=3 of our universe very particular with possible implications regarding the anthropic principle. Our study enters in a long tradition of scientific and philosophical papers who studied how the dimension of space affects the laws of physics.