Thermodynamics with long-range interactions: from Ising models to black-holes

TL;DR

This paper introduces new methods to analyze the thermodynamics of long-range interacting systems, revealing non-extensive entropy, temperature behaviors, and black-hole analogs, with implications for gravitational systems and entropy scaling.

Contribution

It develops a framework for understanding thermodynamics in long-range systems, including entropy non-extensivity, temperature distributions, and black-hole analogs, connecting statistical mechanics with general relativity.

Findings

Long-range systems exhibit non-extensive entropy.

A black-hole analog with similar temperature and entropy properties is proposed.

Entropy scaling transitions from volume to area as interaction strength increases.

Abstract

New methods are presented which enables one to analyze the thermodynamics of systems with long-range interactions. Generically, such systems have entropies which are non-extensive, (do not scale with the size of the system). We show how to calculate the degree of non-extensivity for such a system. We find that a system interacting with a heat reservoir is in a probability distribution of canonical ensembles. The system still possesses a parameter akin to a global temperature, which is constant throughout the substance. There is also a useful quantity which acts like a {\it local temperatures} and it varies throughout the substance. These quantities are closely related to counterparts found in general relativity. A lattice model with long-range spin-spin coupling is studied. This is compared with systems such as those encountered in general relativity, and gravitating systems with Newtonian-type interactions. A long-range lattice model is presented which can be seen as a black-hole analog. One finds that the analog's temperature and entropy have many properties which are found in black-holes. Finally, the entropy scaling behavior of a gravitating perfect fluid of constant density is calculated. For weak interactions, the entropy scales like the volume of the system. As the interactions become stronger, the entropy becomes higher near the surface of the system, and becomes more area-scaling.