# Statistical Physics of the Symmetric Group

## TL;DR

This paper explores the statistical physics of systems modeled by permutations of ordered lists, revealing complex phase behavior with multiple regimes and transitions, and discusses extensions to more complex combinatorial state spaces.

## Contribution

It introduces a novel analysis of the symmetric group as a state space in statistical physics, uncovering unique phase transitions and regimes.

## Key findings

- Identification of five physical regimes with two transition temperatures
- Discovery of a triple point and a quadruple point in the phase diagram
- Extension framework for more complex combinatorial state spaces

## Abstract

Ordered chains (such as chains of amino acids) are ubiquitous in biological cells, and these chains perform specific functions contingent on the sequence of their components. Using the existence and general properties of such sequences as a theoretical motivation, we study the statistical physics of systems whose state space is defined by the possible permutations of an ordered list, i.e., the symmetric group, and whose energy is a function of how certain permutations deviate from some chosen correct ordering. Such a non-factorizable state space is quite different from the state spaces typically considered in statistical physics systems and consequently has novel behavior in systems with interacting and even non-interacting Hamiltonians. Various parameter choices of a mean-field model reveal the system to contain five different physical regimes defined by two transition temperatures, a triple point, and a quadruple point. Finally, we conclude by discussing how the general analysis can be extended to state spaces with more complex combinatorial properties and to other standard questions of statistical mechanics models.