Networks of Coadjoint Orbits: from Geometric to Statistical Mechanics

TL;DR

This paper develops a class of network models with symmetry groups evolving as Lie-Poisson systems, connecting geometric mechanics with statistical mechanics, revealing new phenomena like phase transitions and synchronization.

Contribution

It introduces a novel framework for network models based on geometric mechanics, analyzing equilibrium stability, symmetry breaking, and phase transitions, including a new heavy top-like system.

Findings

Reduction to classical Heisenberg model for $SO(3)$

Discovery of a new symmetry-broken system similar to heavy top

Observation of a triple-humped phase transition

Abstract

A class of network models with symmetry group $G$ that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising `triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.