Tropical Limit in Statistical Physics

TL;DR

This paper explores the tropical limit in statistical physics, where Boltzmann constant approaches zero, revealing simplified properties of complex systems like spin glasses and frustrated systems.

Contribution

It introduces the tropical limit as a new analytical framework for studying systems with degenerated energy levels and analyzes their thermodynamic properties.

Findings

Tropical free energy is piecewise linear in temperature.

Tropical entropy is piecewise constant.

Systems exhibit maximum tropical Gibbs probability at specific energies.

Abstract

Tropical limit for macroscopic systems in equilibrium defined as the formal limit of Boltzmann constant k going to 0 is discussed. It is shown that such tropical limit is well-adapted to analyse properties of systems with highly degenerated energy levels, particularly of frustrated systems like spin ice and spin glasses. Tropical free energy is a piecewise linear function of temperature, tropical entropy is a piecewise constant function and the system has energy for which tropical Gibbs' probability has maximum. Properties of systems in the points of jump of entropy are studied. Systems with finite and infinitely many energy levels and phenomena of limiting temperatures are discussed.