Finite temperature equilibrium density profiles of integrable systems in confining potentials

TL;DR

This paper investigates the equilibrium density profiles of classical integrable systems in confining potentials, using field theory and simulations, highlighting differences between strongly and weakly interacting models.

Contribution

It introduces an analytical field theoretic approach to describe density profiles in integrable systems with strong interactions and compares these with Monte Carlo simulations, also analyzing weakly interacting cases.

Findings

Good agreement between field theory and simulations for hard rods and Calogero models.

Field theory is ineffective for the Toda model due to lack of a thermodynamic length scale.

Harmonic chain model provides insights into Toda model density features.

Abstract

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the inter-particle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and compare them with results from Monte-Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which inter-particle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited due to the lack of a thermodynamic length scale. The density profiles for the Toda model obtained from Monte-Carlo simulations can be understood by studying the analytically tractable harmonic chain model (Hessian approximation of the Toda model). For the harmonic chain model one can derive an exact expression for the density that shines light on some of the qualitative features of the Toda model in a quadratic trap. Our work provides an analytical approach towards understanding the equilibrium properties for interacting integrable systems in confining traps.