# Statistical Mechanics of the Periodic Benjamin-Ono Equation

## TL;DR

This paper explores the statistical mechanics of the periodic Benjamin-Ono equation, demonstrating Gibbs measure properties, soliton solution behavior, and connections to gas dynamics and Euler equations.

## Contribution

It establishes logarithmic Sobolev inequalities for Gibbs measures and links soliton solutions to gas density distributions in the limit.

## Key findings

- Gibbs measures satisfy logarithmic Sobolev inequalities
- Soliton solutions' measures concentrate as n→∞
- Limiting measures relate to solutions of Euler equations

## Abstract

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({\mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $n\rightarrow\infty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.