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Rights statement: Copyright 2019 American Institute of Physics. The following article appeared in Journal of Mathematical Physics 60, 2019 and may be found at http://dx.doi.org/10.1063/1.5091737 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

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## Statistical mechanics of the periodic Benjamin Ono equation

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Article number 093302 19/09/2019 Journal of Mathematical Physics 9 60 25 Published English

### 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.

### Bibliographic note

Copyright 2019 American Institute of Physics. The following article appeared in Journal of Mathematical Physics 60, 2019 and may be found at http://dx.doi.org/10.1063/1.5091737 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.