  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.
