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 parametrized 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 equat