Hydrodynamics of the Polyakov Line in SU$(N_c)$ Yang-Mills

We discuss a hydrodynamical description of the eigenvalues of the Polyakov line at large but finite $N_c$ for Yang-Mills theory in even and odd space-time dimensions. The hydro-static solutions for the eigenvalue densities are shown to interpolate between a uniform distribution in the confined phase and a localized distribution in the de-confined phase. The resulting critical temperatures are in overall agreement with those measured on the lattice over a broad range of $N_c$, and are consistent with the string model results at $N_c=\infty$. The stochastic relaxation of the eigenvalues of the Polyakov line out of equilibrium is captured by a hydrodynamical instanton. An estimate of the probability of formation of a Z(N$_c)$ bubble using a piece-wise sound wave is suggested.