In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap logarithmic Sobolev constants uniform in number particles. The Poincaré inequality is based on work Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) Zegarlinski’s theorem for Gibbs measures, both combi...