On the mean field description of systems of point particles in equilibrium

The free energy of a classical system of interacting point particles confined to a hypersurface Σv-1r in Rv is minimized over the set of all product states corresponding to systems of noninteracting particles in equilibrium. Using techniques previously employed by Tindemans and Capel, it is shown that for separable interactions the exact free energy density is equal in the thermodynamic limit to that given by the minimizing product state. The equation for the minimizing product state has a unique solution analytic in β = (kT)-1 above some temperature Tc, proving absence of phase transitions above Tc. Analogous theorems are proved for infinite- dimensional quantum systems of distinguishable particles. The method is applied to a system of coupled oscillators on a sphere in Rv which, as shown, is equivalent to the Curie-Weiss model.