Topological properties of the mean-field phi4 model

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a ?4 mean-field model. We compare the critical energy vc [i.e., the potential energy v(T) evaluated at the phase transition temperature Tc] with the energy v? at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc?v?, at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=v? [ Casetti, Pettini and Cohen Phys. Rep. 337 237 (2000)]. By direct calculation of the energy vs(T) of the "inherent saddles," i.e., the saddles visited by the equilibrated system at temperature T, we find that vs(Tc)~v?. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than to a change of the topology of the potential energy surface at T=Tc. Finally, we discuss the approximation involved in our analysis and the generality of our method.