Mean-field Potts glass model: Initial-condition effects on dynamics and properties of metastable states

The dynamics of an infinite-range soft-spin Potts glass model is considered. It is shown that, in order to obtain a completely consistent description of the freezing transition, proper account of the initial configuration of spins has to be incorporated into the dynamical theory. By appealing to free-energy considerations, it is argued that canonical weight should be associated with the initial configurations. The solution of the resulting dynamical theory automatically avoids the unstable replica symmetric solution. We also show that the equation of state for the Edwards-Anderson order parameter obtained from the dynamical theory with the memory of initial conditions coincides with that obtained using the mean-field Thouless-Anderson-Palmer (TAP) equations for the Potts model. In addition, an explicit calculation of the number of metastable solutions of the TAP equation shows that it is exponentially large at (and below) the temperature, $T_A$, where the dynamical theory predicts a continuous freezing. At a lower temperature, $T_K$, where there is a true thermodynamic transition, the solution degeneracy becomes nonextensive, enabling one to identify it as the Kauzman temperature for the model.