Order parameters of the spin glass mean field theory and initial conditions

Abstract

The time-dependent order parameter q(t₂-t₁) is re-examined as a function of the time difference t₂-t₁ and number of spins N, for the infinite-range spin glass. In the statistical mechanics limit, t₂-t₁ to infinity before N to infinity , the authors find q(t₂-t₁)=q= integral q(x)dx, where q(x) is the order parameter function of Parisi (1979, 1980, 1983), provided the system has a canonical (Boltzmann) distribution at the earlier time t₁. In the opposite limit, N to infinity before t₂-t₁ to infinity , they obtain q(t₂-t₁)=q(x=1). If the initial distribution differs slightly from a canonical one they obtain q(t₂, t₁)=q(x=0) (i.e. the statistical mechanics limit) and the Sompolinsky solution (1981) appears to be correct under these (non-equilibrium) conditions.