Distribution of Equilibrium Free Energies in a Thermodynamic System with Broken Ergodicity

<Abstract>At low temperatures the configurational phase space of a macroscopic complex system (e.g.,a spin-glass) of N～10^(23) interacting particles may split into an exponential number Ω_s～exp(const×N) of ergodic sub-spaces (thermodynamic states).It is usually assumed that the equilibrium collective behavior of such a system is determined by its ground thermodynamic states of the minimal free-energy density,and that the equilibrium free energies follow the distribution of exponential decay.But actually for some complex systems,the equilibrium free-energy values may follow a Gaussian distribution within an intermediate temperature range,and consequently their equilibrium properties are contributed by excited thermodynamic states.Based on this analysis,the re-weighting parameter y in the cavity approach of spin-glasses is easily understood.Depending on the free-energy distribution,the optimal y can either be equal to or be strictly less than the inverse temperature β.

At low temperatures the configurational phase space of a macroscopic complex system （e.g., a spin-glass） of N - 10^23 interacting particles may split into an exponential number Ωs - exp（const × N） of ergodic sub-spaces （thermodynamic states）. It is usually assumed that the equilibrium collective behavior of such a system is determined by its ground thermodynamic states of the minimal free-energy density, and that the equilibrium free energies follow the distribution of exponentied decay. But actually for some complex systems, the equilibrium free-energy values may follow a Gaussian distribution within an intermediate temperature range, and consequently their equilibrium properties are contributed by excited thermodynamic states. Based on this analysis, the re-weighting parameter y in the cavity approach of spin-glasses is easily understood. Depending on the free-energy distribution, the optimal y can either be equal to or be strictly less than the inverse temperature β.