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

At low temperatures the configurational phase space of a macroscopic complex system (e.g., a spin-glass) of N 10²³ 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.