Distribution of equilibrium free energies in a thermodynamic system with broken ergodicity

TL;DR

This paper investigates the distribution of equilibrium free energies in complex systems like spin-glasses, revealing that they can follow a Gaussian distribution at intermediate temperatures, involving excited states rather than just ground states.

Contribution

It challenges the assumption that equilibrium states are solely determined by minimal free-energy states, showing that excited states can dominate under certain conditions.

Findings

Equilibrium free energies can follow a Gaussian distribution at intermediate temperatures.

Excited thermodynamic states significantly influence the equilibrium properties.

The traditional assumption of exponential decay distribution of free energies may not always hold.

Abstract

At low temperatures the configurational phase space of a macroscopic complex system (e.g., a spin-glass) of $N\sim 10^{23}$ interacting particles may split into an exponential number $\Omega_s \sim \exp({\rm const} \times N)$ of ergodic sub-spaces (thermodynamic states). Previous theoretical studies 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. Here we show that these assumptions are not necessarily valid. 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 {\em excited} thermodynamic states. This work will help improving our understanding of the equilibrium statistical mechanics of spin-glasses and other complex systems.