Statistical Mechanics in the Extended Gaussian Ensemble

TL;DR

The paper introduces the extended gaussian ensemble (EGE), a new statistical mechanics framework that generalizes the canonical ensemble by controlling both mean energy and energy fluctuations, with applications demonstrated on spin systems.

Contribution

It develops the formalism of the EGE from first principles, establishing its thermodynamic structure and stability criteria, and compares it with q-exponential distributions.

Findings

EGE formalism derived from maximum entropy and reservoir analysis.

Legendre transform structure established for EGE thermodynamics.

Application to spin systems demonstrates EGE's practical relevance.

Abstract

The extended gaussian ensemble (EGE) is introduced as a generalization of the canonical ensemble. The new ensemble is a further extension of the Gaussian ensemble introduced by J. H. Hetherington [J. Low Temp. Phys. {\bf 66}, 145 (1987)]. The statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the Maximum Statistical Entropy Principle. The probability of each microstate depends on two parameters $\beta$ and $\gamma$ which allow to fix, independently, the mean energy of the system and the energy fluctuations respectively. We establish the Legendre transform structure for the generalized thermodynamic potential and propose a stability criterion. We also compare the EGE probability distribution with the $q$-exponential distribution. As an example, an application to a system with few independent spins is presented.