Generalized canonical ensembles and ensemble equivalence

TL;DR

This paper discusses a generalized canonical ensemble that can reproduce all microcanonical equilibrium properties, including nonconcave entropy cases, thus extending the equivalence between ensembles beyond traditional limits.

Contribution

It introduces a simplified physical perspective on the generalized canonical ensemble, demonstrating its ability to match microcanonical properties for nonconcave entropies, unlike the standard canonical ensemble.

Findings

Generalized ensemble reproduces microcanonical properties for nonconcave entropies.

Standard canonical ensemble cannot achieve this for nonconcave entropies.

Gaussian ensemble is a specific case with quadratic g-functions.

Abstract

This paper is a companion article to our previous paper (J. Stat. Phys. 119, 1283 (2005), cond-mat/0408681), which introduced a generalized canonical ensemble obtained by multiplying the usual Boltzmann weight factor $e^{-\beta H}$ of the canonical ensemble with an exponential factor involving a continuous function $g$ of the Hamiltonian $H$. We provide here a simplified introduction to our previous work, focusing now on a number of physical rather than mathematical aspects of the generalized canonical ensemble. The main result discussed is that, for suitable choices of $g$, the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties of the many-body system represented by $H$ even if this system has a nonconcave microcanonical entropy function. This is something that in general the standard ($g=0$) canonical ensemble cannot achieve. Thus a virtue of the generalized canonical ensemble is that it can be made equivalent to the microcanonical ensemble in cases where the canonical ensemble cannot. The case of quadratic $g$-functions is discussed in detail; it leads to the so-called Gaussian ensemble.