Generalized canonical ensembles and ensemble equivalence

Abstract

This paper is a companion piece to our previous work [J. Stat. Phys. 119, 1283 (2005)], which introduced a generalized canonical ensemble obtained by multiplying the usual Boltzmann weight factor of the canonical ensemble with an exponential factor involving a continuous function of the Hamiltonian . 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 , the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties of the many-body system represented by even if this system has a nonconcave microcanonical entropy function. This is something that in general the standard canonical ensemble cannot achieve. Thus a virtue of the generalized canonical ensemble is that it can often be made equivalent to the microcanonical ensemble in cases in which the canonical ensemble cannot. The case of quadratic functions is discussed in detail; it leads to the so-called Gaussian ensemble.

Nonequivalent Ensembles and Metastability

“…3 Finally, we could have mentioned our recent work on a generalization of the canonical ensemble which aims at converting unstable and metastable states of the canonical ensemble into stable, equilibrium states of a modified canonical ensemble so as to recover equivalence with the microcanonical ensemble. 8 Research is ongoing on this topic.…”

Termodinâmica do modelo de Ising com interações de alcance infinito via ensemble canônico generalizado

“…Nasúltimas décadas, o desenvolvimento de ensembles generalizados 2 tem constituido uma abordagem promissora no tratamento de uma série de modelos na mecânica estatística [4][5][6][7][8][9]. Computacionalmente, a utilização desses ensembles generalizados está ligadaà busca de algoritmos de amostragem de configurações (por exemplo, o método de Monte Carlo) mais eficientes [10][11][12][13].…”

“…Utilizando a distribuição de probabilidades dada pela expressão (8), calculamos a função de partição a partir da Eq. (6), obtendo assim…”

Hot and Heavy Matters in the Foundations of Statistical Mechanics

“…Still others consider moving to the use of new ensembles. Costeniuc et al [12] define a new "generalized canonical" ensemble by modifying the Lebesgue measure from which the canonical ensemble is crafted. They then show that this 'generalized canonical ensemble' is equivalent to the microcanonical measure in the conventional thermodynamic limit.…”