skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 e{sup -{beta}}{sup 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 often be made equivalent to the microcanonical ensemble in cases in which the canonical ensemble cannot. The case of quadratic g functions is discussed in detail; it leads to the so-called Gaussian ensemble.

Authors:
; ;  [1];  [2]
  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003 (United States)
  2. School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS (United Kingdom)
Publication Date:
OSTI Identifier:
20778700
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; Journal Volume: 73; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevE.73.026105; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ELECTRONS; ENTROPY; EQUILIBRIUM; HAMILTONIANS; MANY-BODY PROBLEM; QUANTUM MECHANICS; STATISTICAL MECHANICS

Citation Formats

Costeniuc, M., Ellis, R.S., Turkington, B., and Touchette, H. Generalized canonical ensembles and ensemble equivalence. United States: N. p., 2006. Web. doi:10.1103/PHYSREVE.73.0.
Costeniuc, M., Ellis, R.S., Turkington, B., & Touchette, H. Generalized canonical ensembles and ensemble equivalence. United States. doi:10.1103/PHYSREVE.73.0.
Costeniuc, M., Ellis, R.S., Turkington, B., and Touchette, H. Wed . "Generalized canonical ensembles and ensemble equivalence". United States. doi:10.1103/PHYSREVE.73.0.
@article{osti_20778700,
title = {Generalized canonical ensembles and ensemble equivalence},
author = {Costeniuc, M. and Ellis, R.S. and Turkington, B. and Touchette, H.},
abstractNote = {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 e{sup -{beta}}{sup 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 often be made equivalent to the microcanonical ensemble in cases in which the canonical ensemble cannot. The case of quadratic g functions is discussed in detail; it leads to the so-called Gaussian ensemble.},
doi = {10.1103/PHYSREVE.73.0},
journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
number = 2,
volume = 73,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2006},
month = {Wed Feb 15 00:00:00 EST 2006}
}
  • It was found that the equivalence of the grand canonical and canonical ensembles for the Coulomb systems is possible only when charged particles of different types in calculating the physical quantities are considered as formally 'independent' ones, and the quasi-neutrality condition is used in the final stage of calculations. The phase equilibrium condition is obtained and the expression is derived for the isothermal compressibility of matter as a two-component Coulomb system, which corresponds to the known limit relations for static structure factors. On this basis, it is demonstrated that the critical point of matter, considering as the Coulomb system ismore » determined from the condition of vanishing mean square of fluctuations of the total charge per unit volume.« less
  • We use the supersymmetric formalism to derive an integral formula for the density of states of the Gaussian Orthogonal Ensemble, and then apply saddle-point analysis to give a new derivation of the 1/N-correction to Wigner's law. This extends the work of Disertori on the Gaussian Unitary Ensemble. We also apply our method to the interpolating ensembles of Mehta–Pandey.
  • We use the supersymmetric formalism to derive an integral formula for the density of states of the Gaussian Orthogonal Ensemble, and then apply saddle-point analysis to give a new derivation of the 1/N-correction to Wigner's law. This extends the work of Disertori on the Gaussian Unitary Ensemble. We also apply our method to the interpolating ensembles of Mehta–Pandey.
  • In this article the Taylor-expansion method is introduced by which Monte Carlo (MC) simulations in the canonical ensemble can be speeded up significantly, Substantial gains in computational speed of 20-40% over conventional implementations of the MC technique are obtained over a wide range of densities in homogeneous bulk phases. The basic philosophy behind the Taylor-expansion method is a division of the neighborhood of each atom (or molecule) into three different spatial zones. Interactions between atoms belonging to each zone are treated at different levels of computational sophistication. For example, only interactions between atoms belonging to the primary zone immediately surroundingmore » an atom are treated explicitly before and after displacement. The change in the configurational energy contribution from secondary-zone interactions is obtained from the first-order term of a Taylor expansion of the configurational energy in terms of the displacement vector d. Interactions with atoms in the tertiary zone adjacent to the secondary zone are neglected throughout. The Taylor-expansion method is not restricted to the canonical ensemble but may be employed to enhance computational efficiency of MC simulations in other ensembles as well. This is demonstrated for grand canonical ensemble MC simulations of an inhomogeneous fluid which can be performed essentially on a modern personal computer.« less
  • The two-dimensional UCLA cumulus ensemble model (CEM), which covers a large horizontal area with a sufficiently small horizontal grid size, is used in this study. A number of simulation experiments are performed with the CEM to study the macroscopic behavior of cumulus convection under a variety of different large-scale and underlying surface conditions. Specifically, the modulation of cumulus activity by the imposed large-scale processes and the eddy kinetic energy (EKE) budget are investigated in detail. In all simulations, cumulus convection is rather strongly modulated by large-scale advective processes in spite of the existence of some nonmodulated high-frequency fluctuations. The modulationmore » exhibits some phase delays, however, when the basic wind shear is strong. This is presumably associated with the existence of mesoscale convective organization. The EKE budget analysis shows that the net eddy buoyancy generation rate is nearly zero for a wide range of cumulus ensembles. 34 refs., 17 figs., 2 tabs.« less