# 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:

- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003 (United States)
- 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}

}