Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Thermodynamics and occupation numbers of a Fermi gas in the canonical ensemble

Kurt Schönhammer

Published Nov 1, 2000• 20 citations
Publisher

Abstract

Exact results for the thermodynamic properties and mean occupation numbers of a system of noninteracting fermions with equidistant level spacing are presented for an arbitrary number of particles. It is discussed quantitatively how the results converge to the corresponding ones in the grand canonical ensemble when the thermodynamic limit is reached. From the simple calculations it also follows that the thermodynamics of an infinite two-dimensional electron gas is identical to that of a one-dimensional harmonic chain with linearized dispersion. The results generalize and simplify previous approaches to this model published in this journal.

DOI
10.1119/1.1286116

Related papers

A soluble one dimensional model of an interacting Fermi gas. I

Eleonora M. Witteles, Gideon Carmi

The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.

Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•1 citation

Quantum Statistics of Closed and Open Systems

P. T. Landsberg

In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit $L$," i.e., in the limit in which the number of particles $N$ and the volume $V$ of the system are allowed to tend to infinity so as to keep $\frac{N}{V}$ a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit $L$ the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.

Physical Review•Mar 15, 1954•5 citations

THERMODYNAMICS OF ONE-DIMENSIONAL FERMI GAS IN THE INFINITE REPULSIVE COUPLING LIMIT

HU ZHAN-NING, ZHOU YU-KUI, 西北大学现代物理研究所,西安,710069

The thermodynamic properties for a gas of one-dimensional fermi particles interacting via a repulsive delta-function potential have been studied under the limit of infinite repulsive coupling using the theory of grand canonical ensemble. All the thermodynamic functions have been found explicitly and the velocity of sound is derived as a function of the fugacity. In particular, the entropy is not zero at the absolute zero temperature for this model. Finally, some results are given for the arbitrary spin gas.

Acta Physica Sinica•Jan 1, 1990•0 citations

Finite-size effects in canonical and grand-canonical quantum Monte Carlo simulations for fermions

Zhenjiu Wang, Fakher F. Assaad, Francesco Parisen Toldin

We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.

arXiv:1706.01874•Physical review. E•Oct 16, 2017•24 citations

Stable Recursion Relation for the Canonical Partition Function of Non-Interacting Fermions

Jiangyong Yu, Hatem Barghathi, Adrian Del Maestro

While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.

Bulletin of the American Physical Society•Mar 5, 2019•0 citations

Theory of noninteracting fermions and bosons in the canonical ensemble

Hatem Barghathi, Jiangyong Yu, Adrian Del Maestro

This work introduces a framework for treating indistinguishable quantum particles in the canonical ensemble that allows for the direct computation of thermodynamic properties including energy levels-occupation probability distributions and their corresponding occupation-numbers and correlations.

arXiv:2007.15661•Physical Review Research•Nov 9, 2020•34 citations
Exact results for the thermodynamic properties and mean occupation numbers of a system of noninteracting fermions with equidistant level spacing are presented for an arbitrary number of particles. It is discussed quantitatively how the results converge to the corresponding ones in the grand canonical ensemble when the thermodynamic limit is reached. From the simple calculations it also follows that the thermodynamics of an infinite two-dimensional electron gas is identical to that of a one-dimensional harmonic chain with linearized dispersion. The results generalize and simplify previous approaches to this model published in this journal.
The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.
In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit LLL," i.e., in the limit in which the number of particles NNN and the volume VVV of the system are allowed to tend to infinity so as to keep NV\frac{N}{V}VN​ a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit LLL the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.
The thermodynamic properties for a gas of one-dimensional fermi particles interacting via a repulsive delta-function potential have been studied under the limit of infinite repulsive coupling using the theory of grand canonical ensemble. All the thermodynamic functions have been found explicitly and the velocity of sound is derived as a function of the fugacity. In particular, the entropy is not zero at the absolute zero temperature for this model. Finally, some results are given for the arbitrary spin gas.
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.