arXiv:2007.15661 · Physical Review Research

Theory of noninteracting fermions and bosons in the canonical ensemble

Hatem Barghathi, Jiangyong Yu, Adrian Del Maestro

Published Nov 9, 2020• 34 citations

Abstract

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.

DOI: 10.1103/physrevresearch.2.043206

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arXiv:2007.15661 · Physical Review Research

Theory of noninteracting fermions and bosons in the canonical ensemble

Hatem Barghathi, Jiangyong Yu, Adrian Del Maestro

Published Nov 9, 2020• 34 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.2.043206

Related papers

a New Theory for the Study of the Collective Excitations and the Thermodynamic Properties of AN Ensemble of Identical Interacting Quantum Mechanical Systems.

Stephen B. Haley

Jan 1, 19700 citations

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

Kurt Schönhammer

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.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.

American Journal of PhysicsNov 1, 200020 citations

Inequivalence of Canonical and Grand Canonical Ensembles for Bosonic Systems

S.K. Mehdi, N. Daoudi, S. Kessal

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.

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We study statistical properties of $N$ noninteracting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$ -point correlation function of occupation numbers $\overline{{n}_{1}\ensuremath{\cdots}{n}_{p}}$. We demonstrate that it can be expressed as a ratio of two $p\ifmmode\times\else\texttimes\fi{}p$ determinants involving the (canonical) mean occupations $\overline{{n}_{1}}$ , ..., $\overline{{n}_{p}}$ , which can themselves be conveniently expressed in terms of the $k$ -body partition functions (with $k\ensuremath{\le}N$). We draw some connection with the theory of symmetric functions and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a one-dimensional harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground-state and excited-state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground-state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme-value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.We study statistical properties of $N$ noninteracting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{{n}_{1}\ensuremath{\cdots}{n}_{p}}$. We demonstrate that it can be expressed as a ratio of two $p\ifmmode\times\else\texttimes\fi{}p$ determinants involving the (canonical) mean occupations $\overline{{n}_{1}}$, ..., $\overline{{n}_{p}}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\ensuremath{\le}N$). We draw some connection with the theory of symmetric functions and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a one-dimensional harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground-state and excited-state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground-state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme-value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.

arXiv:1802.02555Physical Review AMay 21, 20187 citations

Nonequilibrium quantum stochastic thermodynamics for bosons and fermions

Tânia Tomé, Mário J. de Oliveira

The European Physical Journal Special TopicsApr 5, 20232 citations

We study statistical properties of

N

noninteracting identical bosons or fermions in the canonical ensemble. We derive several general representations for the

p

-point correlation function of occupation numbers $\overline{{n}_{1}\ensuremath{\cdots}{n}_{p}}$. We demonstrate that it can be expressed as a ratio of two $p\ifmmode\times\else\texttimes\fi{}p$ determinants involving the (canonical) mean occupations

\overline{{n}_{1}}

, ...,

\overline{{n}_{p}}

, which can themselves be conveniently expressed in terms of the

k

-body partition functions (with $k\ensuremath{\le}N$). We draw some connection with the theory of symmetric functions and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a one-dimensional harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground-state and excited-state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground-state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme-value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.