Physical Review

Quantum Statistics of Closed and Open Systems

P. T. Landsberg

Published Mar 15, 1954• 5 citations

Abstract

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.

DOI: 10.1103/physrev.93.1170

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Physical Review

Quantum Statistics of Closed and Open Systems

P. T. Landsberg

Published Mar 15, 1954• 5 citations

Publisher

Abstract

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.

DOI: 10.1103/physrev.93.1170

Related papers

The Continuous Spectrum Approximation in Quantum Statistics

Peter T. Landsberg

A class $O$ of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class $O$ . A "limit $L$ " is defined as the limiting case which arises of the number of particles $N$ and the volume $V$ of the system are both allowed to tend to infinity so as to keep the ratio $\frac{N}{V}$ a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class $O$ in the limit $L$ , one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.A class $O$ of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class $O$. A "limit $L$" is defined as the limiting case which arises of the number of particles $N$ and the volume $V$ of the system are both allowed to tend to infinity so as to keep the ratio $\frac{N}{V}$ a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class $O$ in the limit $L$, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.

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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

A class

O

of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class

O

. A "limit

L

" is defined as the limiting case which arises of the number of particles

N

and the volume

V

of the system are both allowed to tend to infinity so as to keep the ratio

\frac{N}{V}

a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class

O

in the limit

L

, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.