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Physical Review

Ground-State Occupancy of an Ideal Bose-Einstein Gas Confined to a Finite Volume

D. L. Mills

Published Apr 20, 1964• 25 citations
Publisher

Abstract

The number of particles in the ground state has been computed as a function of temperature for an ideal Bose-Einstein gas confined to a box of finite volume by evaluating the discrete sum over states on a computer. Large deviations from London's bulk-gas result are found when the length of the box is much greater than its width for the range of dimensions investigated here. It is shown that the deviations occur because in this limit the system tends to behave like a one-dimensional system.

DOI
10.1103/physrev.134.a306

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The number of particles in the ground state has been computed as a function of temperature for an ideal Bose-Einstein gas confined to a box of finite volume by evaluating the discrete sum over states on a computer. Large deviations from London's bulk-gas result are found when the length of the box is much greater than its width for the range of dimensions investigated here. It is shown that the deviations occur because in this limit the system tends to behave like a one-dimensional system.
Abstract The Bose‐Einstein energy spectrum of an ideal quantum gas confined in a box is discrete and strongly dependent on the box geometry and temperature, for low product of the atomic mass number, Aat and the adiabatic invariant, TV2/3, i.e. on γ = Aat TV2/3. Even within the approximation of non‐interacting particles in the gas, the calculation of the thermodynamic properties of Bose‐Einstein systems turns out to be a difficult mathematical problem. The present study compares the total number of particles, the total energy and the specific heat obtained by sums over the discrete states to the results of the approximate integrals (for defined values of γ) and shows the noticeable errors of the last ones at low adiabatic invariant (around condensation). Then, more rigorous and precise analytical approximations of sums are found in the case of finite number of atoms (correlated with the existence of a zero energy level) and the finite volume of the gas. The corrected thermodynamic functions depend on γ (for fixed fugacity). The condensation temperature is corrected also in order to describe more accurately the discrete Bose‐Einstein systems. Under these circumstances, the analysis of the specific heat leads to the conclusion that it shows no discontinuity, for the entire range of γ values.
We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of noninteracting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-TTT behavior in contrast to an earlier theory in Feynman's textbook on statistical mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum-mechanical treatment. Furthermore, we derive the finite-size effect of the system.
We study the behaviour of an ideal non-relativistic Bose gas in a three-dimensional space where one of the dimensions is compactified to form a circle. In this case, there is no phase transition like that for the case of an infinite volume, nevertheless Bose-Einstein condensation signified by a sudden buildup of particles in the ground state can occur. We use the grand canonical ensemble to study this problem. In particular, the specific heat is evaluated numerically, as well as analytically in certain limits. We show analytically how the familiar result for the specific heat is recovered as we let the size of the circle become large so that the infinite volume limit is approached. We also examine in detail the behaviour of the chemical potential and establish the precise manner in which it approaches zero as the volume becomes large. © 2006 IOP Publishing Ltd.
We study the behaviour of an ideal non-relativistic Bose gas in a three-dimensional space where one of the dimensions is compactified to form a circle. In this case, there is no phase transition like that for the case of an infinite volume, nevertheless Bose–Einstein condensation signified by a sudden buildup of particles in the ground state can occur. We use the grand canonical ensemble to study this problem. In particular, the specific heat is evaluated numerically, as well as analytically in certain limits. We show analytically how the familiar result for the specific heat is recovered as we let the size of the circle become large so that the infinite volume limit is approached. We also examine in detail the behaviour of the chemical potential and establish the precise manner in which it approaches zero as the volume becomes large.
Abstract This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.
Evading the Mermin–Wagner–Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods, we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically, N ⩾ 35131 . The instability implies that the isobar of the gas zigzags on the temperature–volume plane, featuring supercooling and superheating phenomena. The Bose–Einstein condensation can then persist from absolute zero to the superheating temperature. Without necessarily taking the large N limit, under a constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose–Einstein condensations of harmonically trapped atomic gases are a first-order phase transition that involves a discrete change of the density at the center of the trap.