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arXiv:1105.5697 · Physical Review A

Isobars of an ideal Bose gas within the grand canonical ensemble

Imtak Jeon, Sang-Woo Kim, Jeong-Hyuck Park

Published Aug 29, 2011• 5 citations
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Abstract

We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble for a large yet finite number of particles, $N$. After solving the equation of the spinodal curve, we derive precise formulas for the supercooling and the superheating temperatures that reveal an ${N}^{\ensuremath{-}1/3}$ or ${N}^{\ensuremath{-}1/4}$ power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if $N\ensuremath{\ge}14\phantom{\rule{0.16em}{0ex}}393$. In particular, for the Avogadro's number of particles, the volume expands discretely about ${10}^{5}$ times. Our results quantitatively agree with a previous study on the canonical ensemble within $0.1%$ error.

DOI
10.1103/physreva.84.023636

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We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble for a large yet finite number of particles, NNN. After solving the equation of the spinodal curve, we derive precise formulas for the supercooling and the superheating temperatures that reveal an N\ensuremath−1/3{N}^{\ensuremath{-}1/3}N\ensuremath−1/3 or N\ensuremath−1/4{N}^{\ensuremath{-}1/4}N\ensuremath−1/4 power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if N\ensuremath≥14393N\ensuremath{\ge}14\phantom{\rule{0.16em}{0ex}}393N\ensuremath≥14393. In particular, for the Avogadro's number of particles, the volume expands discretely about 105{10}^{5}105 times. Our results quantitatively agree with a previous study on the canonical ensemble within 0.10.1%0.1 error.
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.
This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation---namely, the fraction of particles in the ground state and the specific heat---are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit---here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.
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.
We conduct a rigorous investigation into the thermodynamic instability of an ideal Bose gas confined in a cubic box, without assuming a thermodynamic limit or a continuous approximation. Based on the exact expression of the canonical partition function, we perform numerical computations up to 106{10}^{6}106 particles. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be 761676167616, the ideal Bose gas subject to the Dirichlet boundary condition reveals a thermodynamic instability. Accordingly, we demonstrate that a system consisting of a finite number of particles can exhibit a discontinuous phase transition that features a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, 7616,7616,7616, can be regarded as a characteristic number of a ``cube,'' which is the geometric shape of the box.
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.