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arXiv:cond-mat/9704087

Dimensional Crossover for the Bose -- Einstein Condensation of an Ideal Bose Gas

Mario I. Molina, Jaime Rösler

Published Apr 10, 1997• 0 citations
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Abstract

We study an ideal Bose gas of N atoms contained in a box formed by two identical planar and parallel surfaces S, enclosed by a mantle of height a perpendicular to them. Calling r0 the mean atomic distance, we assume S >> r0^2 while a may be comparable to r0. In the bidimensional limit (a/r0 << 1) we find a macroscopic number of atoms in the condensate at temperatures T ~1/log(N); therefore, condensation cannot be described in terms of intensive quantities; in addition, it occurs at temperatures not too low in comparison to the tridimensional case. When condensation is present we also find a macroscopic occupation of the low--lying excited states. In addition, the condensation phenomenon is sensitive to the shape of S. The former two effects are significant for a nanoscopic system. The tridimensional limit is slowly attained for increasing (a/r0), roughly at (a/r0) ~ 10^{2}-10^{3}.

DOI
10.48550/arxiv.cond-mat/9704087

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We study an ideal Bose gas of N atoms contained in a box formed by two identical planar and parallel surfaces S, enclosed by a mantle of height a perpendicular to them. Calling r0 the mean atomic distance, we assume S >> r0^2 while a may be comparable to r0. In the bidimensional limit (a/r0 << 1) we find a macroscopic number of atoms in the condensate at temperatures T ~1/log(N); therefore, condensation cannot be described in terms of intensive quantities; in addition, it occurs at temperatures not too low in comparison to the tridimensional case. When condensation is present we also find a macroscopic occupation of the low--lying excited states. In addition, the condensation phenomenon is sensitive to the shape of S. The former two effects are significant for a nanoscopic system. The tridimensional limit is slowly attained for increasing (a/r0), roughly at (a/r0) ~ 10^{2}-10^{3}.
We study Bose-Einstein condensation (BEC) for a non-interacting gas in a very anisotropic trap. Therefore, at enough small temperatures some degrees of freedom "freeze", thus reducing the "effective" dimensionality of the system. We mainly focus in quasi-bidimensional traps characterized by a surface S in the (x, y) plane. We consider two confining mechanisms in the z direction: harmonic and rigid wall potentials. There are not forces parallel to S, excepting the rigid wall at the edges. The most relevant results are: (a) The condensate smoothly sets at T ∼ script O sign(Tc/ log N), where Tc is the tridimensional condensation temperature. (b) When BEC is present, also the low-lying excited states have a macroscopic occupation; this effect is also present in a quasi-onedimensional harmonic "cigar-shape" trap. (c) The condensation process is sensitive to the shape of S.
Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l ( L ≫ l ) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N ( l/L ) 4 ≪ l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t = T/T 1 , where T 1 is a certain characteristic temperature depending only on the longitudinal size L . Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ∼ 1 and acquires the shape of the Bose-Einstein distribution for t ≫ 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N ( l/L ) 4 ≫ 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.
The ideal Bose gas in a highly anisotropic harmonic potential is studied. It is found that Bose-Einstein condensation occurs in two distinct steps as the temperature is lowered. In the first step the specific heat shows a sharp feature, but the system still occupies many one-dimensional quantum states. In the second step, at a significantly lower temperature, the ground state becomes macroscopically occupied. It should be possible to verify these predictions using present-day atom traps. The two-step behavior can occur in a rather general class of anisotropic traps, including the box potential.
Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
For a non-self-interacting Bose gas with a fixed, large number of particles confined to a trap, as the ground-state occupation becomes macroscopic, the condensate number fluctuations remain microscopic. However, this is the only significant aspect in which the grand canonical description differs from canonical or microcanonical in the thermodynamic limit. General arguments and estimates including some of the vanishingly small quantities are compared to explicit, fixed-number calculations for 102{10}^{2}102--106{10}^{6}106 particles. \textcopyright{} 1996 The American Physical Society.