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Physical Review

Finite Geometries and Ideal Bose Gases

David A. Krueger

Published Aug 5, 1968• 30 citations
Publisher

Abstract

It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$ with ${L}_{1}$, ${L}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\infty}>{L}_{3}\ensuremath{\gg}{\ensuremath{\lambda}}_{T}=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}$) is found to go from order $\frac{{\ensuremath{\lambda}}_{T}{T}_{\mathrm{cB}}}{(T\ensuremath{-}{T}_{\mathrm{cB}})}$ to order ${L}_{3}\mathrm{exp}{\frac{1}{2}\sqrt{\ensuremath{\pi}}{\ensuremath{\zeta}}_{\frac{3}{2}}(\frac{{L}_{3}}{{\ensuremath{\lambda}}_{T}})[{(\frac{{T}_{\mathrm{cB}}}{T})}^{\frac{3}{2}}\ensuremath{-}1]}$ as the temperature is lowered through the bulk critical temperature ${T}_{\mathrm{cB}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\zeta}}_{\ensuremath{\chi}} \mathrm{is}\mathrm{the}\mathrm{Riemann} \ensuremath{\zeta} \mathrm{function}.)$ The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.

DOI
10.1103/physrev.172.211

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•Jan 1, 2004•0 citations
It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab (L1\ifmmode×\else\texttimes\fiL2\ifmmode×\else\texttimes\fiL3{L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}L1​\ifmmode×\else\texttimes\fiL2​\ifmmode×\else\texttimes\fiL3​ with L1{L}_{1}L1​, L2\ensuremath→\ensuremath∞{L}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}L2​\ensuremath→\ensuremath∞ and \ensuremath∞>L3\ensuremath≫\ensuremathλT=thethermalwavelength\ensuremath{\infty}>{L}_{3}\ensuremath{\gg}{\ensuremath{\lambda}}_{T}=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}\ensuremath∞>L3​\ensuremath≫\ensuremathλT​=thethermalwavelength) is found to go from order \ensuremathλTTcB(T\ensuremath−TcB)\frac{{\ensuremath{\lambda}}_{T}{T}_{\mathrm{cB}}}{(T\ensuremath{-}{T}_{\mathrm{cB}})}(T\ensuremath−TcB​)\ensuremathλT​TcB​​ to order L3exp12\ensuremathπ\ensuremathζ32(L3\ensuremathλT)[(TcBT)32\ensuremath−1]{L}_{3}\mathrm{exp}{\frac{1}{2}\sqrt{\ensuremath{\pi}}{\ensuremath{\zeta}}_{\frac{3}{2}}(\frac{{L}_{3}}{{\ensuremath{\lambda}}_{T}})[{(\frac{{T}_{\mathrm{cB}}}{T})}^{\frac{3}{2}}\ensuremath{-}1]}L3​exp21​\ensuremathπ​\ensuremathζ23​​(\ensuremathλT​L3​​)[(TTcB​​)23​\ensuremath−1] as the temperature is lowered through the bulk critical temperature TcB\ifmmode⋅\else\textperiodcentered\fi(\ensuremathζ\ensuremathχistheRiemann\ensuremathζfunction.){T}_{\mathrm{cB}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\zeta}}_{\ensuremath{\chi}} \mathrm{is}\mathrm{the}\mathrm{Riemann} \ensuremath{\zeta} \mathrm{function}.)TcB​\ifmmode⋅\else\textperiodcentered\fi(\ensuremathζ\ensuremathχ​istheRiemann\ensuremathζfunction.) The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.
An asymptotic evaluation of the specific heat of an ideal relativistic Bose gas confined to a cuboidal enclosure (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$) is carried out, under periodic boundary conditions, taking into account the possibility of particle-antiparticle pair production in the system. Finite-size corrections to the standard bulk behavior are calculated explicitly in the regions t>0t>0t>0 and t<0t<0t<0, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, such that $|t|\ensuremath{\ll}1$ and $|{L}_{i}t|\ensuremath{\gg}1$. While for t>0t>0t>0 finite-size corrections turn out to be exponential for all geometries, for t<0t<0t<0 this is true only in the case of a film; for other geometries, such as a cuboid or a rectangular channel, these corrections conform to a power law instead. Finally, we consider the situation in the core region, where ∣Lit∣=O(1)|{L}_{i}t|=O(1)∣Li​t∣=O(1), and examine the location t∗{t}^{*}t∗ and the height ${c}_{\ensuremath{\rho}}^{*}$ of the specific-heat maximum; finite-size corrections in this region turn out to be $O({L}_{<}^{\ensuremath{-}1})$, where L<{L}_{<}L<​ denotes the shortest side of the enclosure.
Several authors have proposed criteria for the onset of Bose-Einstein condensation in finite systems. Using mathematical techniques developed by Pathria and collaborators (1972) these various proposals are systematically examined by applying them to an ideal Bose gas confined to a cuboidal enclosure under periodic boundary conditions. The 'transition points' thus obtained have been subjected to a comparative study, which indicates that the most useful general criterion is the macroscopic one based on the maximum of the specific heat C v .
Taking into account the possibility of particle-antiparticle pair production, we have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to restricted geometries. Through an extensive use of the Poisson summation formula, we have carried out an explicit evaluation of the summations over states appearing in the problem, which enables us to make a rigorous analysis of the temperature dependence of the thermogeometric parameter yyy of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\ensuremath{\rho}}_{0}}{\ensuremath{\rho}}$ as a smooth function of temperature from $T\ensuremath{\gtrsim}{T}_{c}$ down to T=0T=0T=0 K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.
The vanishing of the quasiaverages usually associated with superfluidity and superconductivity is shown for arbitrarily interacting Bose and Fermi systems which are confined to a geometry with one or more dimensions finite while one or more dimensions extend to infinity. However, it is suggested that these partially finite geometries are anomalous and are not good approximations to thin film and pore geometries found in the laboratory. The conditions on the box size which are necessary and sufficient for condensation to occur in an ideal Bose gas are also given.
We generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane and approach the classifications of phase transitions in ideal and weakly interacting Bose gases of a finite number of particles, confined in a cubic box of volume L3 with different boundary conditions. For this finite ideal Bose system, by extending the classification parameters to all regions, we predict that the phase transition for periodic boundary conditions is of second order, while the transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the effects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on interaction strengths and particle numbers.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.