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Journal of Mathematical Physics

Low-temperature thermodynamics of a weakly interacting Bose gas confined to restricted geometries

C. S. Zasada, Raj K. Pathria

Published Dec 1, 1979• 1 citation
Publisher

Abstract

The low-temperature internal energy and specific heat of a hard sphere Bose gas confined to a general cuboidal enclosure are calculated under periodic boundary conditions. The special cases of a film and a channel geometry are considered in some detail. It is shown that, in the appropriate limits, one recovers the recent results of Pajkowski and Pathria, and of Zasada and Pathria, for a system of phonons confined to different geometries. Systematic applications of the Poisson summation formula enable us to obtain strongly convergent expansions, in both λ/L≪1 and λ/L≫1 regimes, and to see how the magnitude of the finite size corrections in the asymptotic limit (L/λ≫1) is directly linked with the dimensionality of the system. Special attention is paid to examining the combined influence of the dispersion in the phonon spectrum of the system and the finiteness of the enclosure.

DOI
10.1063/1.524018

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The low-temperature internal energy and specific heat of a hard sphere Bose gas confined to a general cuboidal enclosure are calculated under periodic boundary conditions. The special cases of a film and a channel geometry are considered in some detail. It is shown that, in the appropriate limits, one recovers the recent results of Pajkowski and Pathria, and of Zasada and Pathria, for a system of phonons confined to different geometries. Systematic applications of the Poisson summation formula enable us to obtain strongly convergent expansions, in both λ/L≪1 and λ/L≫1 regimes, and to see how the magnitude of the finite size corrections in the asymptotic limit (L/λ≫1) is directly linked with the dimensionality of the system. Special attention is paid to examining the combined influence of the dispersion in the phonon spectrum of the system and the finiteness of the enclosure.
Through an extensive use of the Poisson summation formula, we have elucidated the description of the phenomenon of Bose-Einstein condensation in finite systems in terms of a "collapse in the thermogeometric space." In particular, we have carried out an explicit evaluation of the temperature dependence of the thermogeometric parameter yyy for a cubical enclosure under periodic boundary conditions, which in turn enables us to evaluate the temperature dependence of the condensate fraction N0N\frac{{N}_{0}}{N}NN0​​ as well as the two-point correlation function $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for cubes of arbitrary sizes. Numerical results are given for two specific sizes, Ll‾=40and100\frac{L}{\overline{l}}=40 \mathrm{and} 100lL​=40and100, where LLL is the edge length of the enclosure and l‾\overline{l}l the mean interparticle distance in the system. In the appropriate limit, our results are in complete agreement with the Fisher-Barber scaling theory for finite-size effects. Application of the Poisson summation formula has also enabled us to extract information of direct physical interest about the growth of long-range order in general cuboidal geometries. The special case of the thin-film geometry has been studied in detail; the resulting formulas provide a considerable improvement over the ones obtained by previous workers.
In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.
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.
In a recent paper we carried out a rigorous, asymptotic analysis of the low-temperature behavior of a Bose gas confined to a finite cubic geometry and subjected to periodic boundary conditions. That analysis is now extended to Dirichlet, Neumann, and antiperiodic boundary conditions. As before, the thermogeometric parameter yyy and the condensate fraction N0N\frac{{N}_{0}}{N}NN0​​ are studied as functions of temperature and are evaluated explicity for cubes with Ll‾=40and100\frac{L}{\overline{l}}=40 \mathrm{and} 100lL​=40and100, where LLL is the edge length of the enclosure and l‾\overline{l}l the mean interparticle distance in the system. The finite-size corrections to the condensate fraction under Dirichlet or Neumann boundary conditions turn out to be qualitatively different from those under periodic or antiperiodic boundary conditions. This difference becomes manifest when one tries to express these corrections in a form consistent with the standard scaling theory for finite-size effects.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.
We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures TTT, we demonstrate that the coefficient in T2{T}^{2}T2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in detail. At T=0T=0T=0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.