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Physical Review A

Low-temperature behavior of Bose systems confined to restricted geometries: Growth of the condensate fraction and spatial correlations

C. S. Zasada, Raj K. Pathria

Published Sep 1, 1976• 30 citations
Publisher

Abstract

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 $y$ for a cubical enclosure under periodic boundary conditions, which in turn enables us to evaluate the temperature dependence of the condensate fraction $\frac{{N}_{0}}{N}$ 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, $\frac{L}{\overline{l}}=40 \mathrm{and} 100$, where $L$ is the edge length of the enclosure and $\overline{l}$ 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.

DOI
10.1103/physreva.14.1269

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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 $y$ 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 $T=0$ 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.

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Physical Review A•Dec 1, 1984•12 citations
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 \ensuremathρ(r\ensuremath→,r\ensuremath→\ensuremath′)\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})\ensuremathρ(r\ensuremath→,r\ensuremath→\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.
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.
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.
We have carried out a rigorous asymptotic analysis of the thermodynamic behavior of an ideal Bose gas confined to an arbitrary, finite cuboidal geometry under periodic boundary conditions. Our investigation, which is based on the grand canonical ensemble, leads to the construction of an abstract, thermogeometric space in which the process of Bose-Einstein condensation appears as a "collapse" of the lattice points of the space towards its origin. In an infinite geometry, the collapse is accomplished in an infinitesimally small interval of temperature; this results in the appearance of mathematical singularities in the thermodynamic functions of the system. In a finite geometry, the "collapse" proceeds gradually and is spread over a temperature range $\ensuremath{\Delta}T$ such that $\frac{\ensuremath{\Delta}T}{{T}_{0}}=O(\frac{\overline{l}}{{L}_{<}})$, where l‾\overline{l}l is the mean interatomic distance while L<{L}_{<}L<​ is the length of the shortest side of the assembly; accordingly, the thermodynamic functions of the system remain smooth throughout. Special events, such as the specific-heat maximum, occur when the "lattice parameters" of our thermogeometric space acquire certain characteristic values which depend only on the shape of the system and not on its actual size. This leads to a new law of corresponding states for Bose-Einstein systems of finite size.
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.
Mathematical singularities, which are known to be at the heart of phase transitions and are responsible for making the thermodynamic functions of the given system nonanalytic, are a consequence of the thermodynamic limit, viz. N and V → ∞ with N/V staying constant. When a system containing a finite number of particles and confined to a restricted geometry undergoes a phase transition, these singularities get rounded off, with the result that all thermodynamic functions become analytic and vary smoothly with the relevant parameters of the problem. Theoretical analysis of such situations requires the use of special mathematical techniques which may vary drastically from case to case.In the present communication we report the results of a rigorous analysis of the problem of "Bose–Einstein condensation in restricted geometries", which has been carried out by making an extensive use of the Poisson summation formula. Particular emphasis is laid on the growth of the condensate fraction [Formula: see text] as the temperature of the system is lowered, and on the influence of the boundary conditions imposed on the wave functions of the particles. The relevance of these results, in relation to the scaling theory of finite size effects, is also discussed.
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.