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Physical Review A

Bose-Einstein condensation in finite noninteracting systems: A relativistic gas with pair production. II

Surjit Singh, Raj K. Pathria

Published Dec 1, 1984• 12 citations
Publisher

Abstract

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 $t>0$ and $t<0$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, such that $|t|\ensuremath{\ll}1$ and $|{L}_{i}t|\ensuremath{\gg}1$. While for $t>0$ finite-size corrections turn out to be exponential for all geometries, for $t<0$ 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 $|{L}_{i}t|=O(1)$, and examine the location ${t}^{*}$ 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 ${L}_{<}$ denotes the shortest side of the enclosure.

DOI
10.1103/physreva.30.3198

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An asymptotic evaluation of the specific heat of an ideal relativistic Bose gas confined to a cuboidal enclosure (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​) 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=(T\ensuremath−Tc)Tct=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}t=Tc​(T\ensuremath−Tc​)​, such that ∣t∣\ensuremath≪1|t|\ensuremath{\ll}1∣t∣\ensuremath≪1 and ∣Lit∣\ensuremath≫1|{L}_{i}t|\ensuremath{\gg}1∣Li​t∣\ensuremath≫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\ensuremathρ∗{c}_{\ensuremath{\rho}}^{*}c\ensuremathρ∗​ of the specific-heat maximum; finite-size corrections in this region turn out to be O(L<\ensuremath−1)O({L}_{<}^{\ensuremath{-}1})O(L<\ensuremath−1​), where L<{L}_{<}L<​ denotes the shortest side of the enclosure.
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 Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=Tc{T}_{c}Tc​, is examined in the context of an ideal relativistic Bose gas confined to a cuboidal enclosure (L1{L}_{1}L1​\ifmmode\times\else\texttimes\fi{}L2{L}_{2}L2​\ifmmode\times\else\texttimes\fi{}L3{L}_{3}L3​) under periodic boundary conditions. Taking into account the possibility of particle-antiparticle pair production in the system, explicit expressions are derived for the free energy, the specific heat, and the condensate density at temperatures close to Tc{T}_{c}Tc​, and the special cases of a cube, a square channel and a film are investigated at length. The various predictions of the Privman-Fisher hypothesis are fully borne out and the scaling functions governing the critical behavior of the system are found to be universal---irrespective of the severity of the relativistic effects. The influence of the latter enters only through the nonuniversal scale factors, C1{C}_{1}C1​ and C2{C}_{2}C2​, which depend on the particle mass m and density \ensuremath{\rho} as well.
Scaling hypothesis on the ``singular'' part of the free-energy density of a finite system is examined in the context of a Bose gas confined to an enclosure of size ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$, with 2<d<4 and d'\ensuremath{\le}2, under periodic boundary conditions. Finite-size effects in the various thermodynamic properties of the system, such as the specific heat, the isothermal compressibility, and the condensate density, are predicted in the regions of both first-order (TTc{T}_{c}Tc​) and second-order (T\ensuremath{\simeq}Tc{T}_{c}Tc​) phase transition. To test these predictions, a detailed analytical study is carried out in the case of an ideal relativistic Bose gas, which includes the possibility of particle-antiparticle pair production in the system. The various predictions of the scaling hypothesis are fully borne out and the scaling functions governing the critical behavior of the system are found to be universal---irrespective of the severity of the relativistic effects. The influence of the latter enters only through nonuniversal parameters, C${\ifmmode \tilde{}\else \~{}\fi{}}_{1}$t\ifmmode \tilde{}\else \~{}\fi{} and C${\ifmmode \tilde{}\else \~{}\fi{}}_{2}$, which depend on the particle mass m and density \ensuremath{\rho} as well.
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.
An asymptotic evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) is carried out, under a variety of boundary conditions, without converting summations into integrations. The theoretical results for the Dirichlet boundary conditions (${\ensuremath{\psi}}_{S}=0$) contain all the significant features of the numerical results obtained earlier by Goble and Trainor; in particular, we reproduce the characteristic length D∗{D}^{*}D∗ at which the specific heat of the system is at an absolute maximum. It turns out that D∗{D}^{*}D∗ is directly proportional to the mean interparticle distance l‾\overline{l}l, with a constant of proportionality $c(\ensuremath{\approx}20\ensuremath{-}30)$. No such characteristic length appears if boundary conditions other than Dirichlet's are employed. The shift, and the rounding off, of the specific-heat maximum are also studied, and the distinguishing influence of the boundary conditions examined.
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 .