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Journal of Physics A Mathematical and General

Finite-size scaling for a relativistic Bose gas in an Einstein universe

S Singh, Raj K. Pathria

Published Dec 21, 1987• 9 citations
Publisher

Abstract

The scaling hypothesis on the 'singular' part of the free-energy density of a finite system is examined in the context of a relativistic Bose gas confined to an Einstein universe of radius R. Finite-size effects in the various thermodynamic properties of the system, such as the free energy, the specific heat, the isothermal compressibility and the condensate density, are predicted in the regions of both first-order (T<T c ) and second-order (T approximately=T c ) phase transitions. To test these predictions, a detailed analytical study is carried out 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 behaviour of the system are found to be universal, irrespective of the severity of the relativistic effects.

DOI
10.1088/0305-4470/20/18/035

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The scaling hypothesis on the 'singular' part of the free-energy density of a finite system is examined in the context of a relativistic Bose gas confined to an Einstein universe of radius R. Finite-size effects in the various thermodynamic properties of the system, such as the free energy, the specific heat, the isothermal compressibility and the condensate density, are predicted in the regions of both first-order (T<T c ) and second-order (T approximately=T c ) phase transitions. To test these predictions, a detailed analytical study is carried out 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 behaviour of the system are found to be universal, irrespective of the severity of the relativistic effects.
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.
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.
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 condensation of a non-relativistic ideal Bose gas in an Einstein universe is investigated. Explicit expressions for the condensate fraction N 0 /N and the specific heat are obtained by using the Poisson summation formula to express the summations as integrations plus corrections. It is shown that the finiteness of the system smoothes out the cusp-like singularity of the infinite system. A rigorous asymptotic analysis of the critical temperature and the specific heat maximum are given, and the relation with the scaling theory of finite size effects is briefly 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.
An ideal gas of relativistic bosons trapped in a D -dimensional power-law potential is explored, where the possibility of particle–antiparticle pair production is taken into account. Expressions for several important parameters, such as the critical temperature of Bose–Einstein condensation (BEC), ground-state fraction and heat capacity, are analytically derived and used to discuss the relevant properties of the system. An important parameter χ is introduced to mark the extent of the relativistic effects on the characteristics of BEC. It is found that the correction of the critical temperature due to the relativistic effects is considerable for the system with a large parameter χ. It is more important to show that the pair production may lead to several novel characteristics for a relativistic Bose gas, such as the high temperature BEC, the rapid increase of the heat capacity at high temperatures and the disappearance of the anomaly of the heat capacity at the critical temperature in the ultrarelativistic limit.