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Physical Review A

Finite-size scaling for a relativistic Bose gas with pair production

Surjit Singh, Raj K. Pathria

Published Jun 1, 1987• 4 citations
Publisher

Abstract

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 (T${T}_{c}$) and second-order (T\ensuremath{\simeq}${T}_{c}$) 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.

DOI
10.1103/physreva.35.4814

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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 Ld\ensuremath−d′{L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}Ld\ensuremath−d′\ifmmode\times\else\texttimes\fi{}\ensuremath∞d′{\ensuremath{\infty}}^{d\mathcal{'}}\ensuremath∞d′, 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~\else˜\fi1{\ifmmode \tilde{}\else \~{}\fi{}}_{1}\ifmmode~\else˜\fi1​t\ifmmode \tilde{}\else ~{}\fi{} and C\ifmmode~\else˜\fi2{\ifmmode \tilde{}\else \~{}\fi{}}_{2}\ifmmode~\else˜\fi2​, which depend on the particle mass m and density \ensuremath{\rho} as well.
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.
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.
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.
We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for T>Tc{\mathrm{T}}_{\mathrm{c}}Tc​(\ensuremath{\infty}) the approach is exponential, for TTc{\mathrm{T}}_{\mathrm{c}}Tc​(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.
An ideal relativistic Bose-gas model with pair production is solved exactly in ddd dimensions. The critical behavior is shown to be the same as the usual nonrelativistic Bose gas. Scaling functions (including nonuniversal parameters) for the equation of state and the correlation functions are obtained in a closed form. The scale-factor universality is verified by the formation of the expected universal ratios. Various limiting cases of the model depending on the relative magnitudes of the three basic length scales, i.e., Compton wave-length, thermal wavelength, and interparticle distance are discussed. In these limiting cases, previously known results are recovered, wherever these are available.