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

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.