Privman-Fisher hypothesis on finite systems: Verification in the case of a relativistic Bose gas with pair production

The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=$T_c$, is examined in the context of an ideal relativistic Bose gas confined to a cuboidal enclosure ($L_1$×$L_2$×$L_3$) 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 $T_c$, 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, $C_1$ and $C_2$, which depend on the particle mass m and density ρ as well.