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Physical Review A

Scaling and universality of thermodynamics and correlations of an ideal relativistic Bose gas with pair production

Surjit Singh, P. N. Pandita

Published Sep 1, 1983• 26 citations
Publisher

Abstract

An ideal relativistic Bose-gas model with pair production is solved exactly in $d$ 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.

DOI
10.1103/physreva.28.1752

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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.
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.
Critical exponents characterize the divergent scaling of thermodynamic quantities near phase transitions and allow for the classification of physical systems into universality classes. While quantum gases thermalizing by interparticle interactions fall into the XY model universality class, the ideal Bose gas has been predicted to form a distinct universality class whose signatures have not yet been revealed experimentally. Here, we report the observation of critical scaling in a two-dimensional trapped quantum gas of essentially noninteracting photons, which thermalize by radiative contact to a reservoir of molecules inside a microcavity. By measuring the spatial correlations near the condensation transition, we determine the critical exponent for the correlation length to be ν=0.52(4)ν= 0.52(4)ν=0.52(4). Our results constitute a first experimental test of the long-standing scaling predictions for the Bose gas universality class.
We discuss the thermodynamics of a nonrelativistic gas of bosons with a local repulsive interaction. In particular, we compute the temperature and density dependences of pressure, energy and entropy densities, superfluid and condensate fractions, correlation length, specific heat, isothermal and adiabatic compressibilities, and various sound velocities. The $T\ensuremath{\rightarrow}0$ limit approaches the quantum phase transition, while the universal critical behavior of a classical second-order phase transition in the O(2) universality class determines the region around the critical temperature. Our nonperturbative flow equations based on exact functional renormalization cover all regions in the phase diagram.
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.
The thermodynamic properties of the noninteracting Bose gas in the neighborhood of its transition are examined in detail. The order parameter is a complex extensive variable, but the thermodynamic properties depend only on its amplitude under simple boundary conditions. As the dimensionality or the single-particle energy spectrum is varied, the critical singularity displays a variety of forms. The equation of state has a simple structure, different from the homogeneous form often discussed for critical systems but asymptotically reducing to the latter except when logarithmic singularities are involved. The correlation function in the critical region is a homogeneous function of the distance and a correlation length. Only for a quadratic energy spectrum is the Ornstein-Zernike theory result valid at the critical temperature. A precise correspondence is noted between the asymptotic properties of the ideal Bose gas transition and those of the spherical model of ferromagnetism.
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.