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Physical Review Letters

Thermodynamics of an Ultrarelativistic Ideal Bose Gas

Howard E. Haber, H. Arthur Weldon

Published Jun 8, 1981• 225 citations
Publisher

Abstract

We discuss the properties of an ideal relativistic Bose gas with nonzero chemical potential $\ensuremath{\mu}$ (i.e., with net charge) but differ from all previous discussions by including the effects of antiparticles. We obtain for the first time the relevant high-temperature expansion, emphasizing the constraint $|\ensuremath{\mu}|<~m$, and show that a box of massless particles can have a net charge even though $\ensuremath{\mu}=0$. Finally, we discuss the properties of Bose-Einstein condensation for both massless and massive bosons in $d$ space dimensions.

DOI
10.1103/physrevlett.46.1497

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We discuss the properties of an ideal relativistic Bose gas with nonzero chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ (i.e., with net charge) but differ from all previous discussions by including the effects of antiparticles. We obtain for the first time the relevant high-temperature expansion, emphasizing the constraint ∣\ensuremathμ∣< m|\ensuremath{\mu}|<~m∣\ensuremathμ∣< m, and show that a box of massless particles can have a net charge even though \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0. Finally, we discuss the properties of Bose-Einstein condensation for both massless and massive bosons in ddd space dimensions.
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.
The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. The limitations and errors of previous calculations which relied on the binomial expansion of the density of states are pointed out. It is shown that the behaviour of the system in the limit of zero temperature is dominated by the non-relativistic effects.
We investigate the thermodynamics of a self-interacting relativistic charged scalar field in the presence of weak disorder. We consider quenched disorder which couples linearly to the mass of the scalar field. After performing noise averages over the free energy of the system, we find that disorder increases the mean-field critical temperature for Bose–Einstein condensation at finite density in the ultrarelativistic limit. In turn, preliminary non-relativistic calculations indicate that the presence of randomness affects the Bose gas in the opposite way in such a limit, i.e. disorder reduces the mean-field condensation temperature. The effect of disorder on the temperature dependence of the chemical potential for a fixed charge density is investigated. Significant differences from the mean-field temperature dependence of the chemical potential are observed as the strength of the noise intensity increases. Finally, the temperature dependence of the chemical potential with fixed total charge and entropy is investigated. It is found that there is no Bose–Einstein condensation for a fixed charge to entropy ratio in the presence of weak disorder. The possible relevance of the findings in the present paper in different areas is discussed.