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American Journal of Physics

Specific heat of an ideal Bose gas above the Bose condensation temperature

Frank Y.-H. Wang

Published Aug 13, 2004• 9 citations
Publisher

Abstract

An approximation for the specific heat of an ideal Bose as an explicit function of temperature above the condensation temperature Tc is derived. The formula yields the exact values of the specific heat and its first derivative at Tc and has an accuracy of at least 0.3% over the full range of temperatures.
DOI
10.1119/1.1764562

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The condensation of a non-relativistic ideal Bose gas in an Einstein universe is investigated. Explicit expressions for the condensate fraction N 0 /N and the specific heat are obtained by using the Poisson summation formula to express the summations as integrations plus corrections. It is shown that the finiteness of the system smoothes out the cusp-like singularity of the infinite system. A rigorous asymptotic analysis of the critical temperature and the specific heat maximum are given, and the relation with the scaling theory of finite size effects is briefly discussed.
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Our previous evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry is extended to second order in (l‾D\frac{\overline{l}}{D}Dl​), where l‾\overline{l}l is the mean interparticle distance. Explicit expressions, involving a dimensionless parameter yyy, are obtained both for the specific heat of the system and for the temperature at which the specific heat is maximum. Occurrence of the parameter yyy, which is a measure of the chemical potential of the system in terms of the spacing of the single-particle energy levels, enables us to cast our expression for the specific heat of the system into a form consistent with the scaling theory of Fisher and Barber. Our results for the specific-heat maximum, as a function of (Dl‾\frac{D}{\overline{l}}lD​), are a considerable improvement over the first-order ones reported earlier, and have come remarkably close to the ones obtained numerically by Goble and Trainor.
We compute the critical temperature T_c of a weakly interacting uniform Bose gas in the canonical ensemble, extending the criterion of condensation provided by the counting statistics for the uniform ideal gas. Using ordinary perturbation theory, we find in first order (Tc−Tc0)/Tc0=−0.93aρ1/3(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}(Tc​−Tc0​)/Tc0​=−0.93aρ1/3, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and ρρρ is the particle number density.