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Analytical computation of generalized Fermi-Dirac integrals by truncated Sommerfeld expansions. (English) Zbl 1305.82058
Summary: For the generalized Fermi-Dirac integrals, \(F_k({\eta},{\beta})\), of orders \(k=-1/2, 1/2, 3/2\), and \(5/2\), we explicitly obtained the first 11 terms of their Sommerfeld expansions. The main terms of the last three orders are rewritten so as to avoid the cancelation problem. If \({\eta}\) is not so small, say not less than 13.5, 12.0, 10.9, and 9.9 when \(k=-1/2, 1/2, 3/2\), and \(5/2\), respectively, the first 8 terms of the expansion assure the single precision accuracy for arbitrary value of \({\beta}\). Similarly, the 15-digits accuracy is achieved by the 11 terms expansion if \({\eta}\) is greater than 36.8, 31.6, 30.7, and 26.6 when \(k=-1/2, 1/2, 3/2\), and \(5/2\), respectively. Since the truncated expansions are analytically given in a closed form, their computational time is sufficiently small, say at most 4.9 and 6.7 times that of the integrand evaluation for the 8- and 11-terms expansions, respectively. When \({\eta}\) is larger than a certain threshold value as indicated, these appropriately-truncated Sommerfeld expansions provide a factor of 10-80 acceleration of the computation of the generalized Fermi-Dirac integrals when compared with the direct numerical quadrature.

MSC:
82D20 Statistical mechanical studies of solids
65D30 Numerical integration
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