非广延量子统计中的玻色-爱因斯坦和费米-狄拉克分布：精确方法和插值方法
Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches
摘要：
非广延量子统计中的广义玻色-爱因斯坦（BE）和费米-狄拉克（FD）分布已经通过最大熵方法（MEM）和基于精确积分表示的最佳拉格朗日乘子来讨论[Rajagopal，Mendes，and Lenzi，Phys. Rev. Lett. { \bf 80}，3907（1998）]。结果表明，精确方法中的$(q-1)$展开式与对$O(q-1)$有效的渐近方法所得到的结果一致。模型计算已与一个统一的电子态密度和声子的德拜模型。基于精确方法的结果，我们提出了广义分布的{ \it插值近似}，它在$O(q-1)$范围内和高低温范围内得到了与精确方法一致的结果.用精确法、插值法、因式分解法和超统计法计算了广义索末菲展开式中的系数，以及低温下的电子和声子比热。四种方法的比较表明，插值逼近在非广延量子统计中具有潜在的用途。对基于精确方法的广义分布的$(q-1)$展开式，利用非归一化最大熵方法作了补充讨论，其结果也与渐近方法的结果一致。
英文摘要：
Generalized Bose-Einstein (BE) and Fermi-Dirac (FD) distributions in nonextensive quantum statistics have been discussed by the maximum-entropy method (MEM) with the optimum Lagrange multiplier based on the exact integral representation [Rajagopal, Mendes, and Lenzi, Phys. Rev. Lett. {\bf 80}, 3907 (1998)]. It has been shown that the $(q-1)$ expansion in the exact approach agrees with the result obtained by the asymptotic approach valid for $O(q-1)$. Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. Based on the result of the exact approach, we have proposed the {\it interpolation approximation} to the generalized distributions, which yields results in agreement with the exact approach within $O(q-1)$ and in high- and low-temperature limits. By using the four methods of the exact, interpolation, factorization and superstatistical approaches, we have calculated coefficients in the generalized Sommerfeld expansion, and electronic and phonon specific heats at low temperatures. A comparison among the four methods has shown that the interpolation approximation is potentially useful in the nonextensive quantum statistics. Supplementary discussions have been made on the $(q-1)$ expansion of the generalized distributions based on the exact approach with the use of the un-normalized MEM, whose results also agree with those of the asymptotic approach.
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