黑洞视界附近非广延玻色-费米态计数中的处方依赖性

中文摘要 AI 辅助

在单函数、非极值、静态、球对称类黑洞视界附近，非广延玻色和费米态计数已被充分研究。计算围绕一个$q$-变形巨正则系综展开，其总配分函数是单能级配分的$q$-乘积。这确定了玻色子占据数求和为主要能级配分，并使我们能够区分两个常被混淆的操作：将该求和替换为几何级数形式，以及对$q$-对数应用普通倒数对数恒等式。一个有限能量隧穿处方给出了由表面引力和视界响应因子控制的近视界态计数主导项。熵匹配随后给出一个与霍金温度成正比的能量尺度；几何因子在归一化的非广延修正和玻色-费米比中相互抵消。我们解析地推导了一阶系数，并数值评估了未展开的$q$-积分。当$q$在单位值的大约百分之二范围内时，线性近似精确到优于百分之一。所有处方都恢复了广延极限，但精确的玻色子能级求和与几何近似对传统$7/8$玻色-费米关系的主导变形给出了相反的符号。结果分离出一个稳健的几何因子化结构，并量化了一个独特的$q$-代数系统不确定性。

英文摘要

Nonextensive Bose and Fermi state is well studied with counting near black-hole horizons in a one-function, nonextremal, static, spherically symmetric class. The calculation is organized around a $q$-deformed grand ensemble whose total partition function is a $q$-product of single-level partitions. This identifies the bosonic occupation-number sum as the primary level partition and allows us to separate two operations that are often conflated: replacing that sum by a geometric-series form and applying the ordinary reciprocal-log identity to a $q$-logarithm. A finite-energy tunneling prescription yields a leading near-horizon state-counting term controlled by the surface gravity and a horizon-response factor. Entropy matching then gives an energy scale proportional to the Hawking temperature; the geometry factor cancels from normalized nonextensive corrections and from the Bose--Fermi ratio. We derive the first-order coefficients analytically and evaluate the unexpanded $q$-integrals numerically. The linear approximation is accurate to better than one percent when $q$ lies within roughly two percent of unity. All prescriptions recover the extensive limit, but the exact bosonic level sum and the geometric approximations give opposite signs for the leading deformation of the conventional $7/8$ Bose--Fermi relation. The result isolates a robust geometry-factorized structure and quantifies a distinct $q$-algebra systematic uncertainty.

发表机构

• School of Science, East China University of Technology（华东理工大学理学院）

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