发表机构
“Vinča” Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade（“文查”核科学研究所 - 塞尔维亚共和国国家研究所，贝尔格莱德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出超统计框架，通过γ分布和q指数函数解析复杂环境中的非马尔可夫量子衰变，给出平均寿命与涨落的收敛条件，并关联安德森局域化与格里菲斯相。
AI 中文摘要
不稳定量子态的指数衰变，如费米黄金规则等标准马尔可夫理论所描述的，假设环境简单且无结构。然而，在具有无序、长程相互作用或强涨落的复杂环境中，局部衰变率会发生涨落，导致非马尔可夫动力学和幂律“长时尾部”。在本文中，我们应用最近提出的超统计框架来解决此类复杂环境中的量子衰变问题。通过考虑介观域上局部衰变率的γ分布，我们推导出由q指数函数控制的宏观存活概率。然后，我们使用q广义伽马函数（通过q指数的梅林变换定义）来计算衰变时间分布的矩。我们证明，对于q<2，平均量子寿命是有限的。而二阶矩的收敛则需要更严格的条件q<3/2。对于1<q<3/2，平均寿命及其方差均有限；对于3/2≤q<2，平均寿命仍有限，但寿命涨落变得无限宽；对于q≥2，平均寿命本身发散。这一结果为极端环境复杂度与安德森局域化和格里菲斯相之间的联系提供了物理解释。
英文摘要
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a $γ$-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a $q$-exponential function. We then use the $q$-generalized Gamma function, defined via the Mellin transform of the $q$-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is $\langle t\rangle = \int_0^\infty P(t)\,dt= [\langleΓ\rangle(2-q)]^{-1},$ and is therefore finite for $q<2$. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition $q<3/2$. These results provide a refined physical interpretation: for $1<q<3/2$ both the mean lifetime and its variance are finite; for $3/2\le q<2$ the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for $q\ge2$ the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
Comments8 pages, revised abstract and section 4.1, added some formulae numbering
