Non-Markovian Quantum Decay in Complex Environments: A Hyperstatistical Approach

Abstract

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 solve 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 finite for $q<2$. The convergence of the second moment instead requires the stricter condition $q<3/2$. 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. This result provides a physical interpretation linking extreme environmental complexity to Anderson localization and Griffiths-like phases.