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 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 ⟨ t⟩ = ∫0∞ P(t) dt= [⟨Γ⟩(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≤ q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
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BibTeX
@misc{fabiano2026nonmarkovian,
title = {{Non-Markovian Quantum Decay in Complex Environments: A Hyperstatistical Approach}},
author = {Nicola Fabiano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22190},
eprint = {2609.22190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22190},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-Markovian Quantum Decay in Complex Environments: A Hyperstatistical Approach
AU - Nicola Fabiano
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.22190
UR - https://arxiv.org/abs/2609.22190
AB - 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 ⟨ t⟩ = ∫_0^∞ P(t) dt= [⟨Γ⟩(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≤ q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
ER -