Evolutionary Quantum Synchronization Theory in Biological Systems

AbstractThis manuscript provides a complete, reproducible framework for studying thestability of quantum coherence and synchronization in biomolecules under environmentalconditions (T = 300K, aqueous environment). The document includesexplicit derivations of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) masterequation, extensions for non-Markovian effects (Nakajima-Zwanzig), full specificationof the Haken-Strobl-Reineker (HSR) and modified Dicke sector, includingthe electron-phonon interaction Hamiltonian He−ph and the Landau-Zener scatteringfunction. The formalism specifies jump operators, the spectral density ofthe environment, and numerical parameters for molecular dynamics (MD) andquantum integration. Entanglement metrics covered include **Wootters Concurrence**,**Logarithmic Negativity**, and the **PPT Criterion**, alongside coherencequantification using the **L1 norm and Hilbert-Schmidt norm**. The dephasingtime constant (τϕ) is used for comparison. Comprehensive statistical validationis included (bootstrap, **Posterior Predictive Checks (PPCs)**, **Kolmogorov-Smirnov test**, **Mann-Whitney U test**, **Welch’s t-test, ANOVA**), and **MonteCarlo Markov Chain (MCMC)** convergence analysis. Code snippets in QuTiP/Python,data examples, and an artifact manifest are provided for independent verification.