Authors: Beckingham, CD, Allan Christopher
Source: Zenodo (CERN European Organization for Nuclear Research) (article, 2025-12-07)
Field: Chemistry — Physical and Theoretical Chemistry

Derivation of Fröhlich Condensate Equations Title: Derivation of Fröhlich Condensate Equations Creators/Authors: Chris Beckingham, CD (Lead Architect; Coherence Dynamic Laboratory | Think Tank Consultancy, Technology, Artificial Intelligence, Leadership, Training, Communication) Description/Abstract: This manuscript provides a rigorous step-by-step derivation of the Fröhlich condensate equations, describing a nonequilibrium Bose-Einstein-like condensation of phonons in biological systems. Proposed by Herbert Fröhlich in 1968, this phenomenon involves metabolic energy pumping leading to preferential occupation of the lowest-frequency vibrational mode at room temperature. The derivation begins with the microscopic Wu-Austin Hamiltonian and evolves into rate equations for phonon occupations, demonstrating a phase transition to condensation under sufficient pumping rates. Key concepts include open quantum system dynamics, Lindblad master equations, and biological applications such as efficient energy transport in proteins and membranes. This work bridges quantum mechanics and biology, offering insights into coherent vibrational states in living systems. The document includes symbolic solutions, solvency proofs, and ties to modern quantum biology experiments, making it a foundational reference for researchers in nonequilibrium quantum phenomena. Publication Date: December 07, 2025 Keywords: Fröhlich condensate; Bose-Einstein condensation; phonons; vibrational modes; nonequilibrium dynamics; Wu-Austin Hamiltonian; Lindblad master equation; quantum biology; biological coherence; phase transition; metabolic pumping; open quantum systems Resource Type: Publication – Working Paper / Technical Report License: Creative Commons Attribution 4.0 International (CC-BY-4.0) Related Identifiers: References: Fröhlich, H. (1968). Long-range coherence and energy storage in biological systems. International Journal of Quantum Chemistry, 2(5), 641–649. (DOI: 10.1002/qua.560020505) References: Wu, T. M., & Austin, S. (1977). Fröhlich's theory of coherent excitation—A critical examination. Journal of Theoretical Biology, 67(4), 741–752. (DOI: 10.1016/0022-5193(77)90254-8) References: Reimers, J. R., et al. (2009). Weak, strong, and coherent regimes of Fröhlich condensation and their applications to terahertz medicine and quantum consciousness. Proceedings of the National Academy of Sciences, 106(11), 4219–4224. (DOI: 10.1073/pnas.0806273106) References: Unified Coherence Law (2025). Zenodo Record https://doi.org/10.5281/zenodo.17847746 Communities: Quantum Physics; Biological Physics; Open Quantum Systems; Theoretical Biology; Coherence Dynamics Funding: Independent (No external funding declared) Version: v1.0 Notes: This 4-page technical derivation is part of a series on coherence in living systems, extending concepts from the Unified Coherence Law (UCL). Supplementary materials include symbolic derivations and potential extensions to phyllotaxis-inspired quantum models. For reproducibility, equations are presented in executable form (e.g., for QuTiP simulations). Contact the lead researcher for collaborations on experimental validations, such as THz spectroscopy in biomolecules. This record emphasizes falsifiability through predicted condensation thresholds in pumped systems. Master equation ; Coherence (philosophical gambling strategy) ; Quantum ; Non-equilibrium thermodynamics ; Open quantum system ; Hamiltonian (control theory) ; Quantum dissipation ; Phonon Advanced Physical and Chemical Molecular Interactions ; Spectroscopy and Quantum Chemical Studies ; Quantum, superfluid, helium dynamics