Evidence Library

This chapter catalogues the established literature on which the mathematics of the Schubart Master Formula rests. It is not a bibliography for any device; it is a map from each term of the formula back to the body of accepted physics that gives that term meaning. The entries are grouped by the part of the formalism they support, and each carries a one- or two-sentence note on why it is relevant and which formula factor it underwrites. A firm distinction is maintained throughout: the works cited here are established physics, broadly agreed and widely taught; the framework that assembles them into $P(t)$ is the proposed contribution of this reference and is not itself claimed to be established. DOIs are given only where they are certain; where a canonical work is well known but a specific identifier cannot be stated with confidence, none is invented. The formula being underwritten is the one at the equation.

Non-equilibrium thermodynamics and open systems

These works underwrite the framework’s core premise — that an open, driven system can convert continuously — and bear on the assumptions behind $\Delta p$ and the overall thermodynamic status (open thermodynamic systems).

• Onsager, L. (1931). Reciprocal Relations in Irreversible Processes, I and II. Physical Review, vol. 37 and vol. 38. Establishes the symmetry relations governing coupled irreversible flows near equilibrium. Relevant to the coupling between the incident flux and the directed response — the conceptual ground beneath treating $\Delta p$ as a driven, cross-coupled flux rather than an equilibrium fluctuation.
• Prigogine, I. (1967). Introduction to the Thermodynamics of Irreversible Processes, 3rd ed. Interscience/Wiley. The standard development of entropy production and steady states in open systems maintained away from equilibrium. Underwrites the framework’s claim that continuous conversion with entropy export violates no law — the premise of the whole formula and directly of the “open, non-equilibrium” assumption.
• de Groot, S. R. & Mazur, P. (1962). Non-Equilibrium Thermodynamics. North-Holland. A comprehensive account of fluxes, forces and entropy production in continuous open systems. Provides the formal setting in which a factor like $\Delta p$ — a directed flux driven by an external force — is a well-defined non-equilibrium quantity.

Fluctuation–dissipation and stochastic rectification (ratchets)

These works underwrite the symmetry-breaking factor $\Delta p$: they establish both that symmetric fluctuations cannot yield directed transport and that broken symmetry can (non-equilibrium energy conversion).

• Callen, H. B. & Welton, T. A. (1951). Irreversibility and Generalized Noise. Physical Review, vol. 83, p. 34. The fluctuation–dissipation theorem, linking equilibrium fluctuations to dissipative response. Relevant because it delimits precisely what equilibrium fluctuations can and cannot do, making clear why a non-equilibrium drive is required for a non-zero $\Delta p$.
• Feynman, R. P., Leighton, R. B. & Sands, M. (1963). The Feynman Lectures on Physics, Vol. I, Ch. 46 (“Ratchet and Pawl”). Addison-Wesley. The canonical demonstration that a ratchet in thermal equilibrium extracts no net work, and that directed motion requires a temperature difference or other non-equilibrium condition. Directly underwrites the ”$\Delta p \to 0$ at equilibrium” property of the framework.
• Reimann, P. (2002). Brownian Motors: Noisy Transport Far from Thermal Equilibrium. Physics Reports, vol. 361, pp. 57–265. A comprehensive review of how broken spatial or temporal symmetry rectifies unbiased fluctuations into directed transport. This is the modern theoretical basis for the framework’s central mechanism: no broken symmetry, no directed $\Delta p$.
• Astumian, R. D. (1997). Thermodynamics and Kinetics of a Brownian Motor. Science, vol. 276, pp. 917–922. A widely cited treatment of stochastic rectification in asymmetric systems. Reinforces the requirement that symmetry breaking, not mere fluctuation, is what permits a net directed contribution to survive averaging.

Scattering theory and cross-sections

These works underwrite the coupling factor $\sigma(E)$: the cross-section is a well-defined, energy-dependent measure of interaction strength (variable definitions, sigma analysis).

• Mott, N. F. & Massey, H. S. W. (1965). The Theory of Atomic Collisions, 3rd ed. Oxford University Press. A foundational treatment of scattering cross-sections and their energy dependence. Underwrites the definition and units of $\sigma(E)$ as an effective interaction area varying with energy.
• Sakurai, J. J. (1994). Modern Quantum Mechanics, rev. ed. Addison-Wesley. Standard textbook development of scattering amplitudes, partial waves and the optical theorem. Provides the quantum-mechanical grounding for $\sigma(E)$ and for the notion that coupling strength depends on the energy of the incident field.
• Newton, R. G. (1982). Scattering Theory of Waves and Particles, 2nd ed. Springer. A rigorous reference for the mathematical structure of scattering and cross-sections. Supports the claim that $\sigma(E)$ is a well-posed quantity, relevant to the well-definedness discussion in scientific questions.

Lattice and phonon transport

These works underwrite the transport factor $v_{\text{ph}}$: the velocity at which the lattice carries transferred momentum is a defined, measurable property of the crystal.

• Ziman, J. M. (1960). Electrons and Phonons: The Theory of Transport Phenomena in Solids. Oxford University Press. The classic account of phonon transport and lattice dynamics. Underwrites the identification of $v_{\text{ph}}$ as a group/sound velocity governing how quickly directed momentum is carried from the interaction site.
• Ashcroft, N. W. & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston. A standard graduate text covering lattice vibrations, phonon dispersion and sound velocity. Provides the textbook basis for treating $v_{\text{ph}}$ as a well-defined material transport parameter.
• Kittel, C. (2005). Introduction to Solid State Physics, 8th ed. Wiley. Widely used introduction to phonons and lattice transport. Reinforces the physical meaning and typical magnitudes of the lattice velocity that enters the framework as $v_{\text{ph}}$.

Efficiency and conversion limits (for comparison)

These works underwrite the bounded conversion factor $\eta_{\text{conv}} \in [0,1]$ and provide the established efficiency ceilings against which the framework is positioned (governing relations).

• Carnot, S. (1824). Réflexions sur la puissance motrice du feu. Bachelier, Paris. The origin of the maximum efficiency of a heat engine operating between two reservoirs. Establishes the archetype of a governing efficiency bound and constrains the interpretation of $\eta_{\text{conv}}$ as a fraction that cannot exceed physical limits.
• Shockley, W. & Queisser, H. J. (1961). Detailed Balance Limit of Efficiency of p-n Junction Solar Cells. Journal of Applied Physics, vol. 32, no. 3, p. 510. doi:10.1063/1.1736034. The detailed-balance efficiency ceiling for single-junction photovoltaic conversion. A concrete, rigorously derived bound of exactly the kind that would enter a photovoltaic application through the value of $\eta_{\text{conv}}$, and the principal comparison case for the framework.
• Landsberg, P. T. & Tonge, G. (1980). Thermodynamic Energy Conversion Efficiencies. Journal of Applied Physics, vol. 51, R1. A thermodynamic treatment of upper bounds on radiation-to-work conversion. Supports the framework’s insistence that $\eta_{\text{conv}}$ is bounded by thermodynamics, relevant to the “too small / bounded output” discussion in scientific questions.

Experimental context for the neutrinovoltaic application

The works above underwrite the general mathematics. The following independently published results are cited specifically in the context of the neutrinovoltaic application — the one application that motivated the formula — and are compiled from the Scientific Advisory Board statement reproduced at scientific questions. They confirm component mechanisms; they do not confirm any integrated device at scale, a distinction that page states in full.

• Freedman, D. Z. (1974). Coherent Effects of a Weak Neutral Current. Physical Review D, vol. 9, p. 1389. First prediction of coherent elastic neutrino–nucleus scattering (CEνNS) — the interaction the application invokes as its coupling channel.
• COHERENT Collaboration (2017). Observation of Coherent Elastic Neutrino–Nucleus Scattering. Science, vol. 357, pp. 1123–1126. First experimental confirmation of CEνNS at above six sigma; the cross-section scales with the square of the neutron number.
• COHERENT Collaboration (2021). First Measurement of Coherent Elastic Neutrino–Nucleus Scattering on Argon. Physical Review Letters, vol. 126, 012002. Independent confirmation on a second target nucleus.
• CONUS+ Collaboration (2025). Reactor CEνNS in the coherent regime. Confirmation of CEνNS with reactor-source neutrinos, an independent flux and detector.
• Thibado, P. M. et al. (2020). Fluctuation-Induced Current from Freestanding Graphene. Physical Review E, vol. 102, 042101. Continuous measurable current from driven lattice motion in a circuit with a load — evidence bearing on the stochastic-rectification assumption behind $\Delta p$ and $\eta_{\text{conv}}$.
• Nobel Committee for Physics (2015). Scientific Background: Neutrino Oscillations. Royal Swedish Academy of Sciences. Establishes neutrino mass, part of the scientific lineage of the name “neutrinovoltaic”.
• Prigogine, I. (1977). Nobel Lecture: Time, Structure and Fluctuations. Royal Swedish Academy of Sciences. The thermodynamics of dissipative structures in open, driven systems — the basis for the second-law argument in scientific questions.
• Particle Data Group (2022). Review of Particle Physics (cosmic-ray muon flux). Establishes one of the ambient input channels as continuously measurable.

How to read this library

Every entry above describes physics that is established and generally accepted. What is not established by any of them, singly or together, is the composite claim of the Schubart Master Formula — that these ingredients combine, with the stated factor values, to yield a given $P(t)$ in a real converter. The literature legitimises each term of the formula; it does not legitimise the assembled prediction, which remains a model projection pending the independent, end-to-end experimental validation called for in scientific questions. Readers extending this library should hold to the same standard applied here: cite only real, widely accepted sources, attach an identifier only when it is certain, and keep the boundary between established physics and proposed framework visible at all times.