P(t) Schubart Master Formula
English

General Equation for Continuous Ambient Energy Conversion

P(t)

Schubart Master Formula

A general mathematical framework for the continuous conversion of ambient energy in open systems.

The canonical equation

P(t)  =  η⋅∫VΦeff(r,t)⋅σeff(E) dVP(t) \;=\; \eta \cdot \int_V \Phi_{\text{eff}}(\mathbf{r},t) \cdot \sigma_{\text{eff}}(E) \, dV

The canonical Schubart Master Formula: the usable power equals the conversion efficiency times the integral, over the converter's interaction volume, of the effective ambient flux density and the effective coupling cross-section. The equation page unpacks this into its expanded physical representation.

Where to begin

01 · The equation Read it term by term The complete formula with every variable explained on hover, in publication typography. 02 · The interpretation Why the framework exists What problem P(t) solves, and why a general, technology-neutral equation is necessary. 03 · The knowledge base Twenty-five chapters From physical motivation to the full derivation, the parameters in depth, and validation. 04 · Interactive Explore the parameters Move each factor on a log scale and watch which lever decides the order of magnitude.

The five factors of the expansion

The expanded physical representation resolves the canonical form into five factors — each one link in the conversion chain, and each with its own chapter, read at three depths: intuition, physics and mathematics.

  1. Radiation flux density How much ambient radiation arrives, per energy and per direction.
  2. Scattering cross-section How strongly the incoming field couples to the converter material.
  3. Momentum transfer The directed push each interaction delivers to the lattice.
  4. Phonon velocity How fast the lattice carries the transferred momentum onward.
  5. Conversion efficiency The fraction of transported momentum that becomes usable power.

P(t) is presented as a general accounting identity, not a description of one device. Established special cases — photovoltaic, thermoelectric and related harvesting limits — are recovered by particular choices of its parameters. Neutrinovoltaics is one possible application, never the definition.

See the full derivation → Read the open questions → Cite this work →

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