The canonical equation

The Schubart Master Formula

The Schubart Master Formula is one canonical reference equation — the compact volume form below. Everything further on this page and across the reference explains, decomposes and deepens it, but never replaces it. One formula, one canonical notation, one unmistakable reference.

The canonical Schubart Master Formula · base equation

The usable power $P(t)$ equals the conversion efficiency $\eta$ times the integral, over the interaction volume $V$ , of the effective ambient flux density $\Phi_{\text{eff}}(\mathbf{r},t)$ and the effective coupling cross-section $\sigma_{\text{eff}}(E)$ .

$P(t)$ Usable power output

The instantaneous usable electrical power delivered by the system, as a function of time.

$\eta$ Conversion efficiency

The dimensionless fraction of the coupled flux that becomes usable electrical output (0 ≤ η ≤ 1).

$\Phi_{\text{eff}}(\mathbf{r},t)$ Effective flux density

The effective ambient radiation flux at position 𝐫 inside the converter and at time t.

$\sigma_{\text{eff}}(E)$ Effective cross-section

The effective coupling of the field to the converter as a function of energy E.

$\int_V \, dV$ Volume integral

Integration over the interaction volume V of the converter.

$\mathbf{r}$ Position vector

The (bold) position vector within the interaction volume V.

$E$ Energy

The energy at which the effective cross-section is evaluated.

Expanded physical representation

To make the physics explicit, the canonical form can be unpacked into its spectral, angular and transport structure. This expanded physical representation — a spectral–transport decomposition — resolves the effective flux and coupling into an integral over energy and incident direction, with the directed momentum transfer and the lattice transport shown as separate factors. It explains and deepens the canonical equation; it is a representation of it, not a second master formula. Every symbol below is live text: selectable, searchable and indexable, never an image.

Expanded physical representation · spectral–transport decomposition

$$P(t) \;=\; \iint \Phi(E,\theta,t)\,\sigma(E)\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\;dE\,d\theta \tag{2}$$

Power as a function of time equals the double integral, over energy $E$ and direction $\theta$ , of the radiation flux density times the scattering cross-section times the momentum transfer times the phonon velocity times the conversion efficiency.

Every factor of the expansion

Each factor is one link in the conversion chain. Read left to right, the expanded representation tells a single story: how much of the invisible ambient field arrives, how strongly it couples to the converter, how much directed momentum each interaction delivers, how quickly the lattice carries it, and what fraction survives as usable power.

$P(t)$ Power output $\mathrm{W}$

P(t) is the quantity the whole framework accounts for: the time-dependent usable power a device extracts from the ambient field. Writing it as an explicit function of time distinguishes this framework from steady-state efficiency limits and makes the balance an open-system, non-equilibrium statement.

$\Phi(E,\theta,t)$ Radiation flux density $\mathrm{m^{-2}\,s^{-1}\,sr^{-1}\,J^{-1}}$

Φ is the differential flux of the ambient field: the number of incident quanta per unit area, time, solid angle and energy. Resolving it in energy E and angle θ — and allowing slow variation in t — is what lets the framework describe an anisotropic, spectrally structured environment rather than a single average intensity.

$\sigma(E)$ Scattering cross-section $\mathrm{m^{2}}$

σ(E) encodes the probability that an incident quantum of energy E interacts with the converter. It is the material- and process-specific coupling term. Because σ depends on E, the useful part of the spectrum is selected by the physics of the interaction, not assumed.

$\Delta p$ Momentum transfer $\mathrm{kg\,m\,s^{-1}}$

Δp is the momentum imparted per interaction. It is the term that breaks symmetry: without a net directed momentum transfer the interactions would average to zero current. Δp is where the converter geometry and asymmetry enter the balance.

$v_{\text{ph}}$ Phonon velocity $\mathrm{m\,s^{-1}}$

v_ph is the group velocity at which the lattice transports the transferred momentum as collective vibrational modes. It converts a per-interaction push into a transport rate, linking the microscopic event to a macroscopic current. High, coherent transport velocities favour conversion.

$\eta_{\text{conv}}$ Conversion efficiency $\text{dimensionless}$

η_conv is the dimensionless fraction of the transported flux that is rectified into directed, usable electrical power rather than lost to heat. It bundles the rectification, extraction and impedance-matching stages. As a bounded factor 0 ≤ η_conv ≤ 1 it keeps the balance physically admissible.

$E$ Energy variable $\mathrm{J}$

E is the spectral integration variable. Integrating over E sums the contributions of every energy channel the field carries, weighted by how strongly the material responds at that energy.

$\theta$ Angular variable $\mathrm{sr}$

θ is the angular integration variable. Integrating over θ sums contributions arriving from every direction, capturing the anisotropy of the ambient field and of the converter's acceptance.

Frequently asked questions

What is the Schubart Master Formula?

The Schubart Master Formula, written P(t), is a general mathematical framework for the usable electrical power a converter draws from continuous ambient energy conversion in an open system. Its one canonical form is the compact volume equation P(t) = η · ∫_V Φ_eff(𝐫,t) · σ_eff(E) dV.

What does P(t) stand for?

P(t) is the instantaneous usable power output as a function of time t, measured in watts. It equals the conversion efficiency η times the integral, over the interaction volume V, of the effective ambient flux density Φ_eff and the effective coupling cross-section σ_eff.

Is the Schubart Master Formula specific to neutrinovoltaics?

No. It is a technology-neutral accounting identity. Established cases such as photovoltaic and thermoelectric harvesting are recovered by particular choices of its parameters; neutrinovoltaics is one possible application, never the definition of the equation.

What is the difference between the canonical form and the expanded physical representation?

The canonical form is the single reference equation, the compact volume integral. The expanded physical representation is a spectral–transport decomposition that unpacks it into an integral over energy and incident direction; it explains the canonical equation but does not replace it and is not itself called canonical.