Full Derivation

This chapter derives the Schubart Master Formula from first principles, building it up one factor at a time rather than presenting it as a finished object. The goal is to make every step inspectable: what quantity we are after, how a single contribution to it is written, how those contributions are summed into an integral, which assumptions that summation rests on, and finally how the full integral collapses to the compact expression that the interactive explorer actually evaluates. The formula is stated in its canonical form at the equation, and every symbol is glossed at the notation reference; here we are concerned with where that form comes from.

Step (a): the target quantity

We seek the instantaneous usable power delivered by a converter that continuously harvests energy from an ambient field. Call this quantity $P(t)$, measured in watts. Two features of the target shape everything that follows. First, it is a power, an energy rate, not an energy — so whatever we integrate must already carry an inverse-time dimension. Second, it is instantaneous, carrying an explicit time argument, because the ambient resource is generally not steady. A converter immersed in a fluctuating environmental flux produces a fluctuating output, and the framework is built to track that curve rather than to report a single average number. The entire construction below can be read as an answer to one question: how much directed, usable power crosses the converter’s control surface at time $t$? The control surface itself is fixed once and for all in system boundaries; everything here is drawn relative to it.

Step (b): the contribution of a single slice

Consider the incident field not as a monolithic beam but as a bundle of contributions labelled by two coordinates: the energy $E$ carried by each quantum of the field, and the direction of arrival $\theta$. Fix a narrow slice at $(E,\theta)$ — a small energy window $dE$ around $E$ and a small angular window $d\theta$ around $\theta$ — and ask how much usable power that one slice contributes. We answer by following the energy through the conversion chain, factor by factor.

The slice first delivers a resource. The flux density $\Phi(E,\theta,t)$ specifies how much field energy arrives per unit area, per unit time, per unit solid angle, per unit energy, from direction $\theta$ at energy $E$ and time $t$. This is the externally imposed boundary condition — a property of the environment, not of the device.

That resource must then couple to the material. The scattering cross-section $\sigma(E)$ sets how strongly the field at energy $E$ interacts with the converter; it has the dimension of area and can be read as the effective target the material presents to the incident field at that energy. Where $\sigma(E)$ is large the flux is intercepted; where it is small the flux passes through untouched.

An intercepted interaction transfers momentum. Only the net directed part matters, and we write it as $\Delta p$ — the momentum deposited into the converter per interaction after the isotropic, cancelling part has been averaged away. This is the symmetry-breaking factor. If the interaction geometry is fully symmetric, the directed part is zero and the slice contributes nothing, no matter how large $\Phi$ or $\sigma$ may be. The physical necessity of $\Delta p \neq 0$ is developed in non-equilibrium energy conversion.

Directed momentum must then be transported away from the interaction site before it randomises. The phonon velocity $v_{\text{ph}}$ characterises how quickly the lattice carries that momentum off, and so acts as the transport rate in the chain.

Finally, the transported, directed energy is rectified into usable electrical power with efficiency $\eta_{\text{conv}}$, a dimensionless number in $[0,1]$ that absorbs every aggregate loss — heat, back-scattering, internal resistance. Because it is bounded above by one, the framework can never report more usable output than the resource, coupling and transport actually supply.

Reading these five factors as a chain, the power contributed by the single slice is their product times the size of the slice:

$$dP(E,\theta,t) = \Phi(E,\theta,t)\,\sigma(E)\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\;dE\,d\theta .$$

The product form is essential: if any one factor vanishes, the slice’s contribution vanishes. There is no way to compensate a dead link in the chain by strengthening the others.

Step (c): integrating over energy and angle

A real converter is not exposed to one slice but to the whole distribution of energies and directions present in the ambient field. The total instantaneous power is the sum of the contributions from every slice. Because $E$ and $\theta$ are continuous, that sum is an integral. Accumulating first over the energies present in the spectrum, then over the directions from which flux arrives, gives the Schubart Master Formula in its expanded physical representation:

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

The energy integral runs over the support of the spectrum — formally from zero to infinity, with the understanding that the integrand is negligible wherever $\Phi(E,\theta,t)\,\sigma(E)$ is negligible. The angular integral runs over the solid angle the incident flux subtends at the converter surface; whatever angular measure a given geometry requires (a $\sin\theta$ factor for a polar angle, for instance) is inserted at this stage without altering the structure. The time argument survives the integration because $\Phi$ carries it and the other factors are treated as quasi-static; $P(t)$ therefore inherits its time dependence entirely from the environment. The dimensional bookkeeping that makes this expression reduce to watts is carried out in full in dimensional analysis and summarised in mathematical architecture.

Step (d): the four governing assumptions

The move from the single slice to the integral is not free; it rests on four assumptions, each of which is a physical statement that could in principle fail.

1. Open, non-equilibrium system. The converter is not isolated. It is continuously coupled to an external reservoir with which it exchanges energy and momentum, and it operates away from thermal equilibrium. This is what permits net work extraction at all: the second law forbids extracting work from a system in equilibrium at a single temperature, but says nothing against a device driven by a directed or non-equilibrium flux. If the system were closed and equilibrated, the framework would correctly report zero, because $\Delta p$ would average to zero. The thermodynamic status is developed in open thermodynamic systems.

2. Continuous, sustained drive. The ambient flux $\Phi$ is present continuously over the timescales of interest, not as a one-off transient. This is what licenses treating $P(t)$ as a persistent power curve rather than the discharge of a finite store. Should the drive switch off, $\Phi \to 0$ and the output follows.

3. Linear, independent interactions. Each interaction event contributes additively and independently of the others, so the total is the integral of the single-slice contributions with no cross-terms. This lets the factors appear as a simple product and the slices as a simple sum. It holds when the converter operates in a regime where the incident field does not saturate the material and interactions do not interfere; strongly non-linear or collective regimes would require correction terms beyond this form.

4. Effective, quasi-static coupling. The material factors $\sigma(E)$, $\Delta p$, $v_{\text{ph}}$ and $\eta_{\text{conv}}$ are effective aggregate quantities that respond effectively instantaneously to changes in flux. Each stands in for a great deal of microscopic detail — $\eta_{\text{conv}}$ alone absorbs all conversion losses — and each is assumed to track the flux without its own lag. If a material parameter drifted on the relevant timescale (say $\eta_{\text{conv}}$ with temperature), it could be given its own time argument without changing the shape of the integral.

These four assumptions are the price of the compact form. They are also its falsifiable content: a measurement that showed net output at equilibrium, or strongly non-additive interactions, would signal that the form needs revising rather than merely re-parameterising.

Step (e): reduction to a representative slice

The double integral is the honest general statement, but for orientation — and for the interactive explorer — it is useful to collapse it to a single representative slice. Suppose the integrand is roughly constant across the effective energy window $\Delta E$ and angular window $\Delta\theta$ that dominate the integral. Then the integral is approximately the integrand evaluated at representative values, multiplied by the size of that window:

$$P \approx \Phi\,\sigma\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\,\Delta E\,\Delta\theta .$$

This is precisely the expression the explorer computes: five physical factors and two window widths, multiplied together. It is not a different formula but the master formula under a single-slice (mean-value) approximation, valid when one dominant band of energy and angle carries most of the contribution. Reading it left to right recovers the whole chain — resource, coupling, directed momentum, transport, rectification, accumulated over a representative band. Its virtue is that every factor is a single tunable number, so one can immediately see how the output responds to each. Its limitation is that it discards spectral and angular structure; when the flux or the cross-section has sharp features, the full double integral of step (c) must be restored, and evaluated by the quadrature and sampling methods of numerical simulation.

Summary of the derivation

The construction proceeds in five moves: name the target power $P(t)$; write the contribution of one $(E,\theta)$ slice as the product of five chained factors times $dE\,d\theta$; integrate over energy and angle to obtain the boxed master formula; note the four assumptions — open non-equilibrium operation, sustained drive, linear independent interactions, effective quasi-static coupling — on which that integration rests; and finally reduce the integral to the representative-slice product that the explorer evaluates. The same logic underlies both the general framework and any specific application of it, such as the neutrinovoltaic case discussed in future extensions, which is obtained simply by supplying concrete forms for the five factors.