Reduction to Known Limits

A general framework earns its name only if it reproduces the established special cases. Any candidate equation for continuous ambient energy conversion must, under the right restrictions, collapse to the laws physics already trusts — otherwise it contradicts settled results rather than generalising them. This chapter performs that test explicitly for three canonical relations. It complements governing relations, which positions $P(t)$ beside those laws; here we show how it becomes them. The Schubart Master Formula, in its expanded physical representation (the equation), is

$$P(t) = \iint \Phi(E,\theta,t)\,\sigma(E)\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\;dE\,d\theta . \tag{1}$$

Passing these reductions is necessary for consistency; it is not, by itself, proof that any device delivers net output. That separation of the settled from the open is kept throughout (scientific questions).

Carnot: the two-reservoir thermal limit

Specialise the ambient field to thermal radiation exchanged between a hot reservoir at temperature $T_{\text{h}}$ and a cold sink at $T_{\text{c}}$, with no other channel present. The absorbed spectrum $\Phi$ is then the blackbody difference between the two reservoirs, and the rectification factor $\eta_{\text{conv}}$ is constrained by the second law: no cyclic conversion of heat drawn from a single hot reservoir into work can exceed the Carnot fraction. Writing the input heat flux as $\dot{Q}_{\text{h}} = \iint \Phi\,\sigma\;dE\,d\theta$, equation (1) factorises into an input flux times an efficiency, and the second law caps that efficiency:

$$\frac{P}{\dot{Q}_{\text{h}}} = \eta_{\text{conv}} \;\le\; 1 - \frac{T_{\text{c}}}{T_{\text{h}}} = \eta_{\text{Carnot}}. \tag{2}$$

In this limit $P(t)$ does not replace Carnot — it inherits it: the ceiling enters through $\eta_{\text{conv}}$, exactly as anticipated in governing relations. The transport factors $\Delta p\,v_{\text{ph}}$ describe how the absorbed energy is carried, but they cannot lift the efficiency above the thermodynamic bound. The reduction is therefore consistent with the second law by construction.

Shockley–Queisser: the single-junction photovoltaic limit

Now specialise the environment to a fixed solar photon spectrum and the absorber to a single bandgap $E_{\text{g}}$. Two restrictions do the work:

• Spectral restriction. $\Phi(E,\theta,t) \to \Phi_{\odot}(E)$, the (time-independent) solar photon flux.
• Absorber restriction. $\sigma(E) \to \sigma_0\,\Theta(E - E_{\text{g}})$, a step that couples only above the gap, where $\Theta$ is the Heaviside step.

The generated photocurrent is then the above-gap photon count,

$$J_{\text{sc}} \;\propto\; \int_{E_{\text{g}}}^{\infty} \Phi_{\odot}(E)\;dE, \tag{3}$$

and the usable power is current times the detailed-balance operating voltage, $P_{\max} = J\,V$, with $\eta_{\text{conv}}$ set to the detailed-balance efficiency of Shockley & Queisser (1961). Maximising over the single free parameter $E_{\text{g}}$ recovers the familiar single-junction ceiling of roughly 30–33% under standard illumination — quoted here as the established order of magnitude, not to more precision than the argument warrants. The point is structural: with these two substitutions, equation (1) is the detailed-balance expression, and its optimum is the Shockley–Queisser limit. The photovoltaic ceiling lives inside $P(t)$ as the value of $\eta_{\text{conv}}$ for the photovoltaic realisation.

Thermoelectric ZT: the diffusive-transport limit

Finally, specialise the drive to a maintained temperature gradient and the transport to diffusive carrier flow. Here the directed factors $\Delta p\,v_{\text{ph}}$ carry the physical content of Seebeck-driven transport, and $\eta_{\text{conv}}$ takes the standard thermoelectric form governed by the dimensionless figure of merit $ZT$:

$$\eta_{\text{conv}} \to \eta_{\text{TE}} = \left(1 - \frac{T_{\text{c}}}{T_{\text{h}}}\right)\,\frac{\sqrt{1 + ZT} - 1}{\sqrt{1 + ZT} + \dfrac{T_{\text{c}}}{T_{\text{h}}}}. \tag{4}$$

As $ZT \to \infty$ the second factor tends to one and $\eta_{\text{TE}} \to \eta_{\text{Carnot}}$, recovering the thermal ceiling of the first reduction — a useful internal check that the three limits are mutually consistent. For finite $ZT$, equation (1) with these choices reproduces the thermoelectric power and its efficiency, with the transport factors $\Delta p\,v_{\text{ph}}$ encoding the material’s ability to carry momentum and charge together.

Reduction table

Established law	Restrict $\Phi$ to…	Restrict $\sigma$ to…	$\eta_{\text{conv}}$ becomes…	$P(t)$ recovers
Carnot	Two-reservoir thermal radiation	Broadband thermal coupling	$\le 1 - T_{\text{c}}/T_{\text{h}}$	Heat-engine efficiency ceiling
Shockley–Queisser	Fixed solar spectrum $\Phi_{\odot}(E)$	Step absorber $\Theta(E - E_{\text{g}})$	Detailed-balance efficiency	Single-junction PV limit ($\sim$33%)
Thermoelectric	Maintained thermal gradient	Diffusive carrier coupling	$\eta_{\text{TE}}(ZT)$	Thermoelectric power and efficiency

What the reductions establish

Each row is a consistency reduction: a set of substitutions under which the general expression (1) collapses onto a law that settled physics already accepts. Passing all three shows that $P(t)$ is dimensionally and thermodynamically compatible with the established ceilings — it contains them rather than contradicting them, and any application-specific bound enters through $\eta_{\text{conv}}$. This is a necessary validity test that a general framework must pass, and $P(t)$ passes it. It is deliberately not a sufficient one: reproducing known limits under idealised restrictions does not demonstrate net output for any particular device, which remains subject to independent full-scale validation (assumptions and limits, scientific questions). What the reductions do secure is the framework’s internal coherence: wherever an established law applies, $P(t)$ agrees with it.