Governing Relations

Physics organises energy conversion around a small number of governing relations — compact statements that tell you what a whole class of devices can and cannot do. Two are canonical: the Carnot efficiency, which caps any heat engine working between two reservoirs, and the Shockley–Queisser limit, which caps single-junction photovoltaic conversion under a given spectrum. This chapter positions the Schubart Master Formula $P(t)$ alongside them — not as a competitor that overturns them, but as a different kind of relation that answers a different question. Understanding what kind of statement $P(t)$ is, by contrast with these two, is the clearest way to see its intended role. The formula itself is stated at the equation.

The two established bounds

The Carnot efficiency states that a reversible heat engine operating between a hot reservoir at temperature $T_{\text{h}}$ and a cold reservoir at $T_{\text{c}}$ can convert at most a fraction

$$\eta_{\text{Carnot}} = 1 - \frac{T_{\text{c}}}{T_{\text{h}}}$$

of the heat drawn from the hot reservoir into work. It is a ceiling: a single dimensionless number that no real engine of that class can exceed, derived from the second law and independent of any construction detail. Its lineage runs to Carnot (1824); see evidence library.

The Shockley–Queisser limit states that a single-junction solar cell under a given illumination spectrum, analysed by detailed balance, has a maximum power-conversion efficiency set by the bandgap — famously around 30–33% for a single junction under standard sunlight (an established result, quoted here as the well-known order of magnitude rather than to more precision than the context warrants). Like Carnot, it is a ceiling: one number bounding a device class, derived by Shockley & Queisser (1961) from detailed-balance arguments.

The Schubart Master Formula is structurally different. It does not return a ceiling; it returns a curve:

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

Given the environment and the material factors, it computes the instantaneous usable power at each moment. Where Carnot and Shockley–Queisser answer “at most how efficient can this class of device be?”, $P(t)$ answers “how much power does this converter deliver right now, given its inputs?”.

Comparison table

Aspect	Carnot efficiency	Shockley–Queisser limit	Schubart $P(t)$
Domain	Any heat engine between two thermal reservoirs	Single-junction photovoltaic cells under a fixed spectrum	Any open-system converter of ambient field energy
Drive / input	A maintained temperature difference $T_{\text{h}} > T_{\text{c}}$	A fixed illumination spectrum on the cell	An ambient flux $\Phi(E,\theta,t)$, generally fluctuating
Output	A dimensionless efficiency ceiling (one number)	A dimensionless efficiency ceiling (one number)	A time-resolved power curve $P(t)$ in watts
Design lever	Increase the reservoir temperature ratio $T_{\text{h}}/T_{\text{c}}$	Match the bandgap to the spectrum	Tune the five factors $\Phi,\sigma,\Delta p,v_{\text{ph}},\eta_{\text{conv}}$ jointly
Availability	Established, second-law theorem	Established, detailed-balance theorem	Proposed framework; components established, composite unvalidated

The last row is deliberately candid: Carnot and Shockley–Queisser are proven theorems of settled physics, whereas $P(t)$ is a proposed accounting framework whose individual terms rest on established physics (evidence library) but whose end-to-end predictions await independent validation (scientific questions). Placing them in one table is not a claim of equal standing; it is a way of showing exactly how they differ in kind.

Three structural differences

1. Continuous rather than intermittent framing

Carnot and Shockley–Queisser are framed around a steady operating point — a fixed temperature ratio, a fixed spectrum — and report a single steady efficiency. They abstract away from time. $P(t)$ is framed around a continuously drawn, generally fluctuating resource and keeps time explicit: the output tracks the environment moment by moment. This is not a cosmetic difference. It means $P(t)$ is the natural object when the resource itself varies — when the interesting question is the shape of the power curve over time, not a single rated figure. Where the established bounds describe the best a device could do under held-fixed conditions, $P(t)$ describes what a device actually does under changing ones.

2. A curve, not an upper bound

Carnot and Shockley–Queisser each deliver an upper bound: a number that output must fall below. $P(t)$ delivers an actual predicted value (conditional on its inputs), not a limit. The two roles are complementary, and importantly the bounds live inside the framework rather than beside it: any efficiency ceiling relevant to a given application enters $P(t)$ through the value assigned to $\eta_{\text{conv}}$. A photovoltaic realisation would carry the Shockley–Queisser bound in its $\eta_{\text{conv}}$; a thermal realisation would carry a Carnot-type bound there. So $P(t)$ does not compete with the ceilings — it consumes them as constraints on one of its factors and then computes the resulting power. A bound tells you the roof; $P(t)$ tells you how high in the room you actually are, given where the roof is.

3. A different design lever

The established relations each expose one dominant lever. For Carnot it is the reservoir temperature ratio; raise $T_{\text{h}}/T_{\text{c}}$ and the ceiling rises. For Shockley–Queisser it is spectral matching of the bandgap; tune the gap to the spectrum and the ceiling rises. $P(t)$ exposes five levers acting multiplicatively — the resource $\Phi$, the coupling $\sigma(E)$, the symmetry breaking $\Delta p$, the transport $v_{\text{ph}}$, and the rectification $\eta_{\text{conv}}$ — so improvements compound and, more usefully, bottlenecks are isolated: because the factors multiply, the smallest factor dominates the output, and the framework points directly at which link in the chain to improve. This multi-lever, bottleneck-revealing character is the framework’s practical contribution over a single-number ceiling; it is developed in engineering interpretation.

What the comparison establishes

Set beside the two canonical governing relations, $P(t)$ occupies a clearly distinct niche: it is a time-resolved, multi-factor power relation for open-system ambient conversion, whereas Carnot and Shockley–Queisser are steady, single-lever efficiency ceilings for their respective device classes. The established bounds are not challenged by this — they are absorbed, entering $P(t)$ through $\eta_{\text{conv}}$ wherever they apply. The honest summary is that $P(t)$ aspires to the role of a governing relation for a broader class of converters, while remaining, unlike Carnot and Shockley–Queisser, a proposed framework rather than a proven theorem until its composite predictions are independently validated. Its value, if it earns one, will be in organising continuous ambient-energy conversion under a single dimensionally and thermodynamically disciplined structure, the way the older relations organise heat engines and solar cells.