architecture

System Boundaries

Defining the control surface for the P(t) framework: what counts as inside, what crosses the boundary, and why the balance is only meaningful relative to it.

Last updated: 2026-07-20

Every energy balance is a statement about a boundary. The quantity $P(t)$ is defined as the power delivered across a specific control surface, and it has no unambiguous meaning until that surface is drawn. This chapter specifies how the system boundary is defined in the Schubart Master Formula, what crosses it, and why careful boundary-drawing is essential for the accounting to be honest. Because the framework applies to open systems that exchange energy and momentum with a reservoir, as set out in open thermodynamic systems, the boundary is where that exchange is measured.

The control surface

The control surface is a closed geometric surface $\partial V$ enclosing the converter volume $V$ . Everything inside is treated as the system; everything outside is the reservoir and the load. Two kinds of quantity are associated with the surface. First, the inward flux: the ambient radiation $\Phi(E,\theta,t)$ crossing the surface from outside carries energy and momentum into the system. Second, the outward useful power: $P(t)$ leaves the system across the surface toward an external load. The master formula computes the second from the first, folding all internal processes — coupling, momentum transfer, transport, and rectification — into the integrand. Drawing the surface fixes exactly which flux is counted as incident and which power is counted as delivered.

The balance can be written as a surface integral to make the boundary-referred character explicit. If $\mathbf{S}(\mathbf{r},t)$ is the local energy-current density and $\hat{\mathbf{n}}$ the outward normal, then the net rate of energy exchange across $\partial V$ is

\oint_{\partial V} \mathbf{S}\cdot\hat{\mathbf{n}}\;dA = -P_{\text{in}}(t) + P(t) + P_{\text{loss}}(t) ,

where the inward-directed part of $\mathbf{S}$ over the illuminated portion of $\partial V$ reconstructs the incident flux $\Phi$ , and the outward-directed parts carry away the useful power and the losses. In the quasi-steady regime the stored energy inside $V$ is stationary, so the surface integral vanishes and the schematic balance $P(t) = P_{\text{in}}(t) - P_{\text{loss}}(t)$ follows directly. The virtue of the surface-integral form is that it makes “delivered power” a property of a specific closed surface, not of a vague region — every quantity in the master formula is, in the end, a flux through $\partial V$ .

What crosses the boundary

Three distinct flows cross a properly drawn boundary, and keeping them separate prevents double-counting. The first is the incident ambient flux entering the system, represented by $\Phi$ . The second is the useful electrical power leaving the system toward the load, which is $P(t)$ itself. The third is the loss flux — heat and non-rectified energy — that also leaves the system but is not useful. The framework accounts for the third implicitly: the conversion efficiency $\eta_{\text{conv}}$ is precisely the fraction of coupled, directed energy that leaves as useful power rather than as loss. Thus the boundary balance can be read as: incident rectifiable energy in, useful power plus losses out, with $\eta_{\text{conv}}$ partitioning the two. This is the concrete, boundary-referred version of the schematic balance $P(t) = P_{\text{in}}(t) - P_{\text{loss}}(t)$ introduced earlier.

Why the boundary must be explicit

A common source of confusion in energy-harvesting claims is an implicit or shifting boundary. If the incident flux is measured on one surface but the delivered power on another, the two are not comparable and apparent gains or losses can appear that are merely artefacts of inconsistent accounting. The framework guards against this by requiring a single, fixed control surface for both the inward flux and the outward power. In particular, the flux $\Phi$ in the integrand must be the flux actually crossing the chosen surface — not the flux far away, and not the flux after it has already been partially attenuated inside the device, unless the surface is placed accordingly. Fixing the surface once removes this ambiguity and makes the reported $P(t)$ reproducible.

An illustrative example shows how a shifting boundary manufactures a spurious result. Suppose a slab converter attenuates the incident flux by a factor $e^{-\mu d}$ over its depth $d$ , where $\mu$ is an effective attenuation coefficient. If one places the “input” surface at the front face (where the flux is $\Phi_0$ ) but computes the interaction rate from the flux deep inside (where it is $\Phi_0 e^{-\mu d}$ ), the two figures differ by the whole attenuation factor. Taking, purely for illustration, $\mu d \sim 1$ , the mismatch is a factor of $e \approx 2.7$ — enough to fabricate an apparent efficiency well above unity or to hide a real loss. The cure is not a better number but a consistently placed surface: whichever surface is chosen, both $\Phi$ and $P(t)$ must be referred to it. This is why the framework treats boundary placement as part of the definition of $P(t)$ , not as an afterthought.

The reservoir side

On the outside of the boundary sits the ambient reservoir. The framework treats it as large enough that harvesting does not appreciably deplete it over the timescales of interest, so that $\Phi$ can be regarded as an externally imposed boundary condition rather than a dynamical variable coupled back to the system. This is the standard reservoir assumption of open-system thermodynamics, and it is what allows the flux to be specified independently of the converter. Where this assumption might fail — for instance if a converter were large enough to shadow its own supply — the boundary condition would have to be solved self-consistently, but such cases lie outside the basic form of the framework and would be treated as extensions.

The reservoir assumption can be given a quantitative validity condition. Let the reservoir supply rectifiable energy at a rate $\dot{E}_{\text{res}}$ into the region seen by the converter, and let the converter extract at rate $P_{\text{in}}$ . The flux may be treated as an unperturbed boundary condition when

P_{\text{in}} \ll \dot{E}_{\text{res}} ,

so that extraction is a negligible perturbation on the local reservoir. For most ambient fields this holds by an enormous margin, because the reservoir is effectively unbounded compared with any laboratory-scale converter; the flux one measures with the converter present is, to excellent approximation, the flux that would be there in its absence. The assumption is worth stating explicitly precisely because it is the hinge on which the whole open-system reading turns: it is what lets $\Phi$ be an input rather than an unknown to be co-solved, and any proposed converter that would violate it must be analysed with the more elaborate self-consistent machinery instead.

Choosing where to draw the surface

Although any consistently applied surface is admissible, some choices are far more convenient than others, and the framework offers guidance on placement. The natural surface is the one where the incident flux is most cleanly defined and most easily measured — typically just outside the physical converter, in the undisturbed ambient field, before any attenuation or reshaping by the device. Placing the surface there makes $\Phi$ equal to the genuine ambient flux and folds all internal attenuation into the integrand, where it belongs. Placing it instead somewhere inside the device forces one to track the already-modified internal flux, which is harder to measure and easier to get wrong. A second consideration is that the surface should enclose all the rectifying and dissipating machinery, so that $\eta_{\text{conv}}$ genuinely accounts for every internal loss; a surface that clips through the middle of the converter would leave some losses uncounted and others double-counted. The general rule is therefore: draw the surface in the clean ambient field, fully enclosing the device and its immediate loss channels, and refer both $\Phi$ and $P(t)$ to it without exception.

The load side and the meaning of “usable”

The other thing outside the boundary is the load. The word usable in the definition of $P(t)$ is defined relative to this load: usable power is power delivered in a form the external circuit can accept, as opposed to energy trapped internally or dissipated as heat. Because the definition is load-referred, the same converter can in principle report different usable outputs into different loads, and a complete specification of $P(t)$ therefore names the load conditions along with the control surface. This is not a weakness of the framework but a reflection of physical reality: delivered power always depends on what it is delivered into.

Momentum crosses the boundary too

Energy is not the only conserved quantity crossing the control surface; momentum crosses it as well, and this is essential to the framework’s non-equilibrium content. The incident flux carries momentum inward, and the directed component of that momentum — the part that survives averaging — is what the term $\Delta p$ in the integrand quantifies. A complete boundary balance therefore tracks a momentum current alongside the energy current. In steady operation the net momentum delivered to the lattice is either rectified into the directed transport that produces useful current or dissipated as the recoil that ultimately appears as heat. Recognising that momentum, not merely energy, crosses the surface is what distinguishes this framework from a purely calorimetric energy balance: it is the directed momentum flux across the boundary that carries the information about symmetry breaking, and hence about whether any net power can be delivered at all. A boundary drawn without regard to the direction of the incident momentum would miss exactly the quantity that determines the sign and magnitude of the output.

Summary

The boundary is the anchor of the whole accounting. It fixes which flux is incident, which power is delivered, and where the reservoir and load begin. With the surface drawn explicitly and held fixed, the master formula — whose structure is detailed in mathematical architecture — becomes a well-posed statement about a specific, reproducible system rather than a vague assertion about “energy from the environment.” The governing relations that follow once the boundary is fixed are collected in governing relations, and the open questions the framework raises are set out in scientific questions.