Future Extensions

Directions in which the P(t) master formula could be generalised: coupled reservoirs, time-dependent parameters, spatial fields, and multi-channel interference.

Last updated: 2026-07-20

The Schubart Master Formula in its expanded physical representation is deliberately explicit: a double integral over energy and angle with quasi-static material factors. That compactness is a strength, but it is also an invitation to generalise. This chapter surveys directions in which the framework could be extended, each of which relaxes one of the simplifying assumptions built into the basic form. All of these are presented as proposed extensions — mathematical possibilities that the framework can accommodate, not established results — and each would require its own careful validation before any physical claim rested on it.

Self-consistent reservoir coupling

The basic framework treats the ambient flux $\Phi(E,\theta,t)$ as an externally imposed boundary condition, on the assumption that the reservoir is large enough that harvesting does not deplete it, as set out in open thermodynamic systems. A natural extension relaxes this. If a converter were large enough to shadow its own supply, or if many converters competed for the same flux, then $\Phi$ would depend on the presence of the converter and the boundary condition would have to be solved self-consistently. Schematically, the imposed flux $\Phi_0$ would be replaced by a corrected flux

\Phi(E,\theta,t) = \Phi_0(E,\theta,t) - \Phi_{\text{abs}}\big[\Phi\big](E,\theta,t) ,

in which the absorbed part $\Phi_{\text{abs}}$ is itself a functional of the flux the converter actually sees — a fixed-point equation rather than a one-way boundary condition. The extended problem couples the flux and the device into a single system of equations, and the master formula becomes one equation in a coupled set rather than a closed expression, typically solved by iteration until $\Phi$ stops changing. In the illustrative weak-shadowing limit, where the absorbed fraction is small, the correction is negligible and the base form is recovered; the self-consistent treatment matters only when a device, or a dense array of devices, removes a non-trivial share of its own supply. This is the appropriate generalisation for dense arrays and for converters that are not small compared with their supply.

Time-dependent material parameters

In the base form, $\Delta p$ , $v_{\text{ph}}$ , and $\eta_{\text{conv}}$ are treated as quasi-static, so that the entire time dependence of the output flows from the flux. A straightforward extension gives these factors their own time arguments. Efficiency may drift with temperature; the transport velocity may change as the lattice heats; the effective momentum transfer may vary with an applied bias. Writing $\eta_{\text{conv}}(t)$ , $v_{\text{ph}}(t)$ , and $\Delta p(t)$ leaves the integral structure intact while allowing the framework to describe converters whose internal state evolves. The extension is genuinely non-trivial when these factors depend on the history of the output rather than only on the instantaneous time — for example when accumulated heating raises the lattice temperature, which in turn changes the transport velocity. In that case the material factors become functionals of the past output, and the once-explicit integral turns into an integro-differential equation coupling $P(t)$ to its own history. This is the minimal generalisation needed to model warm-up transients, thermal feedback, ageing, and dynamic control.

Spatial and field extensions

The basic integral is over energy and angle only; it treats the converter as effectively a point with respect to spatial structure. A richer formulation would add spatial integration, so that the flux, coupling, and rectification could vary across an extended device and the local contributions be summed:

P(t) = \int_{V} \iint \Phi(\mathbf{r},E,\theta,t)\,\sigma(\mathbf{r},E)\,\Delta p(\mathbf{r})\,v_{\text{ph}}(\mathbf{r})\,\eta_{\text{conv}}(\mathbf{r})\;dE\,d\theta\;d^3r .

In this form the framework resembles a field theory of the converter, with $P(t)$ obtained by integrating a local power density over the device volume $V$ as well as over energy and angle. The spatial dependence also lets adjacent regions influence one another — directed momentum generated in one layer can be transported into and rectified by another — so the naive assumption that each volume element contributes independently must be checked rather than assumed. Such an extension is the natural language for graded materials, layered structures, and any device where the internal geometry matters.

Multi-channel interference

The multi-channel treatment in multi-channel energy flux assumes that distinct channels add incoherently — that the total power is the sum of per-channel integrals with no cross terms. Where two channels share a coherent phase relationship, this assumption fails and interference terms appear. A generalised framework would carry these cross terms explicitly, allowing constructive or destructive interference between channels to modify the total. This is a delicate extension, because interference can in principle enhance or suppress output, and any claim of enhancement would need especially careful validation to rule out a bookkeeping error masquerading as a physical effect.

Applications as special cases

A different kind of extension is not mathematical but interpretive: applying the general framework to a specific proposed converter by supplying concrete forms for each factor. Neutrinovoltaics, for example, is one such proposed application in which the ambient flux and the coupling would be given particular physical identities; it is treated at greater length in frequently asked questions. The essential discipline for any application is the same: the general formula is agnostic, and the burden of proof lies entirely on the specific inputs. An application is only as credible as the measurements behind its factors, and until those exist the predicted output remains a model projection.

The role of extensions in the framework

Taken together, these directions show that the base form of the master formula is best understood as the leading-order member of a family. Each extension restores something the basic form omits — reservoir back-reaction, temporal drift, spatial structure, coherent interference — at the cost of additional complexity, and each recovers the base form in an appropriate limit: weak shadowing, quasi-static parameters, a point-like device, incoherent channels. This limiting-case structure is the signature of a well-posed hierarchy of models rather than an arbitrary collection of add-ons. The framework is designed so that these can be added incrementally, without discarding the core factorisation described in mathematical architecture. None of them changes the fundamental commitment of the framework: that reported power is always drawn from an identified non-equilibrium resource and always referred to an explicit boundary. Each extension inherits that discipline, and each also inherits the burden of its own validation — a richer model has more places to be wrong, not fewer.