motivation

Non-equilibrium Energy Conversion

The symmetry-breaking mechanics that let the P(t) integral be non-zero: directed momentum transfer, rectification, and the role of the Δp term.

Last updated: 2026-07-20

If the ambient field were perfectly isotropic and the converter perfectly symmetric, every interaction that pushed the lattice one way would be matched by another that pushed it the opposite way, and the net rectified power would be exactly zero. Continuous energy conversion is therefore possible only under non-equilibrium conditions in which some symmetry is broken. This chapter sets out how the P(t) framework encodes that requirement, and why the momentum-transfer term $\Delta p$ is the mathematical heart of the whole construction.

Directed versus isotropic interactions

Consider an incident flux interacting with a converter. Each interaction transfers some momentum to the lattice. If the transfers are distributed isotropically, their vector sum vanishes: there is heating, perhaps, but no directed current and no usable directed power. To obtain a net effect the interactions must, on average, deposit momentum preferentially in one direction. The framework represents this net directed transfer per interaction by $\Delta p$ , the momentum transfer [kg m/s]. Crucially, $\Delta p$ is defined as a directed quantity: it is the net push that survives after the isotropic part has been averaged away. When the model assumes a symmetric, equilibrium situation, $\Delta p \to 0$ and, as required, the master formula yields no output. The compact statement of the formula, with $\Delta p$ shown as an explicit factor, appears in the equation.

Where the asymmetry comes from

There are several general routes by which a converter can break symmetry, and the framework is agnostic about which is realised in any given device. A converter may be structurally asymmetric, so that forward and backward interactions are geometrically inequivalent. It may be built from a material with a broken inversion symmetry, so that the response to an incoming field is intrinsically directional. Or it may be coupled to an external flux that is itself anisotropic, so that more momentum arrives from one hemisphere of solid angle than another — a possibility made explicit by the angular integration variable $\theta$ in $\Phi(E,\theta,t)$ . In every case the physically meaningful output is proportional to the surviving directed component, and the framework deliberately does not presuppose the mechanism. It only insists that some such mechanism be present for $\Delta p$ to be non-zero.

From microscopic transfers to the directed term

It is useful to see how $\Delta p$ emerges from an average over individual events. Let an interaction with incident state $(E,\theta)$ deposit a momentum vector $\mathbf{q}(E,\theta,\Omega')$ , where $\Omega'$ labels the outgoing channel, and let $w(\Omega'\,|\,E,\theta)$ be the probability of scattering into $\Omega'$ . The mean momentum deposited per interaction, projected onto the collection axis $\hat{\mathbf{n}}$ , is

\Delta p(E,\theta) = \int \big[\mathbf{q}(E,\theta,\Omega')\cdot\hat{\mathbf{n}}\big]\,w(\Omega'\,|\,E,\theta)\;d\Omega' .

If the scattering is symmetric under $\mathbf{q}\to-\mathbf{q}$ — the equilibrium, detailed-balance case — the positive and negative contributions cancel and $\Delta p = 0$ . A non-zero result requires that $w$ itself be asymmetric, i.e. that forward and backward channels carry different weight. This is the precise mathematical content of “symmetry breaking”: it is an inequality between $w(+\mathbf{q})$ and $w(-\mathbf{q})$ , and it is exactly what the structural, material, or flux asymmetries above provide. The scattering weights $w$ are set by the same physics that fixes $\sigma(E)$ ; see scattering cross-section.

Rectification and transport

Breaking symmetry at the point of interaction is necessary but not sufficient. The directed momentum must then be transported through the lattice and converted into a usable electrical current before it is randomised by scattering. The framework separates these steps into two factors. The phonon velocity $v_{\text{ph}}$ [m/s] characterises how quickly the lattice carries the transferred momentum away from the interaction site, setting the rate at which directed momentum can be collected. The conversion efficiency $\eta_{\text{conv}}$ [dimensionless, between 0 and 1] then captures the fraction of that collected, directed energy that is actually rectified into usable output rather than lost to heat, back-scattering, or internal resistance. Together, $v_{\text{ph}}$ and $\eta_{\text{conv}}$ describe the machinery that turns a microscopic directed push into a macroscopic delivered power.

The competition between collection and randomisation can be made quantitative. Let the directed momentum injected at a site relax back toward isotropy with a characteristic scattering time $\tau$ , while transport carries it away at speed $v_{\text{ph}}$ . Over the collection process the surviving directed fraction decays as $\exp(-t/\tau)$ , so the momentum that can be usefully collected before randomisation scales with the ratio of the transport rate to the scattering rate. Writing the collection length as $\ell = v_{\text{ph}}\,\tau$ , the fraction of injected directed momentum that reaches the collection contacts a distance $L$ away behaves, in an illustrative one-dimensional picture, as

f_{\text{collect}} \sim \exp\!\left(-\frac{L}{v_{\text{ph}}\,\tau}\right) .

This shows explicitly why $v_{\text{ph}}$ enters as a factor: a faster lattice response lengthens $\ell$ and lets more of the directed momentum survive the journey. It also shows that $\eta_{\text{conv}}$ and $v_{\text{ph}}$ are not fully independent — the same scattering that limits collection also feeds the losses folded into $\eta_{\text{conv}}$ — which is why the framework keeps them as separate but jointly constrained factors.

Assembling the non-equilibrium chain

The logic of the framework is thus a chain of four physical requirements, each represented by its own factor. A flux must be present ( $\Phi$ ); it must couple to the material ( $\sigma$ ); the coupling must deposit net directed momentum ( $\Delta p$ ); and that momentum must be transported ( $v_{\text{ph}}$ ) and rectified ( $\eta_{\text{conv}}$ ). Because these are multiplied together and then integrated over energy and angle, the master formula

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

vanishes if any single link in the chain is absent. This multiplicative structure is what enforces thermodynamic honesty: no amount of strong coupling can produce output if the symmetry is unbroken, and no amount of directedness helps if there is no flux to couple to. The individual factors are catalogued in variable definitions and the full symbol set in the notation reference.

The chain can also be read as a sequence of conditional probabilities, which sharpens the intuition. Interpret $\Phi\,dE\,d\theta$ as the rate of incident quanta in a spectral–angular cell; $\sigma$ (relative to a geometric area) as the probability that an incident quantum interacts; the directedness encoded in $\Delta p$ as the probability that the interaction deposits net-forward momentum; and $\eta_{\text{conv}}$ as the probability that the deposited, transported momentum survives as useful output. Because these are conditional stages, the joint survival probability is their product, and the overall yield is bounded by the smallest link. This is the probabilistic restatement of the bottleneck principle: the framework can never report a yield larger than that permitted by its least favourable stage, which is exactly the discipline one wants when weighing speculative proposals.

The role of detailed balance

The deepest reason equilibrium yields no output is the principle of detailed balance. In thermal equilibrium every microscopic process is exactly counterbalanced by its time-reverse: for every interaction that deposits momentum $+\mathbf{q}$ there is, on average, an equally frequent interaction depositing $-\mathbf{q}$ . The forward and reverse rates are locked together by the equilibrium occupation of states, and their equality is what defines the absence of net currents. Symbolically, if $W_{i\to f}$ is the rate from state $i$ to state $f$ and $p_i^{\text{eq}}$ the equilibrium occupation, detailed balance requires

p_i^{\text{eq}}\,W_{i\to f} = p_f^{\text{eq}}\,W_{f\to i} ,

from which any net directed transport is guaranteed to vanish. Driving the reservoir out of equilibrium — through a directed flux, a spectral tilt, or a temperature difference — is precisely what breaks this equality of rates and allows a net current to flow. The momentum-transfer term $\Delta p$ in the master formula is therefore best understood as a measure of the departure from detailed balance: it is zero when detailed balance holds and grows with the degree to which the driving breaks it. This ties the framework directly to the foundations of non-equilibrium statistical mechanics rather than leaving symmetry-breaking as an unexplained postulate.

Relation to established converters

It is instructive that familiar devices fit this pattern. In a photovoltaic cell the built-in field of a p–n junction supplies the directional bias; in a thermoelectric device a temperature gradient does. The present framework abstracts the essential ingredient — a directed, non-equilibrium coupling — and expresses it generically, so that both established and proposed converters can be analysed with one formalism. Where a proposed mechanism for generating $\Delta p$ has not been experimentally confirmed — a neutrinovoltaic coupling would be one such speculative case — the resulting output must be treated as a model prediction rather than a demonstrated fact. The thermodynamic setting that permits all of this is described in open thermodynamic systems.

Entropy cost of rectification

Rectifying a directed current is not free of thermodynamic consequence, and the framework’s honesty is reinforced by tracking this. Each rectification event that produces ordered, low-entropy output must be accompanied by the export of at least a compensating amount of entropy to the surroundings, in line with the open-system entropy balance. In a steady state the directed power $P(t)$ leaves as low-entropy electrical work while the unavoidable heat is dumped back into the reservoir at the ambient temperature $T_0$ . The rate of entropy export associated with rejecting a loss power $P_{\text{loss}}$ is of order $P_{\text{loss}}/T_0$ , and this must at least offset the internal entropy production. This is why $\eta_{\text{conv}}$ can never reach unity in any real device: some fraction of the coupled energy must always be sacrificed as rejected heat to satisfy the second law. The momentum-transfer picture and the entropy picture are thus two views of the same constraint — one kinematic, one thermodynamic — and the framework is consistent with both simultaneously. The thermodynamic side is developed further in open thermodynamic systems.

An illustrative estimate of the directed fraction

To feel the scale of the effect, consider an illustrative case in which the incident flux is nearly isotropic but tilted by a small anisotropy $\epsilon \ll 1$ , so that the forward channel carries weight $\tfrac{1}{2}(1+\epsilon)$ and the backward channel $\tfrac{1}{2}(1-\epsilon)$ . If a full head-on interaction would deposit momentum $q_0$ , the surviving directed transfer is

\Delta p \approx \epsilon\, q_0 .

Taking, purely for illustration, a per-event momentum scale $q_0 \sim 10^{-27}\,\mathrm{kg\,m\,s^{-1}}$ and a modest anisotropy $\epsilon \sim 10^{-3}$ gives $\Delta p \sim 10^{-30}\,\mathrm{kg\,m\,s^{-1}}$ per interaction. These numbers are placeholders, not measurements; their purpose is only to show that the directed term is generically suppressed by the smallness of the anisotropy and that any realistic estimate must track that suppression explicitly rather than assume $\Delta p \sim q_0$ . Readers can vary $\epsilon$ and the other factors in the playground.