The mathematician of slow manifolds, or: on work that waits for its readers

A few weeks after we posted the second paper of our kinetic-theory series on arXiv — Richards' equation as a hydrodynamic limit — I received a letter from A. J. (Tony) Roberts, of the University of Adelaide. He had read the preprint, recognized in it a structure he has been building for forty years, and wrote — generously, precisely — to offer "an alternative framework, one that provides complementary illumination." Attached was his recent paper in the Transactions of Mathematics and Its Applications (Roberts, 2025). Reading it, and then following the thread backwards through his earlier work, I had two reactions in quick succession. The first: this is exactly the rigorous scaffolding our derivation needed. The second, more uncomfortable: why had I — why had, as far as I can tell, essentially the whole hydrological and homogenization literature — never engaged with it?

This post is about both reactions. The sociology first, because it carries a lesson beyond this particular case; the substance after, because the substance is what hydrologists should actually take home.

How good work gets stranded

Roberts' program — using the modern theory of invariant manifolds to derive macroscale models from microscale dynamics, with proofs, at the finite scale separations of real physics — should have landed squarely in the homogenization mainstream: the community that computes effective properties of heterogeneous media, the RVE world that every pore-scale modeler implicitly inhabits. It did not, and the reasons are worth naming because none of them concerns the quality of the mathematics.

He arrived from the wrong direction: dynamical systems rather than the calculus of variations, publishing in journals (ANZIAM J., IMA J. Appl. Math., SIAM monographs) that the mechanics community does not routinely scan. His computer algebra runs on Reduce, a system few researchers under sixty have installed. And his 2025 paper confronts the mainstream head-on — by my count it contains thirty-one explicitly flagged points of contrast with standard homogenization practice, nearly all of them, as far as I can judge, technically warranted. But communities metabolize challenges more slowly than contributions, and an outsider's justified critique reads, sociologically, as an outsider's critique first and as justified much later.

There is a subtler reason too: his most distinctive results resist sloganization. "Macroscale models valid down to scale separations of two" contradicts folklore so entrenched — one or two orders of magnitude between micro and macro, says every textbook — that readers assume a special case. "An exact remainder term for the gradient expansion" sounds like bookkeeping until the day you need an error bar. And his "backwards theory" — your reduced model is exact, but for a system provably close to the one you specified (Hochs & Roberts, 2019) — is philosophically the correct validity statement and rhetorically a hard sell, because it sounds weaker than the false statement people prefer to make. The closest parallel I know is Gorban and Karlin's work on exact hydrodynamic manifolds for kinetic equations, which had the same semi-overlooked trajectory until the framing "Hilbert's sixth problem" finally gave it a banner. Roberts never found his banner. Perhaps hydrology, of all fields, can lend him one; we are, after all, professional users of the equation his theory certifies.

The substance, for hydrologists

When I described the framework to a colleague, the reaction was a version of a question I had asked myself: isn't this obvious? A linear operator has a null space; the null space becomes the model; the rest follows. It is worth saying carefully why the rest does not follow, because everything a practicing hydrologist would pay for lives precisely in the part that doesn't.

The null space tells you what the fast processes cannot erase — for soil water, exactly one thing, mass, hence the water content θ. That is a direction, not a model. A model requires that a curved surface exist in the space of all possible pore-filling configurations — one point per value of θ — onto which every soil state slides and along which it then travels. That this surface exists, attracts, and can be computed is a theorem with a hypothesis, and the hypothesis is not the null space: it is the spectral gap, the clean separation between the slowest internal redistribution rate and the rate of the forcing. When the gap holds, "local equilibrium" stops being an assumption and becomes a state the soil demonstrably reaches, at a computable rate — in principle a measurable spin-up time after every irrigation pulse. When the gap closes — at the percolation threshold, when the water phase fragments — the surface does not become inaccurate; it ceases to exist. Richards' equation fails there the way a rating curve fails when the river leaves its banks: the object being parametrized is gone.

Around this central fact, Roberts' theorems deliver things I have not seen stated anywhere in the hydrological literature. That the validity of a macroscale model is local and checkable: the gradient expansion carries an exact remainder (Bunder & Roberts, 2021), so the model is quantitatively fine in the drained profile and quantitatively suspect at the wetting front, with a number attached, instead of a global incantation about ε → 0. That the required scale separation is startlingly small — his worked examples hold down to about twice the microscale (Roberts, 2015) — which should give pause to every campaign that agonizes over REV support volumes. That a state variable of a reduced model need not correspond to any conservation law: the second variable of a dual-permeability model, seen clearly, is not the budget of a second continuum but a wetness contrast between pore populations, which does not balance but relaxes, like the overtone of a struck string dying away under the fundamental. And — the result that genuinely surprised me — that the admissible nonlinearity of a multi-domain model is capped by a ratio of two relaxation rates: if that ratio is modest, an elaborately nonlinear macropore–matrix exchange function is fitting structure the reduced description cannot resolve. One number, two eigenvalues, and a ceiling on how fancy your two-domain model is allowed to be.

There is even an answer to a question we never ask: when pore-network modelers impose periodic boundary conditions on a unit cell, who authorized them? Roberts' phase-shift construction — consider the ensemble of all shifted copies of the medium, in which periodicity becomes a theorem rather than an assumption — is the receipt. (Our kinetic theory, as it happens, never needed the trick: the pore-size axis is separate from space from the start, which is one of the small structural blessings of that formulation.)

What it did to our papers

The test of a framework is whether it changes what you write. Our series — the statistical physics of unsaturated soil water (the kinetic theory itself) and Richards' equation as its hydrodynamic limit (the Chapman–Enskog reduction) — derives the hydraulic conductivity as a Green–Kubo bracket over the relaxation spectrum of the pore network, and the dual-permeability models as a band projection of the same kinetic equation. Roberts' letter, and his theorems, sharpened both in ways that are now in revision. Where we wrote that multiple spectral gaps yield "several Richards equations," the correct statement — his correction, and he is right — is a nested family: each member of the hierarchy rests on a single gap, and reduces to the next by adiabatic elimination, with an exact bookkeeping identity (a sum rule) tying every level to the one conductivity K(θ). Where we invoked the limit Da → 0, the theorems permit the honest, stronger, finite-Da statement: existence of the slow manifold in a finite neighbourhood, attraction at a computable rate, error of the order of the residual. And his two-zone/two-mode equivalence (Roberts & Strunin, 2004) told us something we had not seen: our band description and the spectral description of dual permeability are the same object in two coordinate systems — Gerke–van Genuchten and the eigenmodes of the redistribution operator stop being rivals.

A closing thought on reading, and on how this post came to be

I will confess the obvious: metabolizing forty years of another person's mathematics is slow, and I have not done it alone. This post, and the revisions to our papers that preceded it, were worked out in sustained dialogue with Claude, Anthropic's AI assistant — not as an oracle, but as an interlocutor with whom I could transcribe Roberts' constructions into our own notation, step by step, asking at every turn "what, exactly, does this theorem consume, and what does it deliver?", testing objections, and letting the papers themselves remain the court of appeal. It is a different mode of study than the one I was trained in. It is dramatically faster at one specific thing: locating which of a framework's results are load-bearing for one's own problem, as opposed to true but idle. It does not replace reading the papers — nothing does, and the reading is slower and still under way — but it changed the order of operations: understand first, then read to verify and deepen, rather than read for months hoping understanding arrives.

And it is worth being precise about the causal chain, because none of its links was dispensable. Without Tony Roberts' email, none of this happens: I would not have found his work by searching, since — as the first half of this post argues — the literature's own structure had hidden it from where I was looking. Without the dialogue, his email would have produced a polite acknowledgment and a citation, not a restructured argument: the nested-family correction, the finite-Da certificate, the band–mode equivalence, the sum rule — each of these took days of back-and-forth to extract, verify, and fold into the manuscripts, work that by traditional means would have taken me months, if I had attempted it at all. And without the papers — his and ours — there would have been nothing to connect. A generous correspondent, a tireless interlocutor, and the primary literature: the triangle is the method, and I suspect it is quietly becoming the method of many of us. Better to say so openly than to let the acknowledgments pretend otherwise.

For those who want the patient version: start with the 2025 TMA paper for the panorama, the 2015 IMA paper for the spatial theory, and his SIAM book (Model Emergent Dynamics in Complex Systems, 2015). Hydrology runs, and has always run, on reduced models. It is a strange comfort to learn that there exists a body of theorems about when we are allowed to.

My thanks to Tony Roberts for writing, and for reading us first.

References

Roberts, A. J. (2025). Accurate families of multi-continuum micromorphic homogenisations in multi-D space-time via dynamical systems theory. Trans. Math. Appl. 9, tnaf001. doi:10.1093/imatrm/tnaf001
Roberts, A. J. (2015). Macroscale, slowly varying, models emerge from the microscale dynamics in long thin domains. IMA J. Appl. Math. 80, 1492–1518. doi:10.1093/imamat/hxv004
Roberts, A. J. (2015). Model Emergent Dynamics in Complex Systems. SIAM, Philadelphia. doi:10.1137/1.9781611973563
Bunder, J. E., Roberts, A. J. (2021). Nonlinear emergent macroscale PDEs, with error bound, for nonlinear microscale systems. SN Appl. Sci. 3, 1–28. doi:10.1007/s42452-021-04229-9
Hochs, P., Roberts, A. J. (2019). Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions. J. Differ. Equ. 267, 7263–7312. doi:10.1016/j.jde.2019.07.021
Roberts, A. J., Strunin, D. V. (2004). Two-zone model of shear dispersion in a channel using centre manifolds. Q. J. Mech. Appl. Math. 57, 363–378. doi:10.1093/qjmam/57.3.363
Gorban, A. N., Karlin, I. V. (2014). Hilbert's 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. Bull. Amer. Math. Soc. 51, 187–246. doi:10.1090/S0273-0979-2013-01439-3
Rigon, R. (2026). The statistical physics of unsaturated soil water. arXiv:2607.09416
Rigon, R. (2026). Richards' equation as a hydrodynamic limit: Chapman–Enskog reduction of the continuum kinetic equation for unsaturated soil water. arXiv:2607.17358