In May I posted the talk I gave at EGU 2026 in Vienna, and promised the two papers “in a couple of weeks after EGU.” It took a little longer than that — it always does — but the first one is now public:
The Statistical Physics of Unsaturated Soil Water: kinetic theory and non-commutative pore-water dynamics
R. Rigon, arXiv:2607.09416 [cond-mat.stat-mech], 22 pages, 9 figures, 2 appendices.
It is going to be submitted to Physical Review E. The companion paper — the Chapman–Enskog derivation that recovers Richards' equation as a hydrodynamic limit — follows shortly, and I will post it here when it lands.
The argument in one sentence
Unchanged from the talk, and worth repeating because everything else is a consequence of it:
Richards' equation is not wrong; it is the equilibrium limit of a deeper kinetic theory — in the same sense that Navier–Stokes is the hydrodynamic limit of Boltzmann's equation for a gas.
Mario Putti asked me, twenty years ago, “if not Richards, what else?” This is my attempt at an answer, and it arrives only after many years spent trying to solve Richards' equation properly — first with GEOtop, later with WHETGEO. You have to take an equation seriously for a long time before you earn the right to say what it is missing.
What the theory actually says
The state variable is not θ. It is the pore-occupancy g(r, x, t): the fraction of pores of radius r that are water-filled at position x and time t. Water content is recovered as a moment of it, θ[g] = φ ∫ g(r) f(r) dr — which is precisely the point: θ is an integral of g, so it throws information away. Two soils with the same θ, one wetted by rain (which fills pores by areal exposure, favouring the large ones) and one drained to the same θ (which empties the large ones first), are in genuinely different hydraulic states. They will conduct water differently, and they will respond to the next rainfall differently. Richards' equation cannot see the difference. That is the figure above, and that is the whole motivation.
The theory is built by passing through three scales, and I think this is the part hydrologists will find easiest to trust, because each step is ordinary physics:
- Microscale. A single water transfer between two pores is set by a Hagen–Poiseuille rate and driven by the difference of pore chemical potentials Φ(r, r′) — capillary and gravitational here, but open to adsorptive, osmotic, or thermal refinement without touching the structure of the theory.
- Mesoscale. Averaging over a representative volume gives a master equation — a gain–loss (Boltzmann-type) kinetic equation whose terms relax the occupancy toward its equilibrium, with a connectivity kernel C(r, r′) that encodes which pores can actually talk to which.
- Macroscale. A Chapman–Enskog reduction gives back Richards' equation in the quasi-static limit Da → 0.
Everything the theory needs as input is a geometric property of the pore network — measurable from micro-CT. Nothing is calibrated against macroscopic hydrological data. I want to be blunt about how unusual that is, and how exposed it leaves me: the theory makes parameter-free predictions, and parameter-free predictions can be wrong in public.
Four things that fall out, which I did not put in
This is the part I care about. These were not assumptions; they are consequences.
1. Matric potential and hydraulic conductivity exist only in the limit. ψ and K are not primitive quantities of the theory. They emerge at Da → 0, and K is derived from the connectivity kernel rather than postulated. Below the percolation threshold, K vanishes — not as a fitting choice, but because the water phase stops spanning the medium. Field capacity gets a geometric meaning: θFC ≈ θc.
2. Hysteresis is geometry, not memory. It is the holonomy of a forcing bundle — a geometric phase, arising from the non-commutativity [W, D] ≠ 0 of the wetting and drying operators. Wetting fills by areal exposure; drying empties by capillary ordering; the two operations do not commute, so a closed loop in the forcing does not return you to where you started. Independent-domain and Preisach models posit bistable pores and reproduce the loop. Here the loop is derived, and it comes with a falsifiable prediction: the loop area scales as H ∼ I² with the forcing intensity. Domain models are rate-independent and predict no such thing. That is a clean experimental discriminant, and I would very much like someone to go and measure it.
3. Preferential flow is not a separate process. It is what the same equation does when Da > 1. The molecular-chaos (Stosszahlansatz) closure that underlies the kinetic equation fails exactly when pore occupancies become correlated near the percolation threshold — and that correlated, channelized regime is fingering and preferential flow. So the Richards / preferential-flow dichotomy dissolves into a continuous, Da-controlled crossover. We do not need two domains and a phenomenological exchange term; we need one equation and an honest look at its Damköhler number.
4. Out of the quasi-static limit, g(r) is irreducible. No single scalar — not θ, not ψ — is a complete description. And, as I discovered while revising the manuscript (a lesson in the value of being asked a hard question at the right moment): this is true even at equilibrium. With gravity present in a finite volume, the equilibrium occupancy is not the sharp step H(r* − r) at all; it is a smeared step, because a large pore low in the profile can stay filled while a smaller pore higher up has already drained. Each pore holds water within its own Jurin rise. The retention curve — the last place where the classical scalar picture was supposed to be exact — is not exact either.
Where this connects
The framework absorbs rather than replaces. Capillary-bundle models are its diagonal limit; critical-path models are its spectral limit; Hassanizadeh–Gray is a thermodynamically consistent extension, here resolved pore-class by pore-class; phase-field methods are gradient flow on a free energy, here with explicit network connectivity; dual-permeability models are the Da > 1 regime, without the phenomenology. The same machinery, with capillary pressure replaced by freezing-point depression, is the freezing-soil problem I have worked on with Niccolò Tubini and John Mohd Wani.
This is not a parallel universe to Richards. It contains it.
What I am not claiming
I would rather say this myself than have it said to me. The paper is a construction, not a rigorous reduction from molecular dynamics: the closures are posited on physical grounds and judged by their consequences. The full kinetic equation has not yet been solved on a real soil — the numerics live in the companion paper and in the supplementary demonstrations. And the single most obvious next step is also the hardest and the most interesting one:
directly observing g(r).
Micro-CT can see it. Nobody, as far as I know, has yet used it to test a kinetic theory of soil water. If you work with imaging of pore-scale water and this sounds like a collaboration, write to me.
Materials
- Paper: arXiv:2607.09416
- The EGU 2026 talk (slides, storyboard, and the notebooks behind the figures): the May post — still the gentlest way in, if the paper looks forbidding.
- Code: will be added on GitHub; the OpenPNM notebooks that generate the supporting figures are already in the OSF repository linked from the talk.
Comments, objections, and counterexamples are all welcome — especially the counterexamples. A theory that cannot be attacked is not saying anything.
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