The Statistical physics of unsaturated soil water: kinetic theory and non commutative pore water dynamics

Abstract

We develop a statistical-mechanical theory of water in unsaturated soil whose outcome is a continuum field equation for the pore-occupancy g(r,x,t), the fraction of pores of radius r that are water-filled at position x and time t. The theory is built across three scales: microscopic inter-pore transfers set by Hagen-Poiseuille rates and a driving potential (the difference of pore-class chemical potentials, taken in capillary-gravitational form but open to adsorptive, osmotic, or thermal refinement); a mesoscale master equation relaxing the occupancy toward the equilibrium step g_eq=H(r*-r); and, on contracting the averaging volume to a point, the continuum balance d_t g + div F = C[g] - E - T, of which everything else is a limit, a moment, or a boundary resolution. The kinetic equation is an Onsager gradient flow descending the Gibbs free energy, with an H-theorem for the isothermal unforced system and mass conservation as its zeroth moment. A single dimensionless group, the pore-resolved Damkohler number Da(r,x), organizes the behavior and unifies phenomenologies long modelled separately. A Chapman-Enskog reduction identifies Richards' equation as the quasi-static (Da->0) limit, with matric potential and hydraulic conductivity K emerging only there and K vanishing below the percolation threshold; capillary-bundle and critical-path models are its diagonal and spectral limits. Hysteresis is the holonomy of a forcing bundle, a geometric phase rather than per-pore bistability, with a falsifiable loop-area law H ~ I^2. Preferential flow is what the same equation does where Da>1, so the Richards/preferential-flow dichotomy becomes a continuous Da-controlled crossover. Out of the quasi-static limit g(r) is the irreducible state variable. All inputs are geometric properties of the pore network, measurable from micro-CT and calibrated against no macroscopic data.