A Meta-Dynamics Framework for Non-Equilibrium Soil Moisture and Unified Matrix-Macropore Flow.

Soil moisture dynamics exhibit puzzling regime transitions: under slow processes (evaporation, gentle infiltration), matric potential ψ controls behavior as expected from equilibrium theory; under intense rainfall, macropores activate and dominate flow independent of ψ, with connectivity apparently decoupled from capillary forces. Current models treat these as separate phenomena requiring different physics.

At microscopic level, distribution g(r,t) across pore sizes seeks equilibrium governed by chemical potential μ_w[g]. At macroscopic level, imposed matric potential ψ(t) defines target distribution g_eq(r,ψ) toward which the system evolves on relaxation timescale τ_relax (days to weeks). This dual-level structure generates three regimes depending on Damkhölet number, Da:

For Da << 1 (slow processes) system tracks g_eq(ψ) quasi-statically. Matric potential controls which pores fill/drain via Young-Laplace or adsorption forces. Classical equilibrium models, Richards equation is valid and hysteresis absent (equilibrium limit).

For Da ~ 1 (typical field conditions): Partial level coupling. Gap emerges: Δg = g - g_eq ≠ 0. This decoupling creates memory and path-dependence and hysteresis emerges as natural consequence of non-equilibrium, and non commutativity of the dynamic paths. Standard laboratory measurements (Da ~ 5 in 48-hour protocols) capture quasi-steady states with persistent gaps, explaining lab-field mismatch.

For Da >> 1 (intense rainfall) Water invades network via kinetic percolation—fills largest accessible pores first, independent of local ψ. There is macropores activation when topological connectivity threshold reached (Euler characteristic M_3 > M_3^crit), governed by network geometry not capillary forces.

The meta-dynamics framework unifies these regimes: single physics (g(r,t) evolution toward target g_eq(ψ)) with behavior determined by Da-dependent level coupling.

The apparent dichotomy between “matrix flow” and “macropore flow” reflects degree of meta-dynamic coupling, not different physics.

Based on these theoretical arguments, we generalize Richards equation to track connectivity via Euler characteristic χ(x,t), representing macroscopic signature of microscopic distribution. we show how hydraulic conductivity depends on both water content and connectivity: K = K(θ, χ).

We discuss measurement Implications: Different methods probe system at different Da and sample different aspects of g(r,t). Pressure plate (Da ~ 5) measures quasi-steady states. Rainfall simulators (Da ~ 10-100) capture kinetic regime. Tensiometers sample connected pathways weighted by connectivity, not equilibrium ψ. We provide operational definitions relating measurements to meta-dynamic state and Da regime, explaining systematic method-dependent discrepancies as physics not error.

The framework connects structurally to glass physics (Deborah number = Damköhler, measuring level coupling), plasticity theory (internal state variables bridging scales), and exhibits mathematical parallels to gauge theory (though ψ is control parameter, not gauge field), validated through universal patterns across path-dependent systems. Non-commutativity of wetting-drying [W,D] ≠ 0 emerges as topological property, proving path-dependence unavoidable.

Keywords: Meta-dynamics, path-dependence, dual-level structure, non-equilibrium, Damköhler number, macropore flow, connectivity, soil moisture