The compressible Euler equations describe gas dynamics: shocks, rarefaction waves, compressive blow-ups. Add linear friction — a damping term proportional to momentum — and the system eventually relaxes. The gas slows down. The density homogenizes. This is well understood for smooth initial data.
Add multiplicative white noise. Infinite-dimensional, temporally uncorrelated random forcing that should, by any naive accounting, make things worse. The noise is not small. It is white — delta-correlated in time, meaning every instant brings a completely independent random perturbation.
The damped Euler equations absorb the noise and relax anyway.
The solutions converge almost surely and exponentially to a constant steady state. The density, in the long-time limit, obeys the porous medium equation — the same equation governing groundwater flow through rock. The momentum follows Darcy's law — the same empirical relationship describing seepage. Random gas dynamics with friction, driven by white noise, asymptotically becomes groundwater hydrology.
The mechanism is that friction organizes faster than noise disorganizes. The damping creates an exponential contraction in momentum space that overwhelms the diffusive spreading from the stochastic forcing. The sharp entropy moment estimates that prove this are the technical core — they show that the noise contributes bounded entropy per unit time, while the damping removes entropy at a rate proportional to the current entropy level. The damping wins.
Friction is a stronger organizer than white noise is a disorganizer. This is not obvious from the equations — the noise is infinite-dimensional while the damping is a scalar coefficient. But the damping operates on the integral of the motion, while the noise operates on the differential, and integration smooths what differentiation roughens. The system forgets its turbulent past and converges to slow, deterministic seepage.