The Computable Parameter

Diffusion — particles spreading out over time with displacement growing as the square root of time — is the hallmark of random motion. Brownian motion, the random walk, heat conduction: all diffusive, all stochastic. The connection between randomness and diffusion is so deep that observing diffusion is typically taken as evidence of underlying randomness.

Nizhnik (arXiv:2501.00182) constructs deterministic dynamical systems on tiled phase spaces — regular triangular and hexagonal tilings — that exhibit normal diffusion. No randomness, no noise, no probability. The system is a map from positions and velocities to positions and velocities, completely specified and repeatable. Yet particles spread diffusively, with displacement scaling as √t indistinguishable from a random walk.

The mechanism: the tiling creates a periodic array of scattering events. Each tile boundary deflects the trajectory, and the sequence of deflections, while deterministic, generates enough effective mixing within each tile to produce diffusive transport across tiles. The diffusion constant is computable from the geometry — it’s a property of the tiling, not of any random process.

The structural lesson: diffusion is a macroscopic transport property, not a microscopic mechanism. It tells you how displacements grow with time, not why. A purely deterministic system can produce the same macroscopic signature as a stochastic one, because diffusion is an emergent property of mixing — and mixing can come from geometry as easily as from noise.

The √t is real. The randomness is optional.