Granular Gas Cooling & Haff's Law

Haff's law (1983): a freely cooling granular gas loses energy as T(t) = T₀/(1 + t/τ)², where τ = (√πσ)/(4nσ(1−e²)) depends on number density n, particle size σ, and restitution coefficient e. Each inelastic collision dissipates energy: ΔE ∝ −(1−e²)v_rel². Unlike elastic gases, the velocity distribution develops non-Gaussian tails (high-velocity overpopulation). At late times, the homogeneous cooling state breaks down via clustering instability — visible as density fluctuations that grow as particles clump in cold regions.