KPZ Height Fluctuations

Kardar-Parisi-Zhang equation: ∂h/∂t = ν∇²h + λ(∇h)²/2 + η(x,t)
Height profile h(x,t)
Width w(t) — log-log scaling
t=0 | w=0.00
Nonlinearity λ1.0
Diffusion ν1.0
Noise σ1.0
Grid N200
The KPZ equation ∂h/∂t = ν∇²h + (λ/2)(∇h)² + η(x,t) describes interface growth. The nonlinear term λ(∇h)²/2 is the key difference from Edwards-Wilkinson (λ=0). KPZ scaling: w(L,t) ~ L^α f(t/L^z) with α=1/2, β=1/3, z=3/2 in 1+1D. The height fluctuations are non-Gaussian — they follow the Tracy-Widom distribution (GUE), connecting KPZ to random matrix theory. Set λ=0 for Edwards-Wilkinson (β=1/4) and compare: yellow = KPZ, gray = EW.