Edwards-Wilkinson Surface Growth & KPZ Crossover

The Edwards-Wilkinson (EW) equation ∂h/∂t = ν∇²h + η describes a growing interface with surface tension ν and noise η. The interface width W(L,t) ∝ L^α t^β with α=1/2, β=1/4 in 1D. Adding nonlinear (KPZ) term λ(∇h)²/2 changes the universality class: α=1/2→1/2, β→1/3. Dynamic scaling collapse reveals universality.

Surface tension ν 1.0 Noise σ 1.0 KPZ λ 0.0

Interface statistics:
t = 0
W(t) = 0
W²/t^(2β) = —

EW universality class (1D):
Roughness α = 1/2
Growth β = 1/4
Dynamic z = α/β = 2

KPZ universality class (1D):
Roughness α = 1/2
Growth β = 1/3
Dynamic z = 3/2

Scaling: W(L,t) = L^α f(t/L^z)