Thermal fluctuations roughen a liquid-gas interface. Each Fourier mode k has amplitude ⟨|h_k|²⟩ = k_BT/(γk²L), giving interface width w² = (k_BT/2πγ)ln(L/a) — logarithmic roughening in 2D. Below: watch the stochastic interface evolve via Langevin dynamics.
Interface width w: — px | Capillary length λ_c = √(γ/ρg): — | w² ~ (T/2πγ)ln(L/a) in 2D
Langevin: ∂h/∂t = γ∇²h − ρg·h + η(x,t), <η(x,t)η(x′,t′)> = 2k_BT·δ(x−x′)δ(t−t′)