The Helfrich free energy F = ∫(κ/2)(∇²h)² + (σ/2)|∇h|² dA gives thermal fluctuations ⟨|h_q|²⟩ = k_BT/(κq⁴ + σq²). Controls shape fluctuations of lipid bilayers, vesicles, and cell membranes.
F = ∫(κ/2)(∇²h)² dA
+ (σ/2)|∇h|² dA
⟨|ĥ_q|²⟩ = kT/(κq⁴+σq²)
Persistence: ξ_p ~ e^(2πκ/kT)