Roughness W = — nm |
Modes: — |
⟨|h_q|²⟩ at q₁: —
The Helfrich free energy: F = ∫ [κ/2 (∇²h)² + σ/2 (∇h)²] dx
gives fluctuation spectrum ⟨|h_q|²⟩ = k_BT / (κq⁴ + σq²).
The membrane roughness W² = (1/L) ∫ ⟨h²⟩ dx diverges logarithmically when σ→0 (Helfrich undulation instability).
Red curve: membrane height; spectrum plot below shows mode amplitudes vs wavenumber.