Helfrich Membrane Fluctuations

Helfrich Hamiltonian: The elastic energy of a fluid membrane is H = ½∫[κ(∇²h)² + σ(∇h)²] dA, where h is the height field, κ is the bending rigidity, and σ is membrane tension. In Fourier space, each mode q decouples: E_q = ½(κq⁴ + σq²)|h_q|². By equipartition theorem, ⟨|h_q|²⟩ = k_BT / (κq⁴ + σq²). The mean-square displacement grows as ⟨Δh²⟩ ~ k_BT/(4πκ) × ln(L/a) at zero tension. Typical lipid bilayers have κ ≈ 10–25 k_BT. High rigidity (κ large) suppresses fluctuations; tension flattens the spectrum at low q. The spectrum is displayed as log-log power spectral density vs wavenumber q.