Membrane Fluctuations — Helfrich Undulations

Lipid bilayer membranes fluctuate thermally. The Helfrich Hamiltonian penalizes bending (κ) and tension (σ). Fourier modes are independent with mean-square amplitude set by k_BT.

Bending κ (k_BT)10.0

Tension σ (pN/μm)0.20

Temperature k_BT1.0

Layers3

⟨|h_q|²⟩ = k_BT / (κq⁴ + σq²) · A⁻¹

RMS height: — | Persistence length: —

Each Fourier mode h(q) fluctuates independently with amplitude ∝√(k_BT/E(q)). Stiffening membrane (high κ) suppresses fluctuations. Tension stretches the membrane flat. Persistence length ξ_p = exp(4πκ/3k_BT).