Helfrich Membrane

Lipid bilayer fluctuations: ⟨|h_q|²⟩ = k_BT/(κq⁴+σq²) — bending rigidity κ ≈ 20k_BT

κ = 20 k_BT | σ = 0.00 | ⟨h²⟩ = — nm²

The Helfrich Hamiltonian H = ∫[κ/2(∇²h)² + σ/2(∇h)²]d²r describes membrane shape fluctuations. Each Fourier mode h_q is an independent harmonic oscillator with equipartition ⟨|h_q|²⟩ = k_BT/(κq⁴+σq²). Flicker spectroscopy of red blood cells measures κ ≈ 50k_BT and σ directly. The 1/q⁴ Fourier spectrum is the smoking gun of bending-dominated fluctuations.