The Helfrich free energy: F = ∫(κ/2)(2H)² dA + σ∫dA, where H is mean curvature. For a sphere of radius R, the undulation modes u_q fluctuate with:
⟨|u_q|²⟩ = k_BT / (κq⁴ + σq²)
This gives the Floppy membrane q⁻⁴ spectrum at low σ, crossover to q⁻² under tension. Thermal fluctuations visible in giant unilamellar vesicles (flicker microscopy) give κ ≈ 10–25 k_BT for typical membranes.