Helfrich Elastic Membrane — Thermal Fluctuations

The Helfrich free energy F = ∫ [κ/2·(∇²h)² + σ/2·(∇h)²] dA governs membrane shape fluctuations. High κ suppresses curvature; σ suppresses tilt.

Bending rigidity κ/k_BT 10

Surface tension σ/k_BT·Å⁻² 0.10

Temperature k_BT 1.0

Grid size N 32

Helfrich 1973: membrane described by height field h(x,y) above flat reference.

Mode amplitudes:
⟨|h_q|²⟩ = k_BT / [κq⁴ + σq²]

Persistence length:
ξ_p = a·exp(4πκ/3k_BT)

Mean square displacement:
⟨h²⟩ = k_BT/(4πσ)·ln(L/a) for σ>0

⟨h²⟩: —
⟨(∇h)²⟩: —
ξ_p/a: —

Membrane height h(x,y) — color = height, rendered as 3D surface

Height-height correlation C(q) = ⟨|h_q|²⟩ vs q

2D power spectrum |h̃(q)|² (Fourier space)