Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 3 / a / Solution

For a nearly planar membrane, the quadratic Helfrich energy of a Fourier mode isThe equipartition theorem and $A-A_{\mathrm{proj}}=\int |\nabla h{|}^2{d}^{2}x/2$ giveUsing the stated molecular and system-size cutoffs, $q_{\max }=\pi /a$ and $q_{\min }=\pi /\sqrt{A}$, gives