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

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- Solution a

Solution

For a nearly planar membrane, the quadratic Helfrich energy of a Fourier mode is

E_{q} = \frac{A}{2} (k_{c} q^{4} + σ_{0} q^{2}) ∣ h_{q} ∣^{2} .

The equipartition theorem and

A - A_{proj} = \int ∣\nabla h ∣^{2} d^{2} x /2

give

α (σ_{0}) = \frac{k _{B} T}{2} \int_{q_{m i n}}^{q_{m a x}} \frac{d ^{2} q}{( 2 π ) ^{2}} \frac{1}{k _{c} q ^{2} + σ _{0}} .

Using the stated molecular and system-size cutoffs,

q_{m a x} = π / a

and

q_{m i n} = π / A

, gives

α (σ_{0}) = \frac{k _{B} T}{8 π k _{c}} lo g (\frac{π ^{2} k _{c} / a ^{2} + σ _{0}}{π ^{2} k _{c} / A + σ _{0}}) .

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