 Solution

ID: past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2022/section-b/5/e/solution

For a small opposing force, the relevant parameter is

q = βF δ ≪ 1

e^{q} - 1 - q = \frac{q ^{2}}{2} (1 + \frac{q}{3} + \frac{q ^{2}}{12} + \dots),

j_{0} = \frac{2 D}{δ ^{2}} (1 - \frac{q}{3} + \frac{q ^{2}}{36} + O (q^{3})) ⟶ \frac{2 D}{δ ^{2}} .

In particular the zero-force rate is finite, despite the numerator and denominator of the original expression both vanishing.

For

q ≫ 1

, the exponential dominates the denominator:

j_{0} \sim D β^{2} F^{2} e^{- βF δ} = \frac{D}{δ ^{2}} q^{2} e^{- q} .

Opening a monomer-sized gap is exponentially rare at high opposing force. The algebraic

F^{2}

prefactor belongs to this first-passage model and must not be discarded when quoting the leading asymptotic rate.

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