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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-2/6e/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 6E / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

For a stable one-species linear noise approximation, the normalized stationary fluctuation-dissipation relation for a reaction network becomes

η = \frac{Var x}{m ^{2}} = \frac{⟨ r ⟩}{H m} .

To see the factors explicitly, the linearized restoring rate is

a = H / τ = 4 β m

, while the chemical noise intensity is the rate-weighted squared step size

B = λ \cdot 1^{2} + β m^{2} \cdot (- 2)^{2} = 3 λ .

Stationarity of the variance gives

2 a Var x = B

. Using

λ = 2 β m^{2}

therefore yields

Var x = \frac{3 m}{4}, η = \frac{3}{4 m} = \frac{3}{4 ⟨ x ⟩} .

This is the stationary fluctuation result within the stated negligible-fluctuation/linear-noise approximation, not an exact identity for the boundary-corrected discrete chemistry.

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