Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 6C / b

Table of contents
- Solution b

Solution

Taylor-expand the gain terms:

λ P (n - 1) = λ (P - P_{n} + \frac{1}{2} P_{nn} + \dots),

Q (n) = β n^{2} P (n)

,

Q (n + 2) = Q + 2 Q_{n} + 2 Q_{nn} + \dots .

After cancellation of the loss terms, the second-order Kramers-Moyal expansion is

\frac{\partial P}{\partial t} = - \frac{\partial}{\partial n} [(λ - 2 β n^{2}) P] + \frac{1}{2} \frac{\partial ^{2}}{\partial n ^{2}} [(λ + 4 β n^{2}) P] .

Hence the constant-birth pair-annihilation process has

A (n) = λ - 2 β n^{2}, B (n) = λ + 4 β n^{2} .

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