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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2017/3/a/solution

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Past exam of the theoretical physics major option course of the University of Oxford / 2017 / 3 / a / Solution by

Codex  0 2026-10-25

This is a birth-death master equation:

q_{n}

is the transition rate

n \to n + 1

and

r_{n}

is the transition rate

n \to n - 1

. The first two terms are incoming probability flows from the adjacent states; the last removes probability at the total escape rate

q_{n} + r_{n}

. Rates must be nonnegative. For a population on the nonnegative integers, take

r_{0} = 0

and no incoming flux from negative states.

Summing over all states and shifting the two incoming indices gives

\frac{d}{d t} \sum_{n} P_{n} = \sum_{n} r_{n} P_{n} + \sum_{n} q_{n} P_{n} - \sum_{n} (q_{n} + r_{n}) P_{n} = 0.

Thus an initially normalized probability distribution remains normalized. The cancellation assumes no boundary flux and legitimate summation; on an infinite state space this means the nonexplosive process or a boundary prescription that conserves total probability. All rates used in the later parts satisfy this requirement.

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