Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 1 / a

Solution

Let

w_{ij}

be the transition rate from

i

j

J_{ij} = w_{ij} P_{i} - w_{ji} P_{j} = - J_{ji}

\dot{P}_{i} = - \sum_{j} J_{ij}

. Differentiating the Shannon entropy

S_{sys} = - \sum_{i} P_{i} lo g P_{i}

and symmetrizing gives

\dot{S}_{sys} = \frac{1}{2} \sum_{i, j} J_{ij} lo g \frac{P _{i}}{P _{j}} .

Adding the entropy flow to the environment produces the nonnegative entropy production rate of a Markov chain

\dot{S}_{tot} = \frac{1}{2} i, j \sum J_{ij} lo g \frac{w _{ij} P _{i}}{w _{ji} P _{j}} \geq 0.

The paper's displayed

\sum P lo g P

is the negative of the thermodynamic system entropy, so its derivative has the opposite sign before the environmental contribution is added.

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