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

For the oriented edge $i\to j=i+1(\mathrm{mod}3)$, $J_i=\alpha P_i-\beta P_j$. Substitution in the Markov-chain entropy production givesSince ${\sum }_i{J}_i=\alpha -\beta$,The second term is relaxational and vanishes for the uniform stationary distribution. The steady entropy exported to the environment is therefore