= Solution
Let $w_{ij}$ be the transition rate from $i$ to $j$ and define the probability current $J_{ij}=w_{ij}P_i-w_{ji}P_j=-J_{ji}$. The <master equation> is $\dot P_i=-\sum_jJ_{ij}$. Differentiating the Shannon entropy $S_{\rm sys}=-\sum_iP_i\log P_i$ and symmetrizing gives
$$
\dot S_{\rm sys}=\frac12\sum_{i,j}J_{ij}\log\frac{P_i}{P_j}.
$$
Adding the entropy flow to the environment produces the nonnegative <entropy production rate of a Markov chain>
$$
\boxed{\dot S_{\rm tot}=\frac12\sum_{i,j}J_{ij}
\log\frac{w_{ij}P_i}{w_{ji}P_j}\geq0.}
$$
The paper's displayed $\sum P\log P$ 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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