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What are the known results on the change in Shannon entropy $\Delta H_{k} = H(\vec{p}_{k}) - H(\vec{p}_{k-1})$ of the $k$-th step in a process governed by a finite state discrete time Markov chain with a general transition matrix $T$ (left stochastic matrix) and initial probability distribution $\vec{p}_{0}$, such that $\vec{p}_{k+1} = T \vec{p}_{k}$? Here, $H(\vec{p})=\sum_{i}-p_{i}\log(p_{i})$ denotes the Shannon entropy of the probability distribution denoted by the vector $\vec{p}$ with elements $\{p_{i} \}$.

In particular, what results are known for its limiting/long term behaviour $\lim_{k\to\infty}\Delta H_{k}$?

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