# Change in Shannon Entropy in Markov Chain Process

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 $$\lim_{k\to\infty}\Delta H_{k}$$?