The Entropy of Markov

Absftact-The idea of thermodynamic depth put forth by Lloyd and Pageh requires the computation of the entropy af Markov tqjectorim Toward this end we consider an irreducible finite state Markov chain with transition matrix P and pssoeiated entropy rate H ( X ) = E,,, p,P,, log P,,, where p is the stationary distribution given by the solution of p = p P . A wectory T,, of the Markov chain is a path with initial state I, 6 4 state 3, and no intervening states equal to j. We show that the entropy H(T, , ) of the random tqiectory originating and termhating in state z is given by H(T, , ) = H ( X ) / p t . Thus the entropy of the random tqiectory T,, is the pmduct of the expected number of steps l/p, to return to state i and the entropy rate H ( X ) per step for the stationary Markov chain. A general closed form solution for the entropies H(T,,) is given by