The Entropy of Markov llajectories

ANract-The idea of thermodynamic depth put forth by Lloyd and Pagels requires the computation of the entropy of Markov trajectories. Toward this end we consider an irreducible finite state Markov chain with transition matrix P and associated entropy rate H(X) = c; j p;I’;j log P;j, where p is the stationary distribution given by the solution of /I = MI’. A trajectory T;j of the Markov chain is a path with initial state i, final state j, and no intervening states equal to j. We show that the entropy H(Z’;;) of the random trajectory originating and terminating in state i is given by H(T;i) = H(X)/p;. Thus the entropy of the random trajectory Ti; is the product 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(Tij) is given by