The entropy of Markov trajectories

The idea of thermodynamic depth put forth by S. Lloyd and H. Pagels (1988) requires the computation of the entropy of Markov trajectories. Toward this end, the authors consider an irreducible finite state Markov chain with transition matrix P and associated entropy rate H(X)=- Sigma /sub i,j/ mu /sub i/P/sub ij/ log P/sub ij/ where mu is the stationary distribution given by the solution of mu = mu P. A trajectory T/sub ij/ of the Markov chain is a path with initial state i, final state j, and no intervening states equal to j. It is shown that the entropy H(T/sub ii/) of the random trajectory originating and terminating in state i is given by H(T/sub ii/)=H(X)/ mu /sub i/. Thus the entropy of the random trajectory T/sub ii/ is the product of the expected number of steps 1/ mu /sub i/ 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/sub ij/) is given by H=K-K+H/sub Delta /, where H is the matrix of trajectory entropies H/sub ij/=H(T/sub ij/); K=(I-P+A)/sup -1/ (H*-H/sub Delta /); K is a matrix in which the ijth element K/sub ij/ equals the diagonal element K/sub jj/ of K; A is the matrix of stationary probabilities with entries A/sub ij/= mu /sub j/; H* is the matrix of single-step entropies with entries H*/sub ij/=H(P/sub i/)=- Sigma /sub k/ P/sub ik/ log P/sub ik/; and H/sub Delta / is a diagonal matrix with entries (H/sub Delta /)/sub ii/=H(X)/ mu /sub i/.>