a process is defined by a unique transition probability matrix R, which has the time-independent elements Rxnxn+1 = r(xn+1/xn). Consistently with the property in Eq. (D2), the conditional probabilities are uniquely defined for p and q: (D17) Next, we apply the chain rule in Eq. (D16): D[p(xn+1,xx)llq(xn+1,xx)] = D[p(xx)llg(xx)] + D[p(xn+1/Xn)llq(xn+1/Xn)] = D[p(xx+1)llq(xx+1)] + (D18) Substituting Eq. (D17) in Eq. (D18), we obtain D[p(xx)||q(xx)] + (D19) = D[p(xx+1)llq(xx+1)] + D[p(xn/Xn+1)llq(xn/Xn+1)] or equivalently, since D[r|r] = 0: D[p(xx+1)]|9(xx+1)] = D[p(xx)llq(xx)] - D[p(xn/Xn+1)llq(xn/Xn+1) (D20) Considering the property D[pl|q] > 0, Eq. (D20) shows that D[p(xx)llq(xx)]. This result means that in a time-invariant Markov process, the relative entropy or distance between any two distributions can only decrease over time. In particular, we can choose q = 9st to be a stationary solution of the Markov process. If this solution is unique, then its distance for any other distribution p decreases over time. This means that p converges to the asymptotic limit defined by 9st (it can be shown, although it is not straightforward, that D[pl|qst] = 0 or p~qst in this limit). Assume next that the stationary solution of the Markov process is a uniform distri- bution, which we shall call Ust (namely, ust(x) = 1/N,x E X). From the definition of Appendix D distance (Eq. (D15) applied to single-variable distributions), we obtain (D21) = Hmax - H(X), with Hmax = log N. Since the distance decreases over time while staying positive, the above result means that the system entropy H(X) increases over time towards the upper limit Hmax. This demonstration could be considered to represent one of several possible proofs of the second law of thermodynamics. We should not conclude that the second law implies that the stationary solution of any physical system must be uniform! What was shown is simply that this condition is sufficient, short of being necessary. Appendix E (Chapter 6) From discrete to continuous entropy In this append