the argument that a real physical system has no reason to be initiated with a uniform distribution, giving maximum entropy for initial conditions. In this case, and if the stationary distribution is uniform, then the entropy will grow over time, which represents a simplified version of the second law, as we shall see in the second part. Note that the stationary distribution does not need to be uniform for the entropy to increase. The condition Hoo > H(X)t=t is sufficient, and it is in the domain of physics, not mathematics, to prove that such a condition is representative of real physical systems. Proving the second law of thermodynamics The second part of this appendix provides an elegant information-theory proof of the second law of thermodynamics.2 The tool used to establish this proof is the concept of relative entropy, also called the Kullback-Leibler distance, which was introduced in Chapter 5. Considering two joint probability distributions p(x, y), q(x,y), the relative entropy is defined as the quantity: (D15) T.M. Cover and J.A. Thomas, Elements of Information Theory (New York: John Wiley & Sons, 1991), Ch. 2. Markov chains and the second law of thermodynamics In particular, it was shown that the relative entropy obeys the chain rule: D[p(x, y)||q(x, y)] = D[p(x)llq(x)] + D[p(y)x)||q(y|x)] (D16) D[p(y)llq(y)] D[p(x|y)||g(x|y)], where D[p(.|.)||q(.1.)] is a conditional relative entropy. Finally, an important property is that the relative entropy is always positive (regardless of the arguments being joint or conditional probabilities), except in the specific case p=q, where it is zero (thus, D[pl|q] > 0 if p # q and D[pl|p] = D[q||q] : 0). We shall apply the above properties to the case of Markov chains. In this analysis, the variables Xn and Xn+1 are substituted for the variables x and y, which define the system events from a single source X that can be observed at two successive instants (Xn, Xn+1 E X). Let us assume now that the system evolution is characterized by a time-invariant Markov process. Such 