al PDF solution (C28) (C29) A0=exp(20-1) = A1 = = exp(21) A1 = exp(A2) (C30) An=exp(An). The , the n + 1 constraints in Eq. (C25) can only be found numerically. An example of a resolution method and its PDF solution in the case n = 2, the event space X being the set of integer numbers, can be found in.4 In these references, it is shown that the photon statistics of optically amplified coherent light (i.e., laser light passed through an optical amplifier) is very close to the PDF solution of maximal entropy. It is straightforward to show that in the general case, the maximum entropy is given by the following analytical formula:5 Hmax=1-(20+21(x)+22(x2y = =1-EMix (C31) Further discussion and extensions of the continuous PDF case of the entropy- maximization problem can be found in.6 4 E. Desurvire, How close to maximum entropy is amplified coherent light? Opt. Fiber Technol., 6 (2000), 357. E. Desurvire, Erbium-Doped Fiber Amplifiers, Device and System Developments (New York: John Wiley & Sons, 2002), Ch. 3, p. 202. We have = - An) = -(20-1+21(x)+22(x2)+..+n(x"> T.M. Cover and J.A. Thomas, Elements of Information Theory (New York: John Wiley & Sons, 1991), Ch. 11, p. 266. Appendix D (Chapter 5) Markov chains and the second law of thermodynamics In this appendix, I shall first introduce the concept of Markov chains, then use it with the results of Chapter 5 concerning relative entropy (or Kullback-Leibler distance) to describe the second law of thermodynamics. Markov chains and their properties Consider a source X of N random events x with probability p(x). If we look at a succession of these events over time, we then observe a series of individual outcomes, which can be labeled Xi (i = 1...n), with Xi E X". The resulting series, which is, thus, denoted X1 Xn, forms what is called a stochastic process. Such a process can be characterized by the joint probability distribution p(x1, X2, , Xn). In this definition, the first argument X1 represents the outcome observed at time t = t1, the second represents the outcome ob