towards the same Gaussian distribution, as a consequence of the central-limit theorem (Chapter 1). 15 It is beyond the scope of these chapters to discuss the parallels between entropy in Shannon's information- theory and entropy in physics. Further and accessible considerations concerning this (however com- plex) subject can be found, for instance, in: www.tim-thompson.com/entropy1.html, www.panspermia. org/seconlaw.htm, http://en.wikipedia.org/wiki/Entropy. 16 A moment of order k is, by definition, the mean value of xk, namely, (xk) = Expi. 17 E. Desurvire, How close to maximum entropy is amplified coherent light? Opt. Fiber Technol., 6 (2000), 357. See also, E. Desurvire, D. Bayart, B. Desthieux, and S. Bigo, Erbium-Doped Fiber Amplifiers: Device and System Developments (New York: John Wiley & Sons, 2002), p. 202. 4.5 Exercises The fact that several discrete (and continuous) random physical processes are ruled by exact or near-maximum entropy distributions can be explained by the following: if a system X offers a large number of possible random arrangements Xi (called microstates), and if such arrangements are equally probable (all independent of each other or past microstate history), then the system's entropy H(X) is a maximum, as I have shown earlier. In some physical processes (like amplification of coherent light), there is no reason for microstates to be equiprobable, which explains the discrepancy between observed and maximal entropies. The discrepancy, however, vanishes when the number of possible microstates becomes infinite. In some other physical processes (such as the electron occupation of atomic energy levels, or the spontaneous emission of photons by single atoms), the microstates are strictly equiprobable and the system's entropy is maximum, as we have seen. The so-called maximum entropy principle, 18 is used in several domains of statisti- cal sciences, from engineering and physics to computer vision, image processing and reconstruction, language analysis, urban design, marketing, elections, bu