s. A first type of constraint is to require that the PDF have a predefined mean value (x) = N. In this case, the following conclusions can be reached: If the event space X = {X1,X2,...,XX} is the infinite set of integer numbers 0,x2==,1,x3=2...),k 8, the optimal PDF is the discrete-exponential distribution, also called the Bose-Einstein distribution (see Chapter 1); If the event space X = {x1,x2, xk} is a finite set of nonnegative real numbers, the optimal PDF is the (continuous) Boltzmann distribution. The Bose-Einstein distribution characterizes chaotic processes, such as the emission of light photons by thermal sources (e.g., candle, light bulb, Sun, star) or the spontaneous emission of photons in laser media. The Boltzmann distribution describes the random arrangement of electrons in discrete atomic energy levels, when the atomic systems are observed at thermal equilibrium. As shown in Appendix C, the (maximum) entropy corresponding to the Boltzmann distribution, as defined in nats, is: (4.16) where hv is the quantum of light energy (photon), kT is the quantum of thermal energy (phonon) and (m) is the mean number of phonons at absolute temperature T and oscillation frequency V (h = Plank's constant, kB : Boltzmann's constant). The quantity AE = (m) hv is, thus, the mean electromagnetic energy or heat that can be radiated by the atomic system. The ratio \E/(k) is the number of phonons required to keep the system in such a state. The maximal entropy Hmax, which represents the average information to define the system, is just equal to this simple ratio! This feature Entropy establishes a nice connection between information theory and atomic physics. Letting S = which is consistent with the physics definition of entropy, we note that S = \E/T, which corresponds to the well known Clausius relation between system entropy (S), heat contents (AE), and absolute temperature (T). 15 Consider next the possibility of imposing an arbitrary number of constraints on the probability distribution function (PDF) moments, (x), (