Q, we calculate the mean as defined by N= ipi, which develops into: of the BE distribution is defined by: H = - PQ log(POi) Elementary calculation and substitution of the definitions of P and Q yields H = (N + 1) log(N +1) - N log N, which can also be written H = the limit of large N (or 1/N E 0), we have H 2 log{N[1 + (N + 1) &]} 22 log N. 2 E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (New York: John Wiley & Sons, 1994), Ch. 3, p. 154. E. Desurvire, How close to maximum entropy is amplified coherent light? Opt. Fiber Technol., 6 (2000), 357. Maximum entropy of discrete sources Figure C1 PDF obtained with 1u = 0.1, k = 5, and m1 = 0.1, m2 = 3.9, m3 = 7.1, m4 = 11.5, m5 = 15.3. (a) m1, m2,..., mk are ordered positive real numbers with m1 = 0; (b) k is finite; (c) u < 0 (again to ensure denominator convergence in Eq. (C19). In this case, the PDF solution is (C21) Figure C1 shows a plot of this PDF obtained, for example, with lu = 0.1, k = 5, and m = 0.1, m2 = 3.9, m3 : 7.1, m4 = 11.5, m5 = 15.3. In physics, this distribution characterizes the atomic populations of electrons within a set of k energy levels, when the atoms are in a state of thermal equilibrium. To be specific, let u = -hv/kBT, with: hv = photon energy at frequency V (h = Planck's constant), KB T = phonon energy at absolute temperature T(kB = Boltzmann's constant), Mi = energy of atomic level i divided by hv. The distribution in Eq. (C21) then takes the form: (C22) which is known as the Maxwell-Boltzmann distribution. This distribution shows that at thermal equilibrium, the electron population is highest in the lowest energy level, and decreases exponentially as the energy of the level increases. In particular, the population Appendix C ratio between two atomic levels i and j is given by (C23) where \Eij = (mi - mj)hv is the energy difference between the two levels. The inter- esting conclusion of this analysis is that at thermal equilibrium electrons randomly occupy the atomic energy levels according to a law of maximal e