ntropy. It can be shown that:3 (C24) exp (1st)- This results expresses in nats (1 nat = 1.44 bit) the average information contained in an atomic system at temperature T with energy levels separated by AE = hv, with hv >> KBT. The same result holds for a two-level atomic system (k = 2). The quantity (m) hv represents the mean thermal energy stored in the atomic system. The ratio (m) hv/ kB T represents the mean number of thermal phonons required to bring the atom into this mean-energy state. 3 By definition, the source entropy is H = - pi logp By convention, we take here the natural logarithm SO that the unity of entropy is the nat. With solution p = Q"i/P (and P = (QQi), we obtain: which gives H = (m) log Q + log P. We now develop the second term: log(Qm1+Q2.+Qk) = log[Qm1(1+Qm2-m1+...+Q" Substituting Q = exp(- mj)hv the preceding, we obtain: For simplicity, we can assume that the energy levels are all equidistant, i.e. = and E (k - 1)hv. Using the geometric series formula 1+q+q2++.++++ q), the entropy is: (-KAST) Maximum entropy of discrete sources Discrete distributions maximizing entropy under additional constraints In this last section, we consider a general method of deriving discrete PDFs, which maximize entropy while being subject to an arbitrary number of constraints. Assume, for instance, that the constraints correspond to the different PDF moments = (x"), which can be expressed according to: so = 1 - (C25) The functional f to be minimized with the Lagrange multipliers 20, A1, ,22,...,An (C26) Taking the derivative of f with respect to pj yields the development: -logpj-1+2021x2x+x (C27) Considering a two-level atomic system (k = 2), we have, in particular; 1 exp If we assume as well that hv/(kBT) is large enough that the exponential can be neglected, we obtain H 2 is valid for all systems with k > 2. It is then straightforward to determine (m), from the definition (m) = :m1p1+m2p2- The result is the well known "mean occupation number" of Boltzmann's distribution: Appendix C which corresponds to the gener