We claim that the famous exponential decrease of probability with energy is natural, the v...

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of 11 oscillators sharing a total energy of just 5hω0. In the symbols of Section 9.3, N = 11 and M = 5.

(a) Using equation (9-9), calculate the probabilities of ni being 0, 1,2, and 3.

(b) How many particles, Nn would be expected in each level? Round each to the nearest integer. (Happily, the number is still 11, and the energy still 5ω0.) What you have is a distribution of the energy that is as close to expectations as possible, given that numbers at each level in a real case arc integers.

(c) Entropy is related to the number of microscopic ways the macrostate can be obtained, and the number of ways of permuting particle labels with N0, N1 N2, and N3 fixed and totaling 11 is 11 !/(N0!N1!N2!/N3). (See Appendix J for the proof.) Calculate the number of ways for your distribution

(d) Calculate the number of ways if there were 6 particles in n = 0,5 in n = 1, and none higher. Note that this also has the same total energy.

(e) Find at least one other distribution in which the 11 oscillators share the same energy, and calculate the number of ways.

(f) What do your findings suggest?