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Physics 427 Introduction to Astrophysics Problem Assignment #13 Due: Monday, November 25, 2013 BOLTZMANN STATISTICS Suppose that five identical objects are to be placed, one at a time, in either bin A, bin B or bin C according to the following procedure: For each object a die is cast. If the roll of the die produces a “1”, “2”, or “3” the object is placed in bin A. For a “4” or “5” the selected bin is B. If a “6” is cast the object is placed in bin C. (a) What are the normalized statistical weights of the bins, gA, gB and gC? (b) How many distinguishable ways are there of distributing the five objects among the three bins? Let η represent the number of ways. (c) Find the probability of each of the distributions of part (b). Express your results as exact fractions. Use the notation W(NA,NB,NC) = probability of NA particles in bin A, . . . . (d) Test your result in part (c) by verifying that the probabilities W are normalized so that 5 Σ 5 - NA Σ W (NA, NB, 5 - NA - NB ) = 1. NA = 0 NB = 0 (e) If 120 objects are distributed according to the described procedure determine Wmax and compare it with the probability of the unlikely equipartitioned distribution, W(40,40,40), and the extremely unlikely W(0,0,120). (Assume that Ni,max = giN, where Ni,max is the number in the ith cell for which W has its maximum value. This can be proven true if giN is an integer.) (f) Cast a die 120 times and record your distribution (NA,NB,NC). Calculate the probability of your experimental distribution, W(NA,NB,NC). Hint: If necessary you can estimate the values of large factorials using the approximation: ln (N!) = N ln N - N + ½ ln (2πN) + 1/(12N) - 1/(360N 3) + 1/(1260N 5) - . . . .