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) - . . . .