Statistical Mechanics - Physics at Oregon State University

1.8. PROBLEMS FOR CHAPTER 1 21
Problem 3.
A system has N sites. The probability that a given site is occupied by an
atom is ξ. Only one atom at a time is able to occupy a site.
(a) Find the probability p(M, N, ξ) that exactly M sites out of N are occupied.
Assume that on the average A sites are occupied, hence ξ = A
N .
(b) Find the probability p(M,A) that M sites are occupied when the average
number of sites occupied is A in the limit N → ∞. A and M remain finite!
(c) Show that ∞
M=0
< M 2 >= ∞
p(M, A) = 1, < M >= ∞
M
M=0
2p(M, A) = A2 + A.
M=0
Mp(M, A) = A, and that
When A is very large, the distribution p(M,A) becomes very sharp around A.
(d) Show that in that case the distribution near A can be approximated by a
Gaussian of width √ A.
Problem 4.
Use the following approximations derived from Stirling’s formula:
(I) log(N!) ≈ N log(N)
(II) log(N!) ≈ N log(N) − N
(III) log(N!) ≈ N log(N) − N + 1
2 log(N)
(IV) log(N!) ≈ N log(N) − N + 1
1
2 log(N) + 2 log(2π)
For each of these approximations find the smallest value of N for which the
relative error in N! becomes less than 1%. ( Relative error: .)
Problem 5.
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A standard game show problem is the following. You have to choose between
three doors. Behind one of them is a car, behind the others is nothing. You
point at one door, making your choice. Now the game show host opens another
door, behind which there is nothing. He asks you if you want to change your
choice of doors. Question: should you change? Answer: Yes!!!!! Question:
Calculate the ratio of the probabilities of your original door hiding the car and
the other, remaining door hiding the car.