Statistical Mechanics - Physics at Oregon State University

Appendix A
Solutions to selected
problems.
A.1 Solutions for chapter 1.
Problem 4.
There are several ways to attack this problem. Use a computer, or use the
formula in the notes:
N! = N N e −N √ 1 −
2πNe 12
where R(N) is of order N −2 .
The values we use in this problem are
(I) N! ≈ N N
(II) N! ≈ N N e N
(III) N! ≈ N N e −N√ N
(IV) N! ≈ N N e −N√ 2πN
1
N +R(N)
This gives for the relative error ɛ = |1 − approximation
N! |:
N 1 1
(I) ɛ1 ≈ |1 − e √ + e 12
2πN
(II) ɛ2 ≈ |1 − 1 1
√ + e 12
2πN
(III) ɛ3 ≈ |1 − 1 1
√ + e 12
2π
1 + (IV) ɛ4 ≈ |1 − e 12
1
1
1
1
N −R(N) |
N −R(N) |
N −R(N) |
N −R(N) |
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