Statistical Mechanics - Physics at Oregon State University

232 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS.
g(N, M) =
N−S,N−S+1,···,NS−1,NS
N
N−S
N−S
NS N
· · · δ S
NS
s=−S Ns,N δ S
s=−S sNs,M
In the limit N → ∞ the log of the multiplicity function is again approximated
by the log of the largest term, and hence we need to find the maximum of
T (N−S, N−S+1, · · · , NS−1, NS) =
S
s=−S
Ns(log(N) − log(Ns))
with the conditions S s=−S Ns = N and S s=−S sNs = M. This can be done
by introducing two Lagrange multiplyers, and we need to minimize:
S
s=−S
U(N−S, N−S+1, · · · , NS−1, NS, α, β) =
Ns(log(N) − log(Ns)) + α(
S
s=−S
Ns − N) + β(
S
s=−S
sNs − M)
Taking the derivatives with respect to the variables Ns and equating these to
zero gives:
or
with α and β determined from
(log(N) − log(Ns)) − 1 + α + βs = 0
N =
M = xN =
Ns = Ne −1+α+βs
S
s=−S
Ns = N
S
s=−S
Ns = N
Therefore, the value of T at this point is:
Tmax =
S
s=−S
S
s=−S
e −1+α+βs
S
s=−S
se −1+α+βs
Ns(1 − α − βs) = (1 − α)N − βxN
which is proportional to N indeed. In order to find how this depends on x we
need to solve the equations for α and β, which are (after dividing by N):
1 = e −1+α
S
e +βs
s=−S