1.4 Statistical Ensembles
21
probability distribution will be quite sharply peaked for large values of N and that
the value n corresponding to the position of the peak is also quite large. For N =
10 12 say, n 5 × 10 11 or so, and changing n by one unit, i.e., by 1 part in 10 12 , will
make little change in p(N, n). This is a way of saying that we have an expression
that reminds us of the condition for a differential, namely,
|p(N, n + 1) − p(N, n)| | p(N, n) .
This also suggests that we may profit from approximating this distribution by
an equivalent continuous distribution so that we can utilize the calculus for our
analysis of its behaviour. As we are actually only really interested in the region
of the maximum in p(n, N), let us henceforth focus upon it. From the calculus, the
condition for a maximum is
dp
dn
= 0 ,
with p(n) ≡ p(n, N) given by Eq. (1.4.7). As this expression has several factors
that depend upon n, it suggests that it would be useful to employ the logarithmic
derivative here. Thus, let us work with ln p, which is given by
ln p(N, n) = ln N! − ln n! − ln(N − n)! + n ln p + (N − n) ln q ,
rather than with p itself. The appropriate derivative expression is hence
−
d
dn
ln n! −
d
dn
ln(N − n)! + ln p − ln q = 0 ,
or, equivalently,
d
dn
ln n! +
d
dn
ln(N − n)! = ln
p
q
.
(1.4.8)
If we now employ the Stirling approximation (see Appendix C) for ln x!, i.e., ln x! !
x ln x − x, which implies that (d/dx) ln x! ! ln x, we obtain the result
d
dn
ln p(N, n) = 0 ln
N − n
n
+ ln
p
q
.
We may further simplify this expression into
ln
N − n
n
p
q
= 0 ,
from which we obtain the result
21
probability distribution will be quite sharply peaked for large values of N and that
the value n corresponding to the position of the peak is also quite large. For N =
10 12 say, n 5 × 10 11 or so, and changing n by one unit, i.e., by 1 part in 10 12 , will
make little change in p(N, n). This is a way of saying that we have an expression
that reminds us of the condition for a differential, namely,
|p(N, n + 1) − p(N, n)| | p(N, n) .
This also suggests that we may profit from approximating this distribution by
an equivalent continuous distribution so that we can utilize the calculus for our
analysis of its behaviour. As we are actually only really interested in the region
of the maximum in p(n, N), let us henceforth focus upon it. From the calculus, the
condition for a maximum is
dp
dn
= 0 ,
with p(n) ≡ p(n, N) given by Eq. (1.4.7). As this expression has several factors
that depend upon n, it suggests that it would be useful to employ the logarithmic
derivative here. Thus, let us work with ln p, which is given by
ln p(N, n) = ln N! − ln n! − ln(N − n)! + n ln p + (N − n) ln q ,
rather than with p itself. The appropriate derivative expression is hence
−
d
dn
ln n! −
d
dn
ln(N − n)! + ln p − ln q = 0 ,
or, equivalently,
d
dn
ln n! +
d
dn
ln(N − n)! = ln
p
q
.
(1.4.8)
If we now employ the Stirling approximation (see Appendix C) for ln x!, i.e., ln x! !
x ln x − x, which implies that (d/dx) ln x! ! ln x, we obtain the result
d
dn
ln p(N, n) = 0 ln
N − n
n
+ ln
p
q
.
We may further simplify this expression into
ln
N − n
n
p
q
= 0 ,
from which we obtain the result