Biomedical Engineering Reference
In-Depth Information
A standard method for tackling
constrained variational
problems of this kind is
Lagrange's method of undetermined multipliers
, which is discussed in detail in all the
advanced texts. I will just outline the solution without justification.
The first step in our derivation is to take logarithms of either side of expression (7.6):
p
ln
W
=
ln
N
!−
ln
N
i
!
(7.8)
i
=
1
For large values of
n
it is appropriate to make use of the Stirling formula for factorials
n
ln
n
O
1
n
2
1
2
ln 2π
1
2
1
12
n
+
ln
n
!=
+
+
−
n
+
+···
(7.9)
In practice all terms except
n
ln
n
are usually omitted and we have
p
ln
W
=
ln
N
!−
(
N
i
ln
N
i
!−
N
i
)
(7.10)
i
=
1
We therefore look for a maximum of ln
W
. We know from elementary differential
calculus that
p
∂ ln
W
∂
N
i
dln
W
=
d
N
i
=
0
(7.11)
i
=
1
at a stationary point. We cater for the constraints by differentiating Equations (7.7)
0
=
d
N
1
+
d
N
2
+···+
d
N
p
d
N
p
ε
p
multiplying by arbitrary undetermined multipliers that I will call α and
0
=
d
N
1
ε
1
+
d
N
2
ε
2
+···+
−
β and then adding:
p
p
p
∂ ln
W
∂
N
i
dln
W
=
d
N
i
+
α
d
N
i
−
β
ε
i
d
N
i
(7.12)
i
=
1
i
=
1
i
=
1
All the d
N
i
can now be treated as independent variables and at a stationary point
∂ ln
W
∂
N
i
+
α
−
βε
i
=
0
(7.13)
All that remains is to use the Stirling formula, and the final result turns out as
N
i
=
exp (α) exp (
−
βε
i
)
(7.14)
Next we have to evaluate the undetermined multipliers α and β; if we sum over all the
allowed quantum states then we have
p
p
N
=
N
i
=
exp (α)
exp (
−
βε
i
)
i
=
1
i
=
1
which gives
N
exp (α)
=
(7.15)
i
=
1
exp (
p
−
βε
i
)
