Biomedical Engineering Reference
In-Depth Information
Ten throws of 100 coins
60
40
20
S1
0
1
2
3
4
5
6
7
8
9
10
Throw
Figure 7.8
One hundred coins tossed 10 times
safety in numbers; if we use 100 coins we can be pretty sure of getting a result between 40
and 60 heads.
To return to the energy problem, because of the very large number of particles that go
to make up a macroscopic sample, we can be very certain that fluctuations about the mean
are negligible.
7.5 Partition Function
As discussed above, I can write Boltzmann's law as
exp
ε
i
k
B
T
−
N
i
N
=
i
=
1
exp
(7.18)
ε
i
k
B
T
p
−
The denominator in the Boltzmann expression turns out to be a useful quantity and so it is
given a special name and symbol:
exp
p
ε
i
k
B
T
Q
=
−
(7.19)
i
=
1
Q
is the partition function. Some authors use
Z
rather than
Q
, and lowercase symbols
z
and
q
are often encountered when the emphasis is on individual particles rather than the
macroscopic collection of particles.
These formulae are correct even when the energy levels are degenerate (i.e. each energy
level corresponds to a number of quantum states), but of course the sums run over the
quantum states and not the energy levels. In the above formulae,
N
i
is the number of
particles in one of the quantum states and many authors choose to write the sums over the
energy levels. For this purpose, they introduce a degeneracy factor
g
i
for each energy level,
and so
g
i
exp
ε
i
k
B
T
−
N
i
N
=
i
=
1
g
i
exp
(7.20)
ε
i
k
B
T
n
−
