Biomedical Engineering Reference
In-Depth Information
8.7 Configurational Integral
Returning now to the canonical partition function, Equation (8.3)
exp
E
i
k
B
T
Q
=
−
i
the first thing to note is that all points in phase space contribute to the sum, and the summation
has to be replaced by an integral. For an ideal monatomic gas the expression becomes
exp
d
p
d
q
1
N
1
h
3
N
E
k
B
T
Q
=
−
(8.15)
!
The equation is often written with the Hamiltonian
H
replacing
E
, for the reasons discussed
above.
The
N
! term is needed in situations where the particles are completely indistinguishable
from one another; for particles that can be distinguished there is no
N
! term. The integrals
have to be done over the spatial variables of all the
N
particles, and also the momentum
variables of the
N
particles. The integral is therefore a 6
N
-dimensional one.
The energy (the Hamiltonian) is always expressible as a sum of kinetic and potential
energies, and I have written the mass of each particle
m
:
N
p
i
2
m
+
E
=
Φ (
q
1
,
q
2
, ...,
q
N
)
(8.16)
i
=
1
Kinetic energies depend on the momentum coordinates
p
. All the potential energies we
will meet depend on the spatial coordinates
q
but not on the momenta and so the partition
function can be factorized into a product of a kinetic part and a potential part:
exp
d
p
exp
d
q
h
3
N
N
1
N
1
1
k
B
T
p
i
2
m
Φ
k
B
T
Q
=
−
−
(8.17)
!
i
=
1
The kinetic integral has to be done over the momentum coordinates of all
N
particles, and
it can be seen to be a product of
N
identical three-dimensional integrals of the type
exp
d
p
1
1
k
B
T
p
1
2
m
−
Each of these is a product of three identical standard integrals of the type
exp
d
p
x
1
k
B
T
p
x
2
m
−
and the final result is
2π
mk
B
T
h
2
3
N
/2
exp
d
q
1
N
Φ
k
B
T
Q
=
−
(8.18)
!
The 3
N
-dimensional integral over the position variables is often referred to as the
con-
figurational integral
. For an ideal gas Φ
0 and so the configurational integral is
V
N
,
where
V
is the volume of the container. Some authors include
N
! in the definition of the
configurational integral.
=
