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7 Classical Statistical Mechanics
Example 7.1 External gravitational field.
A good example of an external field imposed upon a system is afforded by the consideration of the internal energy per particle, u(T ), of an ideal gas of structureless
particles of mass m under the influence of a gravitational field characterized by an
acceleration g acting in the negative z-direction. The single-particle Hamiltonian is
then
H =
p 2
2m
+ mgz .
This Hamiltonian has three quadratic momentum terms describing the translational
motion of the particle along the three Cartesian directions plus a non-quadratic term,
mgz, describing the gravitational contribution to the energy of a particle.
Let us consider a cylindrical column (height H ) of this gas, so that the three
quadratic momentum terms corresponding to Eq. (7.1.11b) thereby give accordingly
a contribution
p
2
=
1
2
p ·
∂H
∂p
=
3
2 k B T
to the internal energy u(T ). Similarly, the non-quadratic term mgz then corresponds
to Eq. (7.1.15), with a = 0, b = H , and V ext (z) = mgz, so that Eq. (7.1.15) gives
z
∂V ext
∂z
H
0
= k B T
⎡
⎢
⎢
⎣ 1 − βmg
H e −βmgH
H
0
e
−βmgz dz
⎤
⎥
⎥
⎦
= k B T
1 − βmgH
e −βmgH
1 − e −βmgH
.
If we now define a dimensionless parameter η via η ≡ βmgH , then we may write
the generalized equipartition result of Tolman [1] as
z
dV ext
dz
H
0
= k B T
1 −
η
e η − 1
.
The sum of these two components thus gives the internal energy per particle, u(T ),
of a structureless ideal gas as
u(T ) = =
1
2 p
2 /m +
z
dV ext
dz
H
0
=
3
2 k B T + k B T
1 −
η
e η − 1
.