7.1 Introduction
363
p 1 in an interval [, u] such that the first term in expression (7.1.9) vanishes, we see
that
Ce
−βH βp 1
∂H
∂p 1
N
i=1
dr i dp i ≡ β
p 1
∂H
∂p 1
= 1 ,
in which · · · · is defined as
· · · =
· · · e
−βH
N
i=1
dr i dp i
e
−βH
N
i=1
dr i dp i
.
(7.1.10)
We shall re-express this result as
p 1
∂H
∂p 1
= k B T ,
(7.1.11a)
which Tolman [1] termed the general equipartition principle. Note that if p 1 appears
quadratically in H, then Eq. (7.1.11a) reduces to the usual equipartition statement,
namely p 2
1 =
1
2 k B T . This generalization of the equipartition result turns out to be
quite useful when an external potential, such as that associated with a gravitational
field, is applied to a gas. The three-dimensional (3D) version of this generalized
equipartition principle is
p ·
∂H
∂p
= 3k B T .
(7.1.11b)
The generalization (7.1.11a) of the classical equipartition principle is predicated
upon a vanishing of the first term in Eq. (7.1.9), most commonly due to unboundedness of the domains of integration. The result (7.1.11a) makes clear the role
played by boundary conditions (via vessel walls) even for the case of a quadratic
dependence of the Hamiltonian upon position coordinates. For example, as the
Hamiltonian for an ideal gas of 3D simple harmonic oscillators of mass m can be
written as
H =
i
H i ,
with
H i =
1
2 kr
2
i +
1
2 p
2
i /m ,
the classical equipartition principle gives
H i = =
1
2 kr
2
i + +
1
2 p
2
i /m = 3k B T ,
363
p 1 in an interval [, u] such that the first term in expression (7.1.9) vanishes, we see
that
Ce
−βH βp 1
∂H
∂p 1
N
i=1
dr i dp i ≡ β
p 1
∂H
∂p 1
= 1 ,
in which · · · · is defined as
· · · =
· · · e
−βH
N
i=1
dr i dp i
e
−βH
N
i=1
dr i dp i
.
(7.1.10)
We shall re-express this result as
p 1
∂H
∂p 1
= k B T ,
(7.1.11a)
which Tolman [1] termed the general equipartition principle. Note that if p 1 appears
quadratically in H, then Eq. (7.1.11a) reduces to the usual equipartition statement,
namely p 2
1 =
1
2 k B T . This generalization of the equipartition result turns out to be
quite useful when an external potential, such as that associated with a gravitational
field, is applied to a gas. The three-dimensional (3D) version of this generalized
equipartition principle is
p ·
∂H
∂p
= 3k B T .
(7.1.11b)
The generalization (7.1.11a) of the classical equipartition principle is predicated
upon a vanishing of the first term in Eq. (7.1.9), most commonly due to unboundedness of the domains of integration. The result (7.1.11a) makes clear the role
played by boundary conditions (via vessel walls) even for the case of a quadratic
dependence of the Hamiltonian upon position coordinates. For example, as the
Hamiltonian for an ideal gas of 3D simple harmonic oscillators of mass m can be
written as
H =
i
H i ,
with
H i =
1
2 kr
2
i +
1
2 p
2
i /m ,
the classical equipartition principle gives
H i = =
1
2 kr
2
i + +
1
2 p
2
i /m = 3k B T ,