360
7 Classical Statistical Mechanics
in which the Hamiltonian H j is defined via H j ≡ H(p j 1 , · · · , p js ; q j 1 , · · · , q js ).
By relabelling the conjugate momenta and coordinates for the molecules so that
the variables p nk and q nk for molecule n (for k = 1, · · · , s) are mapped onto the
variables p i and q i as p ns+k , q ns+k , we may write Z N (V , T ) as
Z N (T , V ) =
1
N!h sN
· · ·
e
−β
j H j
sN
i=1
dp i dq i
=
1
N!h sN
· · ·
e
−βH
sN
i=1
dp i dq i ,
in which H ≡
j H j is the Hamiltonian for the N-molecule system.
7.1.1 Energy Equipartition in Classical Mechanics
For a structureless classical ideal gas of N point particles, each of mass m, the
Hamiltonian H is typically given as
H
r
N , p
N
=
1
2m
N
i=1
p
2
i ,
(7.1.3)
in which r N = (r 1 , · · · r N ) and p N = (p 1 , · · · p N ) are vectors in the 6Ndimensional classical phase space. The absence of interparticle potential energy
terms in expression (7.1.3) is, of course, consistent with the concept of an ideal gas:
the point particles are assumed to undergo only elastic collisions with one another
via delta-function repulsive interactions. We may note, however, that H will depend
upon r N should an external potential energy function V ext (r N ) be imposed upon the
gas.
For the present discussion, we shall focus only upon a structureless classical
ideal gas to which no such external field has been applied. When such a gas in
a container of volume V is placed in thermal contact with its surroundings and
allowed to equilibrate via inelastic collisions with its energy-permeable (but massimpermeable) walls, we may represent it in terms of a canonical ensemble of N
particles that is characterized by volume V and temperature T .
The probability p(r N , p N ) that an N-particle system in thermal equilibrium at
a temperature T be in a state in which its N particles have coordinates {r i } and
momenta {p i }, i = 1, · · · N, lying within respective volume elements {dr i } and
{dp i } is
p
r
N , p
N
N
i=1
dr i dp i = Ce
−βH(r N ,p N )
N
i=1
dr i dp i
(7.1.4a)
7 Classical Statistical Mechanics
in which the Hamiltonian H j is defined via H j ≡ H(p j 1 , · · · , p js ; q j 1 , · · · , q js ).
By relabelling the conjugate momenta and coordinates for the molecules so that
the variables p nk and q nk for molecule n (for k = 1, · · · , s) are mapped onto the
variables p i and q i as p ns+k , q ns+k , we may write Z N (V , T ) as
Z N (T , V ) =
1
N!h sN
· · ·
e
−β
j H j
sN
i=1
dp i dq i
=
1
N!h sN
· · ·
e
−βH
sN
i=1
dp i dq i ,
in which H ≡
j H j is the Hamiltonian for the N-molecule system.
7.1.1 Energy Equipartition in Classical Mechanics
For a structureless classical ideal gas of N point particles, each of mass m, the
Hamiltonian H is typically given as
H
r
N , p
N
=
1
2m
N
i=1
p
2
i ,
(7.1.3)
in which r N = (r 1 , · · · r N ) and p N = (p 1 , · · · p N ) are vectors in the 6Ndimensional classical phase space. The absence of interparticle potential energy
terms in expression (7.1.3) is, of course, consistent with the concept of an ideal gas:
the point particles are assumed to undergo only elastic collisions with one another
via delta-function repulsive interactions. We may note, however, that H will depend
upon r N should an external potential energy function V ext (r N ) be imposed upon the
gas.
For the present discussion, we shall focus only upon a structureless classical
ideal gas to which no such external field has been applied. When such a gas in
a container of volume V is placed in thermal contact with its surroundings and
allowed to equilibrate via inelastic collisions with its energy-permeable (but massimpermeable) walls, we may represent it in terms of a canonical ensemble of N
particles that is characterized by volume V and temperature T .
The probability p(r N , p N ) that an N-particle system in thermal equilibrium at
a temperature T be in a state in which its N particles have coordinates {r i } and
momenta {p i }, i = 1, · · · N, lying within respective volume elements {dr i } and
{dp i } is
p
r
N , p
N
N
i=1
dr i dp i = Ce
−βH(r N ,p N )
N
i=1
dr i dp i
(7.1.4a)