84 Microcanonical ensemble
As an example of the use of the virial theorem, consider the choicexi=pi, a
momentum component, andi=j. IfH=
∑
ip
2
i/^2 mi+U(r^1 ,...,rN), then according
to the virial theorem,
〈
pi
∂H
∂pi〉
=kT
〈
p^2 i
mi〉
=kT
〈
p^2 i
2 mi〉
=
1
2
kT.Thus, at equilibrium, the kinetic energy of each particle must bekT/2. By summing
both sides over all 3Nmomentum components, we obtain the familiar result:
∑^3 Ni=1〈
p^2 i
2 mi〉
=
∑^3 N
i=1〈
1
2
miv^2 i〉
=
3
2
NkT. (3.3.12)3.4 Conditions for thermal equilibrium
Another important result that can be derived from the microcanonical ensemble and
that will be needed in the next chapter is the equilibrium state reached when two
systems are brought into thermal contact. By thermal contact, we mean that the
systems can exchange only heat. Thus, they do not exchange particles, and there is
no potential coupling between the systems. This type of interaction is illustrated in
N , V , EH ( x )Heat-conducting
divider(^1 1 1)
1 1
N , V , E
H ( x )
2 2 2
(^2 2)
Fig. 3.1Two systems in thermal contact. System 1 (left) hasN 1 particles in a volumeV 1 ;
system 2 (right) hasN 2 particles in a volumeV 2.
Fig. 3.1, which shows two systems (system 1 and system 2), each with fixed particle
number and volume, separated by a heat-conducting divider. If system 1 has a phase
space vector x 1 and system 2 has a phase space vector x 2 , then the total Hamiltonian
can be written as
H(x) =H 1 (x 1 ) +H 2 (x 2 ). (3.4.1)
Additionally, we let system 1 haveN 1 particles in a volumeV 1 and system 2 haveN 2
particles in a volumeV 2. The total particle numberNand volumeVareN=N 1 +N 2