156
3 Ensembles: Systems of Particles
in which 1 , , 2 represent the energies of the two subsystems, and 12 represents
the energy of interaction between them. This energy of interaction will, as usual,
be assumed to be so small in comparison with 1 and 2 that it may be neglected
(weak contact). Further, the total number of particles N is related to the numbers of
particles N 1 and N 2 in the two subsystems by
N = N 1 + N 2 .
(3.3.3)
If for subsystem 1, which has N 1 particles, we consider the restrictions 1 →
1 +d 1 , then a certain number of states, which we shall designate by 1 (( 1 , N 1 )d 1 ,
of the combined system satisfy these restrictions in addition to those for the total
energy and number of particles. At the same time we have for subsystem 2 the
corresponding energy relation − 1 → − 1 + , in which d 2 , for
the N − N 1 particles in the subsystem. The corresponding number of states is given
by 2 (( − 1 , N − N 1 ))). The approximation of d 2 by is connected with
the assumption that the reservoir (subsystem 2) is much larger than the system of
interest, so that the vast majority of resides in the reservoir. The probability
p(( 1 , N 1 )d 1 is obtained by taking the ratio of the total number of states available
when the energy and number of particles are partitioned as just described, divided
by the total number of states available for all such partitions: hence, we may write
p(( 1 , N 1 )d 1 =
1 (( 1 , N 1 )dd 1 2 (( − 1 , N − N 1 )))
((, N)))
,
in which
((, N))) =
N 1
0
1 (( 1 , N 1 )) 2 (( − 1 , N − N 1 )d 1 .
(3.3.4)
We can recognize that the total number of microstates available to the entire system
at energy can be obtained by summing over N 1 and integrating over 1 the
expression for the number of microstates for an individual partition. The expression
for p(( 1 , N 1 ) can be simplified to
p(( 1 , N 1 ) =
1 (( 1 , N 1 )) 2 (( − 1 , N − N 1 )
((, N)
.
(3.3.5)
To obtain a more useful explicit expression for p(( 1 , N 1 ) we proceed in much
the same way as we did for the canonical ensemble: the most probable energy is
found from the condition
∂p(( 1 )
∂∂ 1
= 0 ⇒ 1m .
(3.3.6)
3 Ensembles: Systems of Particles
in which 1 , , 2 represent the energies of the two subsystems, and 12 represents
the energy of interaction between them. This energy of interaction will, as usual,
be assumed to be so small in comparison with 1 and 2 that it may be neglected
(weak contact). Further, the total number of particles N is related to the numbers of
particles N 1 and N 2 in the two subsystems by
N = N 1 + N 2 .
(3.3.3)
If for subsystem 1, which has N 1 particles, we consider the restrictions 1 →
1 +d 1 , then a certain number of states, which we shall designate by 1 (( 1 , N 1 )d 1 ,
of the combined system satisfy these restrictions in addition to those for the total
energy and number of particles. At the same time we have for subsystem 2 the
corresponding energy relation − 1 → − 1 + , in which d 2 , for
the N − N 1 particles in the subsystem. The corresponding number of states is given
by 2 (( − 1 , N − N 1 ))). The approximation of d 2 by is connected with
the assumption that the reservoir (subsystem 2) is much larger than the system of
interest, so that the vast majority of resides in the reservoir. The probability
p(( 1 , N 1 )d 1 is obtained by taking the ratio of the total number of states available
when the energy and number of particles are partitioned as just described, divided
by the total number of states available for all such partitions: hence, we may write
p(( 1 , N 1 )d 1 =
1 (( 1 , N 1 )dd 1 2 (( − 1 , N − N 1 )))
((, N)))
,
in which
((, N))) =
N 1
0
1 (( 1 , N 1 )) 2 (( − 1 , N − N 1 )d 1 .
(3.3.4)
We can recognize that the total number of microstates available to the entire system
at energy can be obtained by summing over N 1 and integrating over 1 the
expression for the number of microstates for an individual partition. The expression
for p(( 1 , N 1 ) can be simplified to
p(( 1 , N 1 ) =
1 (( 1 , N 1 )) 2 (( − 1 , N − N 1 )
((, N)
.
(3.3.5)
To obtain a more useful explicit expression for p(( 1 , N 1 ) we proceed in much
the same way as we did for the canonical ensemble: the most probable energy is
found from the condition
∂p(( 1 )
∂∂ 1
= 0 ⇒ 1m .
(3.3.6)