Quantum Statistics
213
7.2 STATISTICAL DISTRIBUTIONS
The number of distinguishable arrangements for a given distribution of a mixture
of the three classes of particles is
P(q i , r i , s i ) = (Q i )
(
–1 )!
!
!
!( –1)!
!(
)!
i
q
i
i
i
i
i
i
i
i
i
i
i
f
r g
h
q
r g
s h s
∞ 
 


+
Π 
 


−





(7.11)
The number of particles of each class should be conserved. Assuming that
the total energy of the system is E,
i
i
q
∑ = ,
,
i
i
i
i
Q
r R
s S
=
=
∑
∑
(7.12)
i
∑ ε i (q i + r i + s i ) = E
(7.13)
The most probable distribution corresponds to the maximum of P (q i , r i , s i ),
subject to the conditions (7.12) and (7.13).
In practice, it is more convenient to maximize ln P (q i , r i , s i ). The calculations
are greatly simplified by using the following approximation (Stirling’s formula):
ln n ! = ln 2 + ln 3 + ...+ ln n
=
1/2
1
ln
0(1)
n
x dx
+
+
∫
≈ n ln n – n
(7.14)
where for large n only the first two leading terms have been retained. Then
keeping only the leading terms gives
ln P = ln
Q Q Q
− +
i
∑ (q i ln f i – q i ln q i + q i )
+
i
∑ [ (r i + g i ) ln (r i + g i ) – (r i + g i )
–r i ln r i + r i – g i ln g i + g i ]
+ i
∑ [ h i ln h i – h i – s i ln s i + s i
– (h i – s i ) ln (h i – s i ) + (h i – s i )]
(7.15)
At the maximum, this expression has to be stationaryfor small but arbitrary
changes in q i , r i and s i subject to the constraints (7.12) and (7.13). Taking the
differential of ln P, one gets