232
C. Tang (唐晨宇) and Y. Wang (王延颋)
Z Nose =
1
N !
dp s dsd
p
N d r
N
δ(H Nose − E)
=
1
N !
dp s dsd
p
N d r
N
δ
N
i=1
p
2
i
2m i
+ E p ( r
N
) +
p
2
s
2Q
+
l
β
ln s − E
=
1
N !
dp s dsd
p
N d r
N
δ(h)
(5.6.26)
The Hamiltonian of the actual system is
H ( p
, r) =
N
i=1
p
2
i
2m i
+ E p ( r
N
)
(5.6.27)
Thus, we have
h(s) = H ( p
, r) +
p
2
s
2Q
+
l
β
ln s − E
(5.6.28)
To solve
h(s 0 ) = 0
(5.6.29)
We have
s 0 = exp
−
β
l
H ( p
, r) +
p
2
s
2Q
− E
(5.6.30)
and
h
(s) =
dh
ds
=
l
βs
(5.6.31)
δ[h(s)] =
δ(s − s 0 )
|h (s 0 )|
(5.6.32)
We can rewrite (6.26) to be
Z Nose =
1
N !
dp s dsd
p N d r N βs 3N +1
l
δ
s − exp
−
β
l
H (p , r) +
p 2
s
2Q
− E
=
1
N !
β
l
exp
3N + 1
l
E
dp s exp
−β
3N + 1
l
p 2
s
2Q
d
p N d r N exp
−β
3N + 1
l
H (p , r)
=
C
N !
d
p N d r N exp
−β
3N + 1
l
H (p , r)
(5.6.33)