266
X. Liu et al.
The diagonal mass matrices are given by
M 1 = 0
M j =
j
j − 1
PM
j = 2, P
.
(27)
Define
φ
ξ 1 , . . . , ξ P
=
1
P
P
j=1
U
x j
ξ 1 , . . . , ξ P
.
(28)
Its derivatives satisfy the chain rule
∂φ
∂ξ 1
=
P
i=1
∂φ
∂x i
=
1
P
P
i=1
U
(x i )
∂φ
∂ξ j
=
∂φ
∂x j
+
j−2
j−1
∂φ
∂ξ j−1
j = 2, P
.
(29)
Adding fictitious momenta (p 1 , . . . , p P ) into Eq. (26) leads to
Z
ξ 1 ≡x 1
= lim
P→∞
P
4π 2 2
3N P/2
|M|
P/2
⎛
⎝
P
j=1
˜
M j
⎞
⎠
−1 / 2
⎛
⎝
P
j=1
dξ j dp j
⎞
⎠
× exp
−β H eff
ξ 1 , . . . , ξ P ; p 1 , . . . , p P
(30)
with the effective Hamiltonian of the form
H eff
ξ 1 , . . . , ξ P ; p 1 , . . . , p P
=
P
j=1
1
2
p
T
j
˜
M
−1
j p j + U eff
ξ 1 , . . . , ξ P
,
(31)
where
U eff
ξ 1 , . . . , ξ P
=
P
j=1
1
2
ω
2
P ξ
T
j M j ξ j + φ
ξ 1 , . . . , ξ P
.
(32)
The fictitious masses are defined as
˜
M 1 = M
˜
M j = M j =
j
j − 1
PM
j = 2, P
(33)
such that all staging modes
ξ 2 , . . . , ξ P
move on the same frequency. The estimator
of the thermodynamic property in Eq. (17) is