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ρ
Middle - NHC
=
1
Z
N
exp
−β
1
2
p
T
(M − A
2
4
)
−1 p +
1
2
x − x eq
T A
x − x eq
+
3N
i=1
M NHC
j=1
p
2
η
(i)
j
2Q j
⎞
⎠
⎤
⎦ ,
(15)
where Z
N is the normalization constant. The stationary marginal distribution is
obtained by integration over p η in Eq. (15) for the phase space variables (x, p),
which is the same as Eq. (11). Similarly, the stationary state marginal distribution of
the physical phase space variables (x, p) for the harmonic system obtained by either
“side-NHC” or “end-NHC” is the same as Eq. (12).
2.2 Simulation results
2.2.1 Quartic potential
We test the quartic potential U (x) = x
4
/4 as a typical example in the anharmonic
region, where the mass is 1 au and the temperature parameter is β ≡
1
k B T
= 8 au.
Fig. 1 MD results for the
average potential energy (of
the quartic potential system)
with various time intervals
(Reproduced with
permission from Ref. [9])
0.031
0.0315
0.032
0.0325
0.033
0.0335
0.034
0.0345
0.035
0
0 . 2
0.4
0.6
0.8
1
U(x)=x
4
/4
β=8
middle-Andersen
middle-Langevin
middle-NHC
side-Andersen
side-Langevin
side-NHC
Potential energy (au)
Time interval (au)