Chemistry Reference
In-Depth Information
limit
all statistical ensembles are equivalent and can be transformed into
each other via Legendre transformations, for finite
ðN ! 1Þ
these ensembles are not
equivalent [
220
]. If one wishes to study phase transition and phase coexistence,
the use of the microcanonical ensemble is somewhat cumbersome [
221
].
There have been many methods suggested to carry out MD simulations at
given temperatures
T
rather than at given energy
E
. The first approach that was
used was based on velocity rescaling, e.g., the velocities were changed until all
velocities satisfied the relation following from the Maxwell Boltzmann distribu-
tion,
m
i
h
v
i
i¼
N
2. Of course, such a velocity rescaling simulation destroys
one of the advantages of MD, namely the possibility to get detailed accurate
information on time-displaced correlation functions
3
k
B
T
=
t
0
h
A
ðf
r
i
ð
t
ÞgÞ
A
ðf
r
k
ð
t
þ
ÞgÞi
of
the variables; moreover this technique does not lead to a distribution of variables
according to the canonical
N
VT
ensemble of statistical mechanics [
87
]. Alterna-
tively, one can couple the system to “thermostats” [
87
,
218
]. Although the popular
Berendsen thermostat [
222
] does not correspond strictly to the
N
VT
ensemble,
and hence we do not recommend its use, the correct
VT
ensemble is obtained
implementing the Nose Hoover thermostat [
223
,
224
]. In this technique, the
model system is coupled to a heat bath, which represents an additional degree
of freedom represented by the variable
N
zð
t
Þ
. The equation of motion then
becomes:
d
~
r
i
=
dt
¼~
v
i
ð
t
Þ;
m
i
d
~
v
i
ð
t
Þ=
d
t
¼r
i
U
ðf~
r
j
gÞ zð
t
Þ
m
i
~
v
i
ð
t
Þ ;
(38)
so this coupling enters like a friction force. However,
zð
t
Þ
can change sign because
it evolves according to the equation:
!
1
N
m
i
v
i
d
zð
t
Þ=
d
t
¼ð
2
M
b
Þ
N
k
B
T
:
3
(39)
i¼
1
M
b
is interpreted as the “mass of the heat bath”. For appropriate choices of
M
b
, the
kinetic energy of the particles does indeed follow the Maxwell Boltzmann distri-
bution, and other variables follow the canonical distribution, as it should be for the
N
VT
ensemble. Note, however, that for some conditions the dynamic correlations
of observables clearly must be disturbed somewhat, due to the additional terms in
the equation of motion [(
38
) and (
39
)] in comparison with (
35
). The same problem
(that the dynamics is disturbed) occurs for the Langevin thermostat, where one adds
both a friction term and a random noise term (coupled by a fluctuation dissipation
relation) [
75
,
78
]:
m
i
d
2
~
r
i
ð
t
Þ
d
~
r
i
dt
þ
W
i
ð
¼r
i
U
ð
f~
r
i
ð
t
Þg
Þ z
t
Þ ;
(40)
d
t
2
h
W
i
ð
Þ
W
j
ð
t
0
t
0
t
Þi ¼ d
ij
dð
t
Þ
6
k
B
Tz :
(41)
