Environmental Engineering Reference
In-Depth Information
used in MD simulation. One is the deterministic type, such as Nose-
Hoover (NH) heat bath [25, 26], and the other is the stochastic type,
such as Langevin heat bath [27].
The evolution of the atoms in thermal contact with NH heat bath
isruled bythe following equations
d
q
i
dt
=
∂
dt
=−
∂
p
i
,
d
p
i
H
H
q
i
−
ζ
p
i
,
(1.25)
∂
∂
where
H
is the Hamiltonian of the system,
p
i
and
q
i
are the
momentum and coordinate of atom
i
,respectively,and
ζ
is an
auxiliary variable modeling the microscopic action of the heat bath.
The dynamics of
ζ
isgoverned bythe following equation:
⎛
⎝
⎞
⎠
i
=
1
N
p
i
·
p
i
m
i
3
Nk
B
T
−
1
d
dt
=
1
τ
,
(1.26)
2
where
T
and
τ
aretheaimedtemperatureandresponsetimeofheat
bath,respectively,
m
i
isthemassofatom
i
,and
N
isthetotalnumber
of atomsthat are in contact withheat bath.
With Langevin heat bath, the equation of motion can be
described as
dt
=−
∂
H
d
p
i
∂
q
i
+
ξ
−
λ
p
i
,
(1.27)
where the random force
ξ
and the dissipation rate
λ
are introduced
into the system simultaneously. This kind of stochastic excitation
is consistent with the microscopic picture of Brownian motion.
The random force
ξ
follows the Gaussian distribution with zero
mean value and variance of 2
m
λ
k
B
T
according to the fluctuation-
dissipation theorem [28].
1.4.2
Quantum Correction
In a classical system, all the modes (degrees of freedom) are equally
exited. Based on the equipartition theorem between the kinetic and
potential energy, each mode has a constant energy of
k
B
T
, and thus
the total energy ofa classical system with
N
atoms is given by
E
C
=
3
Nk
B
T
C
,
(1.28)
