Environmental Engineering Reference
In-Depth Information
2.3 The Dissipative Particle Dynamics Thermostat
The Dissipative Particle Dynamics thermostat improves the Langevin thermostat and
solves the screening problem. In this way, it allows one to simulate hydrodynamics
phenomena at the mesoscale in equilibrium and in presence of flows. It is very
similar to the Langevin thermostat, described in the previous section. Dissipative
and stochastic forces are applied, but to pairs of particles. The thermostat is also
Galilean invariant because it damps relative velocities and not the absolute velocity
of a single particle. The stochastic forces are applied also to pairs of interacting
particles in opposite direction, such that Newton's third law is satisfied. In this way
the total “thermostating” (external and non-conservative) force to a pair of particles is
zero and the momentum is conserved locally. The new expressions for the dissipative
and random forces are
F
i
D
F
ij
;
F
ij
=−
ʳˉ
D
=
(
r
ij
)(
dž
r
ij
·
v
ij
)
dž
r
ij
,
j
(
=
i
)
F
i
R
F
ij
;
F
ij
=
˃
R
ˉ
R
=
(
r
ij
)ʸ
ij
r
ij
,
(3)
j
(
=
i
)
where
r
ij
≡
r
i
−
r
j
=
r
ij
dž
r
ij
and
v
ij
≡
v
i
−
v
j
are the relative positions and
velocities, respectively.
ʳ
is the friction constant and
˃
R
the noise strength. Friction
2
and noise obey the relation
˃
R
=
2
k
B
T
ʳ
, exactly in the same of way as the Langevin
D
R
thermostat does.
ˉ
(
r
ij
)
and
ˉ
(
r
ij
)
are weight functions that have to satisfy the
relation
R
2
D
[
ˉ
]
=
ˉ
.
(4)
in order to fulfill the fluctuation-dissipation theorem (Español and Warren
1995
;
Español
1995
).
ʸ
ij
stands for a random variable with zero mean and second moment
t
t
ʸ
ij
(
t
)ʸ
kl
(
)
=
(ʴ
ik
ʴ
jl
+
ʴ
il
ʴ
jk
)ʴ(
t
−
).
(5)
The standard weight functions found in the literature are:
2
(
1
−
r
/
r
c
)
,
r
<
r
c
,
R
2
D
[
ˉ
]
=
ˉ
=
(6)
0
,
r
≥
r
c
where
r
c
is the cut-off radius for a given molecular model. It is emphasized that Eq. (
6
)
is just the typical choice when the DPD thermostat is employed in conjunction with
“soft” potentials. Any choice satisfying the first equality in Eq. (
6
) would be a suitable
option for the weight functions. In Fig.
1
, we present a sketch with the direction of the
dissipative and stochastic forces of the DPD thermostat on a pair of fluid particles.
The equations of motion including the DPD thermostat are often integrated using
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