372
M. Gonzales and N. N. Thadhani
also be invoked to construct interatomic potentials for use in MD simulations and
has been successfully deployed to simulate shock compression, cf. [52, 53].
Further coarse-graining via dissipative particle dynamics (DPD) as described
in the works by Español et al. [18–20] and Hoogerbrugge and Koelman [39]
provides a truly mesoscopic simulation technique which links micro-level processes
to meso-level processes via the fluctuation-dissipation theorem [34, 80]. The method
combines lattice gas automata techniques and their time-stepping algorithms from
Frisch et al. [23], which allowed a nonlinear system of particles to evolve microscopically and recover the Navier-Stokes equations, with MD to recover Galilean
invariance and flow isotropy. The original method as employed by Hoogerbrugge
and Koelman [39] involves N particles in a domain V , evolving the system by
initializing positions r i and momenta p i and updating with discrete timesteps δt
for the nth timestep via:
p
n+1
i
= p
n
i +
j
ij e ij
(1)
e ij =
r i − r j
r i − r j
(2)
r
n+1
i
= r i +
δt
m i
p i ,
(3)
where m i is the mass of particle i, e ij is the unit vector pointing from particle
i to particle j , and ij
1 represents the momentum transferred from particle j to
particle i. The function ij must be suitably selected to ensure that a homogeneous
equilibrium state is possible and that recovers Galilean invariance. The form
selected by Hoogerbrugge and Koelman:
ij = W
r i − r j
ij − ω(p i − p j ) · e ij
,
(4)
satisfies these conditions. The weight function W (r) is dimensionless and nonnegative W (r) ∈ W , where the set W is defined by:
W =
W (r)
W (r) ∈ R,
N
V
W (r)dr = 1, W(r ≥ r c ) = 0
.
(5)
The general form of the DPD method is based on Newtonian mechanics [20],
where spheres interact in a force field:
f i = m i ¨
r i
(6)
1 ij must be symmetric, i.e., ij = ji to ensure conservation of momentum. Also, the authors
restricted momentum transfer to a ball of radius r c , ij = 0 if
r i − r j
> r c .
M. Gonzales and N. N. Thadhani
also be invoked to construct interatomic potentials for use in MD simulations and
has been successfully deployed to simulate shock compression, cf. [52, 53].
Further coarse-graining via dissipative particle dynamics (DPD) as described
in the works by Español et al. [18–20] and Hoogerbrugge and Koelman [39]
provides a truly mesoscopic simulation technique which links micro-level processes
to meso-level processes via the fluctuation-dissipation theorem [34, 80]. The method
combines lattice gas automata techniques and their time-stepping algorithms from
Frisch et al. [23], which allowed a nonlinear system of particles to evolve microscopically and recover the Navier-Stokes equations, with MD to recover Galilean
invariance and flow isotropy. The original method as employed by Hoogerbrugge
and Koelman [39] involves N particles in a domain V , evolving the system by
initializing positions r i and momenta p i and updating with discrete timesteps δt
for the nth timestep via:
p
n+1
i
= p
n
i +
j
ij e ij
(1)
e ij =
r i − r j
r i − r j
(2)
r
n+1
i
= r i +
δt
m i
p i ,
(3)
where m i is the mass of particle i, e ij is the unit vector pointing from particle
i to particle j , and ij
1 represents the momentum transferred from particle j to
particle i. The function ij must be suitably selected to ensure that a homogeneous
equilibrium state is possible and that recovers Galilean invariance. The form
selected by Hoogerbrugge and Koelman:
ij = W
r i − r j
ij − ω(p i − p j ) · e ij
,
(4)
satisfies these conditions. The weight function W (r) is dimensionless and nonnegative W (r) ∈ W , where the set W is defined by:
W =
W (r)
W (r) ∈ R,
N
V
W (r)dr = 1, W(r ≥ r c ) = 0
.
(5)
The general form of the DPD method is based on Newtonian mechanics [20],
where spheres interact in a force field:
f i = m i ¨
r i
(6)
1 ij must be symmetric, i.e., ij = ji to ensure conservation of momentum. Also, the authors
restricted momentum transfer to a ball of radius r c , ij = 0 if
r i − r j
> r c .