Dissipative Particle Dynamics: A Method to Simulate Soft Matter Systems in Equilibrium and Under Flow - Selected Topics of Computational and Experimental Fluid Mechanics - page 53

dynamics of each particle of the simulation. The integration scheme and most of the
MD program implementation remains exactly the same, except that two force terms
are added to the conservative forces already present in the MD simulations. The first
order equations of motion for the particle i are

p i
m i ,

r i Ǚ =

F c i + F i D F i R p i Ǚ = + ,

where r i and p i are the positions and momenta of the particles. F c stands for all the
conservative forces of the system, with arbitrary molecular complexity. These are
called the molecular interaction model. The Langevin thermostat adds a dissipative
and a stochastic force to the conservative forces of the system. It can be rationalized
by thinking that a particle is coupled to an implicit fluid, which acts as a thermal
bath (Pastorino et al. 2006 , 2007 ). The dissipative force on particle i is given by
F i D

=− ʳ ʳ

is a friction coefficient fixed in the simulation, and v i is the
velocity of the particle. The random force, F i

v i , where

, is chosen such that it has zero mean

value and its variance satisfies

F i μ ( F j ʽ ( t ) = ˃

2

t ), t )

i ʴ ij ʴ μʽ ʴ(

t − (2)

2
i

where ˃ i is the noise strength. The relation ˃ = k B T ʳ/ m i , couples the friction

ʳ ˃ i in a particular way such that the fluctuation-dissipation

theorem is satisfied and the system is simulated in the canonical ensemble ( NVT ).
An elegant derivation of this can be obtained from a Fokker-Planck equation for
this system (Dünweg 2006 ). This way of thermostating, also known as Stochastic
Dynamics, allows particle simulations at constant temperature. However, this algo-
rithm violates Galilean invariance because it damps the absolute velocities of the
particles, thus assuming as “special” the laboratory frame. The Langevin thermostat,
while allowing the simulation of correct thermodynamic conditions, it is of little use
for hydrodynamic phenomena. Taking a simple liquid, it was shown by Dünweg and
Kremer ( 1993 ) and Dünweg ( 2006 ) that this unphysical behavior can be thought as
a screening length for the hydrodynamic correlations

and the noise strength

ʷ
ˁʳ/

1 / 2

ʾ = ,

m

where the friction constant

and m the mass of each particle (Dünweg and Kremer 1993 ). The propagation of
momentum is screened and dampened, as compared to the correct hydrodynamics
behavior.

ʷ is the viscosity of the fluid, ˁ the number density, ʳ