Dissipative Particle Dynamics: A Method to Simulate Soft Matter Systems in Equilibrium and Under Flow - Selected Topics of Computational and Experimental Fluid Mechanics - page 52

to warrant that all particles will always be in the simulation box. From a physical
point of view, this bulk simulation should avoid any surface effect (Frenkel and Smit
2002 ).
Assuming ergodicity, a time average over the integrated numerical trajectory of
the N -particle system of any quantity depending on the dynamical variables positions
{ r i } and velocities { v i } is equivalent to an ensemble average

˄

1
˄

A ens ≡ A ( { r i ( t ), v i ( t ) } ) dt ,

0

where A stands for any physical quantity as a function of the dynamics variables of
the system. From the point of view of the statistical mechanics, the simple integrating
of the Newton equations corresponds to a micro-canonical ensemble in which the
number of particles N , the volume of the system V and the total internal energy of
the system E are held constant.

2.2 Simulating at Constant Temperature: The Langevin
Thermostat

The microcanonical ensemble, while useful theoretically is not usually used in exper-
iments in which exchange of heat, particles and or volume is usually the case. A typ-
ical ensemble in experiments, which is widely used in simulations, is the canonical
ensemble, in which the constant thermodynamic variables are the number of particles
N ,thevolume V and the temperature T . Keeping the temperature constant means, of
course, fluctuations of the energy of the system due to a heat exchange with a “ther-
mal bath”. To simulate the system at constant temperature, some additional terms
must be added to the original dynamical equations ( 1 ). In the last thirty years, sig-
nificant effort has been done to extend the Newton equations of a classical system to
obtain constant temperature. The different thermostats can be classified conceptually
as those which provide constant temperature by stochastic relaxation (i.e. Langevin
thermostat or Brownian dynamics simulations), stochastic coupling (Andersen ther-
mostat), extended Langrangian schemes (i.e. Nosé-Hoover thermostat), temper-
ature constraining (Woodcock and Hoover-Evans thermostats) and weak coupling
(Berendsen thermostat) (Hünenberger 2005 ). All these schemes have advantages and
drawbacks and find areas of application according to the systems, physical conditions
and phenomena to be addressed. We will not review this, and suggest the excellent
review by Hünenberger ( 2005 ) and the classical textbooks on MD simulations by
Frenkel and Smit ( 2002 ), Allen and Tildesley ( 1987 ), and Rapaport ( 2004 ). We will
give some details of the Langevin thermostat, also termed Brownian Dynamics, as a
first step, to describe then the DPD thermostat. In this scheme instead of integrating
the Newton equation of motion, we integrate a Langevin equation that describes the