8.9 Dynamical quantities: nonequilibrium molecular dynamics 251
The systematic interaction force between the particles in the solvent will affect
the friction which these particles are subject to through hydrodynamic effects.
This coupling is usually neglected, but a method including these effects has been
proposed and implemented [68]. We mention the dissipative particle dynamics
(DPD) method which is based on these ideas [69].
An algorithm for simple Langevin dynamics can be formulated starting from the
methods given in Section 8.4.1. Suppose the random force is constant during one
integration step. Denoting the force during the interval[0,h]byR+and that during
the interval[−h,0]byR−, the random force can directly be included into(8.40):
x(h)[ 1 +γh/ 2 ]+x(−h)[ 1 −γh/ 2 ]= 2 x( 0 )+h^2 [F( 0 )+R+/ 2 +R−/ 2 ].
(8.160)Therefore, at each step a new value of the random force during the new interval
must be drawn from a Gaussian random generator, and this force is to be used
together with the random force generated at the previous step in order to predict
the new position. This is, however, not always a satisfactory procedure. Normally,
the integration time stephis determined by the requirement that the systematic
forceFcan be assumed to be reasonably constant over a time intervalh. This
means that the time over which we take the random force to be constant depends
on the smoothness of the systematic force. In fact we would prefer to allow for
a rapidly varying random force combined with a large time step allowed by the
systematic force. This turns out to be possible. Using the statistical properties of
the random force, equations of motion can be obtained which are somewhat similar
to the ones given here, but with more complicated correlations between the random
contributionsatsubsequentsteps–fordetailsseeRef. [70].
It is straightforward to develop a Langevin program for a molecule in a fluid or a
gas, using the simple algorithm presented here. For molecules containing chains of
at most three chemically bonded atoms, torsion is absent, which reduces the number
of forces considerably. Examples are molecules with a tetrahedron conformation,
such as CH 4 (methane) and CF 4 , and two-dimensional molecules. In Problem 8.9
the construction of a Langevin molecule for methane is considered.
8.9 Dynamical quantities: nonequilibrium molecular dynamics
In the molecular dynamics method, the equations of motion of a classical
many-body system are integrated numerically. There is no reason to restrict the
applicability of this method to systems in equilibrium. MD is the method of choice
for dynamic phenomena in equilibrium or nonequilibrium systems. We speak of
nonequilibrium molecular dynamics (NEMD). We consider two examples very
briefly here.