Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 473
whose calculation at each step of the simulation constitutes the bulk of the computational
demand of the calculation. The first formulation of the method, due to Alder, 75 referred to a
system of hard spheres: in this case, the particles move at constant velocity between perfectly
elastic collisions, so that it is possible to solve the problem without making any approximations.
More difficult is the solution of the equations of motion for a set of
Lennard-Jones particles: in fact, in this case an approximated step-by-step procedure is
needed, since the forces between the particles change continuously as they move.
Let us consider a point Γ in the phase space and suppose that it is possible to write the
instantaneous value of some property A as a function A(Γ). As the system evolves in time, Γ
and hence A(Γ) will change. It is reasonable to assume that the experimentally observable
“macroscopic” property A obs is really the time average of A(Γ) over a long time interval:
obs = =
time
=
time t→∞
t
∫
0
A A A t
( Γ() ) ( Γ()
)
lim
t 1
A t dt
[8.98]
The time evolution is governed by the well-known Newton equations, a system of differential
equations whose solution is practical. Obviously it is not possible to extend the integration
to infinite time, but the average can be reached by integrating over a long finite
time, at least as long as possible as determined by the available computer resources. This is
exactly what is done in a MD simulation, in which the equations of motion are solved
step-by-step, taking a large finite number τ of steps, so that:
Aobs A(
() )
= ∑
1
Γ τ
[8.99]
τ
τ
It is worth stressing that a different choice in the time step is generally required to describe
different properties, as molecular vibrations (when flexible potentials are used),
translations and rotations.
A problem arising from the methodology outlined above is whether or not a suitable
region of the phase space is explored by the trajectory to yield good time averages (in a relatively
short computing time) and whether consistency can be obtained with simulations
with identical macroscopic parameters but different initial conditions. Generally, thermodynamically
consistent results for liquids can be obtained provided that careful attention is
paid to the selection of initial conditions.
Monte Carlo
As we have seen, apart from the choice of the initial conditions, a MD simulation is entirely
deterministic. By contrast, a probabilistic (stochastic) element is an essential part of any
Monte Carlo (MC) simulation.
The time-average approach outlined above is not the only possible: it is in fact practical
to replace the time average by the ensemble average, being the ensemble a collection of
points Γ distributed according to a probability densityρ(Γ) in the phase space. The density is
determined by the macroscopic parameters, NPT, NVT, etc., and generally will evolve in
time. Making the assumption that the system is “ergodic”, the time average in eq. [8.98] can
be replaced by an average taken over all the members of the ensemble at a particular time:
() ()
A = A =∑A Γ ρ Γ
[8.100]
obs ensemble ensemble
Γ