Biomedical Engineering Reference
In-Depth Information
In this text, we will generally be concerned with situations where the potentials (and so
the forces) are pair wise additive. If we focus on one particular particle A, then the mutual
potential energy of A with all the other particles
U
A
is found from
U
A
=
U
A
j
j
=
A
and we can find the force on particle A,
F
A
, by differentiating
U
A
with respect to the
coordinates of particle A:
F
A
=−
grad
U
A
For example, if we consider a pair of Lennard-Jones particles A and B where
4ε
σ
R
AB
6
12
σ
R
AB
U
AB
=
−
we note that the potential only depends on the distance between the particles. The expression
for grad
U
is particularly simple in this case so that
∂
U
∂
R
A
R
BA
R
BA
F
A
=−
which gives
24ε
2
σ
R
AB
6
R
AB
R
AB
12
σ
R
AB
F
A
=
−
Newton's second law connects force and acceleration by
d
2
R
A
d
t
2
and in principle we could study the time development of a system by solving this second-
order differential equation, one such equation for each of the particles in our system.
Calculating the trajectories of
N
particles therefore appears to involve the solution of a set
of 3
N
second-order ordinary differential equations.Alternatively, we could use an advanced
method such as Hamilton's to solve 6
N
first-order differential equations. For any set of
N
particles it is always possible to find three coordinates that correspond to translation of
the centre of mass of the system, and, if the particles have 'shape', three coordinates that
correspond to rotations about three axes that pass through the centre of mass.
Most of the early molecular dynamics studies were directed at the problem of liquid
structure, so that is where we will begin our discussion.
F
A
=
m
A
10.1 Radial Distribution Function
Of the three states of matter, gases are the easiest to model because the constituent particles
are so far apart on average that we can ignore intermolecular interactions apart from during
their brief collisions. This is why researchers were able to bring the kinetic theory of gases
to such an advanced stage by the end of the nineteenth century (and before the existence
of a satisfactory theory of molecular structure).
