Basic idea of Molecular Dynamics: The basic concept underlying any attempt to understand any system in all of its microscopic detail is, in principle, easily stated. Consider a system of N atoms in a container of any shape having a volume, V. The system need not be homogeneous, and there can be some chemical bonding pattern between the atoms, however, we do not need to assume this. The system is prepared in some initial state and allowed to evolve in time from this state. How do we follow the evolution of the system and how do we determine the properties of the system knowing how it will evolve?
Experimentally, we would use some apparatus to probe the system. For example, we might scatter X-rays or neutrons from the system in order to probe its structure, or expose the system to electromagnetic radiation (e.g. from a laser) in order to excite vibrations or electronic excitations and watch the relaxation. We might also subject the system to other external mechanical forces such as a shear force or a drift force and watch the response in order to study transport phenomena. In addition, the experiment might be repeated many times from different initial states in order to confirm the reproducibility of the results and perform averages over all initial preparations.
Theoretically, we would need to compute the time evolution of the system from some initial state using the fundamental laws of physics, i.e. the microscopic evolution equations of the system. Given that we can do this, we could then perform such a calculation many time starting from different initial states and calculate averages over all initial preparations.
.
However, the task of finding the wave function of the system for N more than
4 or 5 is extraordinarily difficult. This can be seen by noting that the representing
of an n-dimensional function on a grid with M points in each dimension requires
points. For n=3N+1 for N larger than 4 or 5 becomes exponentially large,
beyond the storage capacity of modern-day computers, for example.
and a velocity vector, 
.
There will be position and velocity vectors for each particle in the system,
so we shall index them 1,...,N as 
and 
.
In addition, since the particles are constantly in motion, the position and
velocity vectors are functions of time, t: 
,
and 
. Recall that the position and velocity are
actually related by
where we have introduced the overdot notation to represent differentiation
with respect to time. Therefore, an classical initial state is represented by
specifying the positions and velocities at some instant in time, which we shall
call t=0 and denote 
and 
.
From this initial state, we then need a procedure for obtaining the
positions and velocities at any time, 
. This is prescribed by
the laws of classical mechanics, or Newton's laws of motion.