MOLECULAR DYNAMICS SIMULATIONS OF SAND

We all like (or liked) to play with sand once in a while.
Therefore pouring, shearing or vibrating of sand
seem to be rather familiar experiences.
Modern science, however, has much less understanding for
the motion of sand than for instance for fluids
(``hydrodynamics'') or solids (``e.g. elasticity''):
while for fluids and solids we can describe the velocities
and displacements by differential equations no such equations
are known for sand or in general for ``granular media'' 
(This includes cement, grains, asteroids, pills, rocks, gravel, etc.).
The only reliable way today to describe the motion of sand
is a direct simulation on the computer.

In this section we want to show how the behavior
of sand can be simulated using ``Molecular Dynamics''.
Then we will present various practical cases where
easy hands-on experiments will also be proposed that
can be performed in parallel.
We will investigate how an hourglass works. 
We will see how we can get the Brazil nuts out
of a mixture of nuts just by shaking,
Further we project the following applications:
Avalanches of sand down a dune, convection rolls in
vibrating sand and density waves of sand flowing through a
narrow pipe.

1. Inelastic collisions via Molecular Dynamics

As opposed to collisions between atoms or molecules
collisions between the grains of sand are ``inelastic''
which means that energy is lost. In fact, all the collisions
we encounter in daily life (between cars, billiard balls or
people) are inelastic since always some energy will be transformed
into heat. Let us consider for instance tennis balls, 
called ``particles'' in the following, as
a rather good (although a little simplified) representation
of sand grains and let us try to calculate on the computer
their trajectories and collisions by ``Molecular Dynamics''.

Molecular Dynamics is a direct calculation of the position 
x(t+dt) of
the particles at time t+dt knowing the position x(t) of the
particles at time t. First we calculate all the forces that
act at time t which besides gravity are all the forces due
to collisions either with walls or with other particles.
Dividing the sum of all the forces that act on a given particle
by its mass gives the accelation a(t) (Newton's law).
From that we calculate the velocity v(t+dt) 
at time t+dt by adding
a(t)*dt to the velocity v(t) at time t and the positions
x(t+dt) by adding v(t)*dt to x(t).

Which are the forces that act? On one hand one always has gravity.
On earth the acceleration due to gravity is 9.8 m/sec
but it could be fun to watch how things would behave on the moon
or on Jupiter and for that reason we will always leave the option
to change the accelation g of the gravity.
All other forces act when particles collide either with other
particles or with a wall. 

