Hard spheres are one of the simplest models of molecular fluids and have represented for many years the testing ground for the predictions of the kinetic theory, thanks to the pioneering efforts of physicists who have developed hard spheres simulations on the old computers which were huge in encumbrance and very small in power (the work of Alder and Wainwright is considered the foundation of this subject [2,3,1,4,5]). Nowadays liquids and gases are almost always simulated with different tools and models (e.g. Lenard-Jones potentials or others), i.e. typically soft spheres models. Nevertheless the study of granular materials has again awakened the interest in hard spheres molecular dynamics, as the geometric character of the grain-grain interaction seems to be better modeled by an hard core interaction. Here we define the hard core potential and give expressions for the quantities calculated in the previous paragraphs.
Two hard spheres in 3D (hard disks in 2D, hard rods in 1D) of diameters
and
interact by means of a discontinuous
potential
of the form:
where
is the distance of the
centers of the spheres at contact. The potential in
Eq. (2.30) can be taken as a definition of hard
spheres systems. In this case the deflection angle is given by:
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(2.31) |
and the dependence from the initial relative velocity disappears (only geometry determines the deflection angle).
The differential scattering cross section for hard spheres is
calculated from Eq. (2.22) obtaining a very simple
formula:
which can be integrated over the
entire solid angle space giving an expression for the total cross
section
. This result is consistent with the physical
intuition of the cross section: it is the average of the areas of
influence of the scatterer in the planes perpendicular to the
approaching velocities of the incident particles.
Other interesting quantities to be easily calculated for hard spheres systems are the transport cross sections (see definition in Eq. (2.24)). The first two are given here:
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Finally we give a definition of smooth hard spheres (we consider
this model as a paradigm for granular gases): smoothness is the
absence of irregularities on the surface of the spheres, i.e. the
instantaneous collision does not change the rotational degrees of
freedom of the spheres at contact. Therefore, in the study of smooth
hard spheres systems, a complete description of the dynamics requires
only the positions of the centers
and their velocities
. In particular the collision is an instantaneous
transformation of the velocities of two particles
and
at
contact which are ``reflected'' with the following rule (see
Fig. fig_collision_2):
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(2.33) |
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(2.34) |
where
and the primes denote the velocities after the collision. This
collision rule conserves momentum and kinetic energy. It only changes
the direction of the component of the relative velocity of the
particles in the direction of
(normal component),
leaving unchanged the tangential component.