Since we use no tangential forces, we will discuss the normal
direction of a contact only. Considering the collision of two particles,
the situation is modelled
by a spring and a dashpot, see Eqs. 1 and 2
so that the relative acceleration during contact is
, with
.
Due to force balance we set 
what
leads to a differential equation for negative
penetration depth 
:
In Eq. 3, 
,
, and the
reduced mass 
. The solution of
Eq. 3 is for 
:
with the corresponding velocity:
In Eqs. 4 and 5
is the relative velocity before collision
and 
the damped frequency.
As long as 
, the typical duration of
the contact of two particles is:
because the interaction ends when y(t) > 0.
The coefficient of restitution 
is defined as
the ratio of velocities after and before contact
so that Eqs. 5 and 6
lead to
From Eqs. 4 and 5 the maximal penetration
depth 
follows
the condition 
, so that
and
The maximum penetration depth 
is in the case of, say, steel particles
much smaller than the particle diameter. Thus we check in our simulations
that is always orders of magnitude smaller than 
.
The elasticity k in Eq. 1
is e.g. a function of the Young modulus and the Poisson ratio,
which are material dependent and thus fix 
for a given material
in our simplified model. Using the theory of Hertz, a more
complicated dependence of k on the impact velocity, the
elasticity , and the penetration depth is found,
e.g. 
. In Ref. [20], the contact
time of two steel spheres with diameter d = 1.5mm and with
an impact velocity of 
m/s
was evaluated to 
s.
We checked for some situations that the more realistic Hertz model
does not change the results [21] and thus used the
simpler linear model. For a detailed discussion of different
MD models and force-laws see Ref. [22].
For weak dissipation
is proportional to 
, so that an increase
of k by a factor of 100 decreases by a factor of 10. Now taking
physical values for leads to extremely long MD-
computing times for a given simulation time. One has to insure
that the time scales of the system, i.e. , and of the
algorithm, i.e. the integration time step 
, are well separated.
Ideally one should have 
. The MD-simulations
reported here were done with 
.
Using contact times in the range 
s 
s, by
choosing k according to Eq. 6, we have
simulation time steps in the range 
s
s.
In our simulations we have as a typical set of parameters
mm, 
s 
,
s 
and 
s.
These parameters lead with the above
equations to 
s, and
, i.e. rather strong dissipation.
