TrapApp also offers the option of simulating Monte Carlo collisions
with a background gas. This option allows the user to model, on a microscopic
scale,
thermal interaction with a ``background'' gas which
is at constant temperature. The user provides the mean time between collisions and the temperature of the background gas. After each finite difference cycle, TrapApp performs Monte Carlo collisions by comparing a random number with the probability that a collision should occur. In the
case of a collision, TrapApp assigns to the unfortunate ion a new
velocity according to the Maxwell-Boltzmann speed distribution at the
temperature of the background gas.
To aid the number-conscious user in choosing an appropriate collision time for a desired background gas pressure, the following relations are provided:[13]
where 
is the mean free path length between collisions, 
is the mean ion velocity, and 
is the mean time between collisions.
The chance of a collision occurring within a displacement dx is
approximately:
where n is the number density (per unit volume) of the background
gas, 
is the cross-sectional area of a molecule of the background
gas, and dx is an incremental distance. The above definition of 
and
the ideal gas law, 
(where N is the number of
ions, V is the volume occupied, p is pressure) can be employed
to yield an expression for the mean
free path as a function of temperature, pressure, and
cross section:
For example, a plausible experimental set-up[30] might employ a background gas of helium at a background pressure 
mbar 
to cool Mg
ions (
amu 
kg ). If we
assume the radius of helium to be roughly
an Angström, the mean free time between ion collisions with the helium is
ms. Unfortunately, a typical Paul trap simulation uses a time step on the
order of tenths to hundredths of 
s, so effectively cooling a collection
of ions via Monte Carlo collisions at 
ms requires hundreds
of thousands of
time steps.