Collisions
On each timestep the particles are sorted into spatial cells and only particles in the same cell
are allowed to collide. Typically the dimension of a cell is no larger than a mean free path.
All pairs of particles in a cell are candidate collision partners, regardless of their actual trajectories.
The details of how collisions are calculated in DSMC depend on the molecular interaction model;
here we take the hard spheres model, which is the simplest.
In the hard spheres model, the collision probability for the pair of particles, and , is
proportional to their relative speed,
where is the number of particles in the cell and the summations are over particles within the cell.
Because of the double sum in the denominator it can be computationally expensive to use this collision probability directly.
Instead, the following rejection sampling scheme can be used to select collision pairs:
- A pair of candidate particles, and , is chosen at random and their relative speed, , is computed.
- The pair is accepted as collision partners if , where is the maximum relative speed in the cell and is a uniform deviate in [0, 1).
- If the pair is accepted, the collision is processed; the velocities of the particles are reset but positions are unchanged.
- After the collision is processed or if the pair is rejected, return to step 1.
This procedure is correct even if the value
of is overestimated, although it is less efficient
in the sense that more candidates are rejected.
After the collision pair is chosen, their post-collision velocities,
and , are evaluated.
Writing the relative velocity in terms of spherical angles, and
these angles are selected by a Monte Carlo process with distributions given by the collision model.
For the hard spheres model these angles are uniformly distributed over the unit sphere.
The azimuthal angle is uniformly distributed between 0 and , so it is selected as
where is a uniform deviate in [0, 1).
The polar angle is distributed according to the probability density,
Using the change of variable , we have so
The post-collision velocities are set as
Note that by conservation of linear momentum and energy the center of mass velocity
and the relative speed are unchanged in a collision. That is,
and
This process is repeated for every pair of colliding particles.
From the collision frequency, , given by kinetic theory the total
number of hard sphere collisions in a cell during a time is
where is the particle diameter and is the volume of the cell.
Since collision candidates go through a rejection sampling procedure
the ratio of total accepted to total candidates for hard sphere particles is
The number of collision candidates selected in a cell over a time step is
This approach for determining the number of collisions is known as the No-Time-Counter (NTC) method.
If is set excessively high then the algorithm processes the same number of collisions (on average)
but the simulation is inefficient because many candidates are rejected.