Originally developed for dilute gas flow in engineering and in space
science
[Bird 94],
[Nanbu 83].
Recent extensions for dense gases:
[Alexander 95].

Basic idea of the DSMC method for a dilute gas of hard spheres:

Setup:

Divide the sample into cells of volume $V_{c}$, each with
$N_{c} \approx 20-40$ particles with given positions and velocities.

The side length of the cells should be smaller than but of the order
of the mean free path.

Boundary conditions appropriate to
the problem are defined, usually specular
(reflecting wall) and/or periodic boundaries.

A time step $\Delta t$
smaller than the typical intercollision time is assumed.

Translate all particles according to
$\vec{r}_{i} \rightarrow \vec{r}_{i}+\vec{v}_{i}\, \Delta t$,
applying the given boundary conditions.

Within each cell, draw $M_{c}$ pairs of particles $(i,j)$ that are
candidates for a collision:

Let the probability of a pair $(i,j)$ to collide depend only
on their relative speed
$v_{ij} \equiv \left| \vec{v}_{j}-\vec{v}_{i}\right|$
and not to their positions. The argument for this is that all
particles in one cell are within free path range of each other.
The probability for the pair $(i,j)$ to collide is thus simply
proportional to the relative speed: $p_{c}(i,j) \propto v_{ij}$.

Recalling the rejection method of Section
3.2
we draw pairs $(i,j)$ in accordance with this probability density:
assuming the maximum of $v_{ij}$ for all pairs in the cell to be known,
draw a random number $\xi$ from a uniform distribution in $[0,1]$ and
compare it to $v_{ij}/v_{max}$.

Calculating $v_{max}$ would
amount to the expensive scanning of all pairs of particles in the cell.
Therefore, use an estimated
value of $v_{max}'$. If that value is larger than the actual $v_{max}$,
the density $p_{ij}$ is still sampled correctly but with a slightly lower
efficiency.

The total number of collision pairs to be sampled in a cell during one time
step is determined as follows. For a gas of hard spheres with diameter
$d$ the average number of pair collisions within the cell is
$
M_{c} \equiv Z \, V_{c} \, \Delta t =
\frac{\textstyle \rho^{2} \pi d^{2} \langle v_{rel}\rangle}{\textstyle 2} \, V_{c} \, \Delta t
$
where $Z$ is the kinetic collision rate per unit volume,
$\rho = N_{c}/V_{c}$ is the number density, and
$\langle v_{rel}\rangle$ is the average relative speed. In order to have
$M_{c}$ trial pairs survive the rejection procedure of step
(a) we have to
sample

Now perform the collision for the pair $(i,j)$.
Since the post-collision velocities are determined
by the impact parameter which is unknown, they must be sampled in
a physically consistent way. In the hard sphere case this is most
easily done by assuming an isotropic distribution of the
relative velocity $\vec{v}_{ij}$ after the collision.
Since the relative speed
$\left| \vec{v}_{ij}\right|$ remains unchanged, the problem is reduced to
sampling a uniformly distributed unit vector. Marsaglia's recipe may be
used for this (see Figure
3.8).