The Knudsen diffusion algorithm models the movement of molecules in the absence of molecule-molecule collisions. Therefore, it is only valid for high Knudsen numbers. For intermediate Knudsen numbers in the range of 0.01 < Kn < 10, diffusion simulations must also consider molecule-molecule interactions. This can be done by adding molecule-molecule collision to the random walk method.

Again, each single molecule starts with a random, Maxwell-Boltzmann distributed velocity (here, the term “velocity” means the three-dimensional velocity vector, so it includes the direction of the movement). Based on the mean free path length of the gas, an exponentially distributed random distance is determined, after which the particles will collide with another particle. The particle now moves in a straight line until it either hits a wall or it has traveled the distance , whatever comes earlier. If it hits a wall, the molecule is reflected diffusely according to Lambert’s cosine law (see Greenwood) and leaves the wall with a new random, again Maxwell-Boltzmann distributed velocity and a new random distance is determined. If it reached the distance before hitting a wall, it is assumed that the particle collides with another particle. In that case, a new Maxwell-Boltzmann distributed velocity and a new random distance are drawn and the particle continues its movement with the new velocity.

The molecule continues its way until the desired simulation time is reached. The resulting displacement (distance between the molecules’ start position and end position) is compared to its travel time, which results in the diffusivity for this individual molecule. Thus, to obtain a good estimate for the diffusivity of the whole 3D structure, a high number of simulated molecules is needed.

From the displacement, the diffusivity is determined by using Einstein’s formula (175) .