I've seen some phrases like 'free-path', 'free-path distributions', 'mean free path' in some computer graphics resources, but none of them explains what does these phrases mean, does anyone have an explanation of these phrases? Thanks a lot!
The term "free path" usually shows up in the context of Monte Carlo simulation of volumetric scattering—i.e., rendering such things as smoke, fog, or clouds in a path tracer. In this context, interactions between a light ray and the volume are treated as random events: the picture is that the photon flies through the volume "freely" (without interacting) for some random distance, then is absorbed or scattered at a discrete point. If scattered, it then flies off in another direction for another random distance, and may interact a second time, and so forth.
The "free path" is the distance that a photon travels in the volume between interactions. This is a random quantity, so sometimes we see "mean free path", which is the mean distance to the next interaction. In a homogeneous volume, scattering events are modeled as a Poisson process, so the distribution of free path distances is an exponential distribution. In such a case, the mean free path is the inverse of the scattering coefficient. A greater scattering coefficient means the volume interacts with photons more often, so the mean free path gets shorter, and vice versa.
In a path tracer, when a ray enters a volume (or starts inside a volume), we would then sample that distribution to determine how far the ray should go. If it does not exit the volume or strike some other object before the sampled free-path distance, then it interacts with the volume, and we do further sampling to see what direction it scatters, etc., to set up the next path segment.
In heterogeneous volumes, the algorithms get more complicated, but the same principles apply. Also, there is some research dealing with so-called "non-exponential media", in which the free path distribution is something other than the standard exponential one (i.e. breaking the Poisson assumption that interactions at different points in the volume are independent of each other).