  There are also a few unstated assumptions that go into the derivation of this equation. First, the number of molecules must be large enough so that the (discrete) distribution of their velocities can be well approximated by a continuous and differentiable function \(f\). Secondly, \(f\) changes under the effect of binary collisions only. This means that the density of the gas should be low (so that three-particle collisions can be ignored) but not too low (so that collisions would be too infrequent to change \(f\) at all. (The modern procedure to put these requirements in a mathematically precise form is that of taking the so-called Boltzmann-Grad limit.) A final ingredient is that all the above assumptions are not only valid at an instant but remain true in the course of time.