Notes on the Boltzmann Equation

We wish to describe the motion of a rarefied gas, consisting of a very large number of identical particles, moving in a three-dimensional space. For (t, x, ξ) ∈ [0,∞[×IR3 × IR, we consider a function f(t, x, ξ) describing the density of particles at time t, at the point x, having speed ξ. In alternative, we may also think of f(t, x, ξ) as the probability of finding a particle with speed ξ near the point x, at time t. If no collisions occur, the speed ξ of each particle will remain constant in time. A particle with speed ξ and located at the point x at the initial time t = 0 will move to x+ τξ at a later time τ . Therefore f(τ, x, ξ) = f(0, x − τξ, ξ). In this case, f provides a solution to the linear transport equation ∂tf + ξ · ∇xf = 0. (0.1)