A linear Boltzmann equation in dimension two is interpreted as the forward equation for the probability density of a Markov process (K(t),i(t),Y (t)) on (T 2 × {1, 2} × R2). Here (K(t),i(t)) is an autonomous jump process, with waiting times between two jumps with finite expectation but infinite variance, and Y (t) is an additive functional of K, defined as Y (t) = R t 0 ds v(Ks) where |v| ∼ 1 for small k. In a phononic picture, Y (t) describes the trajectory of a phonon. We prove that the rescaled process (N ln N) −1/2Y (Nt) converges in distribution to a Brownian motion. As a consequence, the appropriately rescaled solutions of the Boltzmann equation converge to a diffusion equation.