The existence of Burnett coefficients in the periodic Lorentz gas

The linear super-Burnett coefficient gives corrections to the diffusion equation in the form of higher derivatives of the density. Like the diffusion coefficient, it can be expressed in terms of integrals of correlation functions, but involving four different times. The power-law decay of correlations in real gases (with many moving particles) and the random Lorentz gas (with one moving particle and fixed scatterers) are expected to cause the super-Burnett coefficient to diverge in most cases. Here, we show that the expression for the super-Burnett coefficient of the periodic Lorentz gas converges as a result of exponential decay of correlations and a nontrivial cancellation of divergent contributions.