Normal and Abnormal Diffusion in Ehrenfests' Wind-Tree Model

Abstract

The Ehrenfest wind‐tree model—a special case of the Lorentz model—where noninteracting point particles move in the plane through a random array of square scatterers, is used to study the divergences previously discovered in the density expansions of the transport coefficients. Two cases for which the results are qualitatively different are discussed. When the scatterers are not allowed to overlap, the diffusion of particles through the array of scatterers is normal, characterized by a diffusion constant D. The calculation of D⁻¹ is carried to the second order in the density of the scatterers, and involves a discussion of the above‐mentioned divergences and a resummation of all most‐divergent terms in the straightforward expansion of D⁻¹. If, however, the trees are allowed to overlap, the growth of the mean‐square displacement with time is slower than linear, so that no diffusion coefficient can be defined. The origin and possible relevance of this new phenomenon to other problems in kinetic theory is discussed. The above results have stimulated molecular dynamics calculations by Wood and Lado on the same model. Their preliminary results seem to confirm the theoretical predictions.