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Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards

We study diffusion on a periodic billiard table with infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially-distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [Phys. Rev. Lett. 50, 1959 (1983)].