Abstract
The dynamics of many-body Hamiltonian systems with long-range interactions is studied, in the context of the so-called α-HMF model. Building on the analogy with the related mean-field model, we construct stationary states of the α-HMF model for which the spatial organization satisfies a fractional equation. At variance, the microscopic dynamics turns out to be regular and explicitly known. As a consequence, dynamical regularity is achieved at the price of strong spatial complexity, namely a microscopic inhomogeneity which locally displays scale invariance.