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Effect of the fluctuations around mean field for N-body systems with long range interactions

We study the effect of Chandrasekhar and Holstmark's distribution of field fluctuations on the dynamics of N-body systems interacting via Coulomb or Newton gravitational force. We develop an approach based on statistical dynamics first principles whose mathematical framework is similar to the one used by Chandrasekhar and Holstmark for their field fluctuation theory. We use the Picard iteration method to approximate the Hamiltonian dynamics in the short time limit. Neglecting correlations between particles, carrying the thermodynamic limit and assuming that the system is spatially homogeneous, we find a fractional kinetic equation for the velocity distribution. Both, the fractional derivative order and the asymptotic behavior of the solution appear to be directly connected to the $1/r^2$ behavior of the Coulombian or gravitational interaction force over short distances.