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Kinetic equations of N-Body systems interacting via 1/r potentials

par Chaffi, YASSIN
Président du jury Gilbert, Thomas
Promoteur Brenig, Léon
Co-Promoteur Knaepen, Bernard
Publication Non publié, 2016-06-30

Thèse de doctorat

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Résumé :

In this work, we study the time evolution of systems containing a large number of particles interacting via a $1/r$ binary interaction potential, such as Coulombian and self-gravitating systems. In particular, we study the effect on the dynamics of the Holtsmark-Chandrasekhar theory, which describes the static fluctuations of the total force field around the Vlasov mean-field. We derive these effects by developing a new perturbative theory using the fundamental representation of Statistical Mechanics : The BBGKY hierarchy. This leads to a modification of the Vlasov equation by an additional term involving a fractional Laplacian to the power $3/4$ in velocity space, and a fractional iterated time integral of order $1/2$. We show that one of the consequences of this new term for spatially homogeneous systems is the appearance in the velocity distribution of long tails in $v^{-5/2}$. By extension, similar behaviors can be expected for weakly inhomogeneous systems. These long tails correspond to a universal mechanism related to the divergence of the interaction potential in $1/r$. More specifically, they are induced by the long tails of the total force field distribution as described by the Holtsmark-Chandrasekhar theory. Such a result cannot be obtained from theories based on the weak-coupling between particles, which lead to the Vlasov term, and, the Landau collision operator at the next order. We verify numerically these results by means of molecular dynamics simulations. We study the evolution of the velocity distributions for times very short compared to the violent relaxation time. In this particular time regime, we find, as expected, power laws in $v^{alpha}$ for the velocity distribution tail. In particular, when the regularization parameter of the interaction potential tends to $0$, the exponent in the power law indeed tends from below toward the theoretically predicted value $alpha=-5/2$.