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We are interested in a kinetic equation intended to describe the interactionof particles with their environment. The environment is modeled by a collectionof local vibrational degrees of freedom. We establish the existence of weaksolutions for a wide class of initial data and external forces. We alsoidentify a relevant regime which allows us to derive, quite surprisingly, theattractive Vlasov--Poisson system from the coupled Vlasov-Wave equations.

Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov--Poisson system