Relaxation to Boltzmann equilibrium of 2D Coulomb oscillators

We propose a two-dimensional model of charged particles moving along the z-axis, on which they are focused by a linear attracting field. The particles are organized into parallel uniformly charged wires which interact with a logarithmic potential. The mean field is described by the Poisson–Vlasov equation, whereas Hamiltonʹs equations need to be solved to take into account the effect of collisions. The relaxation to the self-consistent Maxwell–Boltzmann distribution is observed in numerical simulations for any initial distribution and the relaxation time scales linearly with the number N of wires, having fixed the total current. We prove that such scaling holds in the kinetic approach given by Landauʹs theory. To this end, we use an approximation to the cross section of the cutoff logarithmic potential, which is asymptotically exact for large N. The drift term inherits the scaling law of the cross section and provides the required scaling for the relaxation time.