RELAXATION TO BOLTZMANN EQUILIBRIUM OF 2D COULOMB OSCILLATORS C. Benedetti 1 * , S. Rambaldi * and G. Turchetti * * Dipartimento di Fisica Universitá di Bologna and INFN, Bologna, Via Irnerio 46, 40126, Italy Abstract. 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 Hamilton’s 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 Landau’s theory. To this end we use an approximation to the cross section of the cut-off 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. PACS: 29.27.Bd; 02.70.Ns; 41.75.-i; Keywords: Beam dynamics; Collisional effects; Numerical simulations; INTRODUCTION Systems with long range interactions exhibit non standard thermodynamic properties. The presence of scaling laws is useful to investigate analytically the thermodynamic limit and to extrapolate the results of numerical simulations where a rather small number of pseudo-particles has to be used. One dimensional Coulomb systems have been studied.[1] We consider here a model of two dimensional Coulomb oscillators which is suited to describe a beam of charged particles, protons or ions, in a circular accelerator. The effects of Coulomb interaction (space charge effects) are relevant for intense beams at moderate energies and cause the formation of a halo around the beam core.[2] In the storage rings, very long bunches are supposed to circulate for a high number of turns (10 6 ) and the approximation with a continuous circulating current, rendering the problem two dimensional, is justifi ed. In this case the mean fi eld 2D Poisson-Vlasov equations have analytical solutions also when the focusing fi eld varies periodically along the ring (unlikely the 3D case). As a consequence the numerical schemes to solve the 2D mean fi eld equations (PIC codes) can be benchmarked against the analytical solutions. This comparison is necessary to control the effect of numerical noise (due to round off errors), which causes a linear growth of the invariants.[3] Our 2D Hamiltonian model takes into account the small angle and hard collisions, in order to explore the relaxation process to the thermodynamic equilibrium and the mean fi eld limit. Collisional effects (intra beam scattering) can be important in the new generation of high intensity accelerators such as the storage rings for neutron Spallation Sources (SNS) [4], for high energy density physics (SIS100 of the FAIR project at GSI) [5] and for the drivers of heavy ion fusion (HIF)[6], since only very small losses can be tolerated in order to avoid activation of the beam pipe. The main result of our investigation is a scaling law that is supported by Landau’s kinetic theory. The algorithms developed to integrate Hamilton’s equations have a N log N computational complexity, where N denotes the number of pseudo-particles (letting q be their charge per unit length we keep Nq constant). This complexity allows us to simulate a number of pseudo-particles which is anyhow two order of magnitude below the physical value, as a consequence scaling laws are necessary. To analyze the kinetic equations of Landau’s theory one diffi culty is related to the Coulomb cross section, which, unlikely the 3D case, has not an explicit analytic expression. In the asymptotic 1 E-mail: benedetti@bo.infn.it, Tel: +39-051-2091123, Fax: +39-051-247244