J. Plasma Physics: page 1 of 13. c  Cambridge University Press 2014 doi:10.1017/S0022377814000877 1 Collisional relaxation: Landau versus Dougherty operator Oreste Pezzi†, F. Valentini and P. Veltri Dipartimento di Fisica and CNISM, Universit` a della Calabria, 87036 Rende (CS), Italy (Received 18 June 2014; revised 28 August 2014; accepted 17 September 2014) A detailed comparison between the Landau and the Dougherty collision operators has been performed by means of Eulerian simulations, in the case of relaxation toward equilibrium of a spatially homogeneous field-free plasma in three-dimensional velocity space. Even though the form of the two collisional operators is evidently different, we found that the collisional evolution of the relevant moments of the particle distribution function (temperature and entropy) are similar in the two cases, once an ‘ad hoc’ time rescaling procedure has been performed. The Dougherty operator is a nonlinear differential operator of the Fokker-Planck type and requires a significantly lighter computational effort with respect to the complete Landau integral; this makes self-consistent simulations of plasmas in presence of collisions affordable, even in the multi-dimensional phase space geometry. 1. Introduction The longstanding problem of collisions in plasmas is a very fascinating and huge topic in the field of plasma physics and it has always been the subject of a relevant scientific effort. Many authors approached the study of collisional effects in plasmas (Landau 1936; Spitzer 1956; Lenard and Bernstein 1958), by modeling particle interactions through different operators with different physical features and mathematical structures. The ‘natural’ operator that describes the Coulombian interactions between charged particles (in absence of wave-particle resonance) is the Landau integral operator (Landau 1936). The Landau collision integral is a nonlinear, integro-differential and Fokker-Planck type operator which satisfies the H -theorem for the entropy growth (Hinton and Hazeltine 1976). Due to its nonlinear nature and multi-dimensionality, any analytical and numerical approaches to the solution of the Landau integral results extremely complicated. When studying plasma dynamics, collisions are usually considered either negligible (Vlasov model) or dominant such to maintain the distribution function Maxwellian (fluid model). For physical systems, such as the solar wind, that exhibit a weak, almost negligible collisionality, the kinetic and collisional approach is necessary, especially when physical processes of particle heating and consequent entropy growing are considered. In fact, on the basis of the H -theorem, collisions are the unique physical ‘ingredient’ that can thermalize free-energy and produce heating in general thermodynamic sense. † Email address for correspondence: oreste.pezzi@fis.unical.it