3 E. Goals of the paper D. Open problems II. KINETIC THEORY FOR ACCRETION DISC PLASMAS: BASIC ASSUMPTIONS We first discuss what is meant by asymptotic kinetic equilibria in the present study and what are the physical conditions under which they can be realized. An asymptotic kinetic equilibrium must be one ob- tained within the context of kinetic theory and must be described by the stationary Vlasov-Maxwell equations. This means that the generic plasma KDF, f$_{s}$ , must be a solution of the stationary Vlasov equation, as will be the case if f$_{s}$ is expressed in terms of exact first integrals of the motion or adiabatic invariants of the system, which in turn implies that f$_{s}$ for each species must be an ex- act first integral of the motion or an adiabatic invariant. The stationarity condition means that the equilibrium KDF cannot depend explicitly on time, although in prin- ciple it could contain an implicit time dependence via its fluid moments (in which case the kinetic equilibrium The paper is organized as follows. In Section 2 we discuss the conditions for the existence of a kinetic equi- librium and we introduce the basic assumptions for the formulation of the kinetic theory. Section 3 deals with the first integrals of motion and the gyrokinetic adia- batic invariants of the system. In Section 4 we construct the equilibrium KDF for AD plasmas with non-isotropic temperature, giving also a useful asymptotic expansion for this in the limit of strong magnetic fields. Section 5 deals with calculation of the fluid moments of the sta- tionary KDF. The limit of isotropic temperature is then investigated in Section 6, while Section 7 is devoted to analyzing the Maxwell equations. Finally, Section 8 con- tains conclusions and a summary of the main results. The aim of this paper is to extend the investigation of kinetic equilibria developed in an earlier paper [15] to a more general class of solutions. In particular, we pose here the problem of constructing analytic solutions for exact kinetic and gyrokinetic axi-symmetric gravita- tional equilibria (see definition below) in accretion discs around compact objects. The solution presented is ap- plicable to collisionless magnetized plasmas with temper- ature anisotropy and mainly toroidal flow velocity. The kinetic treatment of the gravitational equilibria neces- sarily requires that the KDF is itself a stationary solu- tion of the relevant kinetic equations. Ignoring possible weakly-dissipative effects, we shall assume - in particu- lar - that the KDF and the electromagnetic (EM) fields associated with the plasma obey the system of Vlasov- Maxwell equations. The only restriction on the form of the KDF, besides assuming its strict positivity and it be- ing suitably smooth in the relevant phase-space, is due to the requirement that it must be a function only of the independent first integrals of the motion or the adiabatic invariants for the system. Finally, increasing attention has been paid to the role of temperature anisotropy and the related kinetic insta- bilities. Some recent numerical studies [18, 27] have tried to include the effects of temperature anisotropy but, al- though it is clear that this can give rise to an entirely new class of phenomena, all of these estimates rely on fluid models in which kinetic effects are included in only an ap- proximate way. A kinetic approach is needed rather than a fluid one, in order to give a clear and self-consistent pic- ture, and this needs to be based on equilibrium solutions suitable for accretion discs. Many problems remain to be addressed and solved re- garding the kinetic formulation of AD plasma dynamics. Among them, we focus on the following: 1. The construction of a kinetic theory for AD plas- mas within the framework of the Vlasov-Maxwell description, and the investigation of their kinetic equilibrium properties. 2. The inclusion of finite Larmor-radius (FLR) effects in the MHD equations. For magnetized plasmas, this can be achieved by making a kinetic treatment and representing the KDF in terms of gyrokinetic variables (Bernstein and Catto [28–31]). The gy- rokinetic formalism provides a simplified descrip- tion of the dynamics of charged particles in the presence of magnetic fields, thanks to the symme- try of the Larmor gyratory motion of the particles around the magnetic field lines. Therefore kinetic and gyrokinetic theory are both fundamental tools for treating FLR effects in a consistent way. 3. The determination of suitable kinetic closure con- ditions to be used in the fluid description of the discs. These should include the kinetic effects of the plasma dynamics in a consistent way. 4. Extension of the known solutions to more general contexts, with the inclusion of important effects such as temperature anisotropy. 5. Development of a kinetic theory for stability anal- ysis of AD plasmas. As already mentioned, this could throw further light on the physical mecha- nism giving rise to the effective viscosity and the related accretion processes. This is particularly in- teresting for collisionless plasmas with temperature anisotropy, since only kinetic theory could be able to explain how instabilities can originate and grow to restore the isotropic properties of the plasma.