11 edged (C.C. and J.M.). This work has been partly devel- oped (C.C. and M.T.) within the framework of the MIUR (Italian Ministry for Universities and Research) PRIN Research Program “Modelli della teoria cinetica matem- atica nello studio dei sistemi complessi nelle scienze ap- plicate” and the Trieste Consortium for Magnetofluid Dynamics. model. A possible astrophysical context is provided, for example, by radiatively inefficient accretion flows onto black holes, where the accreting material is thought to consist of a plasma of collisionless ions and electrons with different temperatures, in which the dominant magnetic field is generated by the plasma current density. We have considered here the specific case in which the structure of the magnetic field is locally characterized by a fam- ily of closed nested magnetic surfaces within which the plasma has mainly toroidal flow velocity. For this, we have proved that a kinetic equilibrium exists and can be described by a stationary KDF expressed in terms of the exact integrals of motion and the magnetic mo- ment prescribed by the gyrokinetic theory, which is an adiabatic invariant. Many interesting new results have been pointed out; the most relevant ones for astrophys- ical applications are the following: 1) the possibility of including the effects of a non-isotropic temperature in the stationary KDF; 2) the proof that the Maxwellian and bi- Maxwellian KDFs are asymptotic stationary solutions, i.e. they can be regarded as approximate equilibrium solutions in the limit of strongly magnetized plasmas; 3) the possibility of computing the stationary fluid moments to the desired order of accuracy in terms of suitably pre- scribed flux functions; 4) the proof that a toroidal mag- netic field can be generated in a stationary configuration even in the absence of any net accretion flow if and only if the plasma has a temperature anisotropy. This last point, in particular, is of great interest be- cause it gives a mechanism for generating a stationary toroidal field in the disc, independent of instabilities re- lated to the accretion flow. The consistent kinetic for- mulation developed here permits the self-generation of such a field by the plasma itself, associated with localized poloidal drift-currents on the nested magnetic surfaces as a consequence of temperature anisotropies. This station- ary poloidal motion is made possible in the framework of kinetic theory by the conservation of the canonical mo- mentum as a result of FLR effects. These results reveal and confirm the power of the ki- netic treatment and the necessity for adopting such a formalism in order to correctly understand the physical phenomena occurring in accretion discs. This study may represent a significant step forward for understanding the physical properties of accretion discs in their kinetic equilibrium configurations, and it motivates making fur- ther investigations of the subject aimed at extending the present range of validity to more general physical config- urations. The conclusions presented here may also have important consequences for other applications, and can provide a reference starting point for future work on ki- netic stability analysis of accretion discs. Here Δ$_{n}$$_{s}$ is as given in Eq.(A1), while Δ$_{2}$$_{s}$ and Δ$_{3}$$_{s}$ are given by Proceeding in the same way for evaluating the second moment of the KDF, it can be shown that the first order correction Δ U$_{s}$ to the toroidal flow velocity can be writ- ten as Δ U$_{s}$ ≡ Δ$_{ϕs}$ e$_{ϕ}$ + Δ $^{3}$s B ∇ ψ × ∇ ϕ, where we recall that in which K ≡ [ A$_{1}$$_{s}$ + A$_{2}$$_{s}$ ( Z$_{s}$e Φ eff s − Z s e c ψ Ω$_{s}$( ψ ) T$_{‖}$$_{s}$ − 1 2 )] and A$_{4}$$_{s}$ ≡ ∂α$_{s}$ ∂ψ , with α$_{s}$ ( ψ ) ≡ B $_{Δ$_{T}$$_{s}$}$. Note that here α$_{s}$ ( ψ ) differs from ̂ α$_{s}$ ( ψ ) because of the guiding-center trans- formation of the magnetic field B . Proceeding in the same way for evaluating the second In this appendix we give the detailed expressions for the coefficients which determine the fluid moments of the KDF, obtained by integrating the KDF over the velocity space. For the number density, we have found that n tot s = n$_{s}$ [1 + Δ$_{n}$$_{s}$]. The term Δ$_{n}$$_{s}$ is given by Appendix A: Calculation of the fluid moments where V$_{s}$ = R Ω$_{s}$( ψ ) and Δ$_{n}$$_{s}$ ≡ V$_{s}$ [ γ$_{1}$ + γ$_{3}$ ( T$_{‖}$$_{s}$ M$_{s}$ + 4 T$_{⊥}$$_{s}$ M$_{s}$ + V 2 s )] + (A1) + 2 γ$_{3}$I 2 B 2 ( T$_{‖}$$_{s}$ − T$_{⊥}$$_{s}$ ) V$_{s}$ R $^{2}$M$_{s}$ − γ$_{2}$ B V$_{s}$T$_{⊥}$$_{s}$, γ$_{1}$ ≡ { cM$_{s}$R Z$_{s}$e K + M$_{s}$V$_{s}$ T$_{‖}$$_{s}$ [1 + ψA$_{3}$$_{s}$ ] } , (A2) γ$_{2}$ ≡ cM$_{s}$R Z$_{s}$e A$_{4}$$_{s}$ } , (A3) γ$_{3}$ ≡ { cM 2 $_{s}$R Z$_{s}$e A$_{2}$$_{s}$ 2 T$_{‖}$$_{s}$ , (A4) Δ$_{ϕs}$ ≡ Δ$_{n}$$_{s}$Ω$_{s}$ R + Δ$_{2}$$_{s}$ + Δ$_{3}$$_{s}$ I RB . (A5) Δ$_{2}$$_{s}$ ≡ T$_{⊥}$$_{s}$ M$_{s}$ ( γ$_{1}$ + 3 γ$_{3}$V 2 s ) − γ$_{2}$ 2 T 2 ⊥ s BM$_{s}$ + (A6) + γ$_{3}$T$_{⊥}$$_{s}$ M 2 s ( T$_{‖}$$_{s}$ + 4 T$_{⊥}$$_{s}$ ) , Δ$_{3}$$_{s}$ ≡ Iγ$_{2}$T$_{⊥}$$_{s}$ RB $^{2}$M$_{s}$ 2 T$_{⊥}$$_{s}$ − T$_{‖}$$_{s}$ + (A7) + I ( T$_{‖}$$_{s}$ − T$_{⊥}$$_{s}$ ) RBM$_{s}$ ( γ$_{1}$ + 3 γ$_{3}$V 2 s ) + + Iγ$_{3}$ RBM 2 s ( 3 T 2 ‖ s − 4 T 2 ⊥ s + T$_{‖}$$_{s}$T$_{⊥}$$_{s}$ ) . Helpful discussions with Alexander Schekochihin (Ox- ford University, Oxford, U.K.) are gratefully acknowl- Acknowledgments