10 field without requiring any net accretion and in the ab- sence of any possible instability/turbulence phenomena . This new mechanism results from poloidal currents aris- ing due to the FLR effects and temperature anisotropies which are characteristic of the equilibrium KDF. We re- mark that the self-generation of the stationary magnetic field is purely diamagnetic. In particular, the toroidal component is associated with the drifts of the plasma away from the flux surfaces. In the present formula- tion, possible dissipative phenomena leading to a non- stationary self field have been ignored. Such dissipative phenomena probably do arise in practice and could oc- cur both in the local domain where the equilibrium mag- netic surfaces are closed and nested and elsewhere. Tem- perature anisotropies are therefore an important physi- cal property of collisionless AD plasmas, giving a possi- ble mechanism for producing a stationary toroidal mag- netic field. We stress that this effect disappears alto- gether in the case of isotropic temperatures, as demon- strated in Appendix A. Finally, we consider the ratio between the toroidal and poloidal current densities ( J$_{T}$ and J$_{P}$ ). In the small inverse aspect ratio ordering, ne- glecting corrections of order O ( δ k ) with k ≥ 1, this pro- vides an estimate of the magnitude of the corresponding components of the magnetic field. In fact, in this limit we can write |∇× B $_{T}$| |∇× B $_{P}$| ∼ | B $_{T}$| | B $_{P}$| ∼ | J $_{P}$| | J $_{T}$| and so conclude that, although for the single species velocity, the order- ing | Δ $^{3}$s B ∇ ψ ×∇ ϕ | | V $_{s}$| ∼ O ( ε ) O ( δ$_{T s}$ ) holds (see Sec. VI), this might no longer be the case for the magnetic field, which instead depends on the ratio between the total toroidal and poloidal current densities. In particular, the possi- bility of having finite stationary toroidal magnetic fields is, in principle, allowed by the present analysis, depend- ing on the properties of the overall solution describing the system. potential. Note that this solution for the electrostatic potential Φ ∼ can be shown to be consistent with earlier treatments appropriate for Tokamak plasma equilibria [24, 36]. This can be exactly recovered thanks to the ar- bitrariness in defining the pseudo-densities and by taking the limit of isotropic temperatures and zero gravitational potential, as in the case of laboratory plasmas. In this limit the species pseudo-densities become flux functions [24]. Then, because of this arbitrariness, by taking it follows that Eq.(31) reduces to the form presented in [24], which can only be used to determine the poloidal variation of the potential. Let us now consider the Ampere equation. Adopting the Taylor analytical expansion of the asymptotic equi- librium KDF and neglecting corrections of O ( ε k ) , with k ≥ 2, and O ( ε n +1 M ) , with n ≥ 0, this can be approxi- mately written as follows: where the elliptic operator Δ ∗ is defined as Δ ∗ ≡ R $^{2}$∇ · ( R − $^{2}$∇ ) [51]. The remaining terms in Eq.(34) give the equation for the toroidal component of the magnetic field I ( ψ,ϑ ) R . In the same approximation, this is: Getting a complete understanding of the dynamical properties of astrophysical accretion discs still represents a challenging task and there are many open problems remaining to be solved before one can get a full and con- sistent theoretical formulation for the physical processes involved. where Δ$_{3}$$_{s}$, given in Appendix A, contains the contribu- tions of the species temperature anisotropies. For con- sistency with the approximation introduced, in the small inverse aspect ratio ordering, it follows that ∂I ( ψ,ϑ ) ∂ϑ = 0 + O ( δ k ) , i.e., to leading order in δ : I = I ( ψ )+ O ( δ k ) , with k ≥ 1. This in turn also requires that the corre- sponding current density in Eq.(36) is necessarily a flux function. Then, correct to O ( ε ), O ( ε 0 $_{M}$) and O ( δ 0 ) , the differential equation for I ( ψ ) becomes: The present investigation provides some important new results for understanding the equilibrium proper- ties of accretion discs, obtained within the framework of a kinetic approach based on the Vlasov-Maxwell de- scription. The derivation presented applies for collision- less non-relativistic and axi-symmetric AD plasmas un- der the influence of both gravitational and EM fields. A wide range of astrophysical scenarios can be investi- gated with the present theory, thanks to the possibility of properly setting the different parameters which char- acterize the physical and geometrical properties of the which uniquely determines an approximate solution for the toroidal magnetic field. This result is remarkable be- cause it shows that there is a stationary “kinetic dynamo effect” which generates an equilibrium toroidal magnetic VIII. CONCLUSIONS ∇ I ( ψ, ϑ ) × ∇ ϕ = 4 π c ∑ s = i,e q$_{s}$n$_{s}$ Δ$_{3}$$_{s}$ B ∇ ψ × ∇ ϕ, (36) ∂I ( ψ ) ∂ψ = 4 π c ∑ s = e,i q$_{s}$n$_{s}$ Δ$_{3}$$_{s}$ B , (37) Δ $^{∗}$ψ$_{p}$ = − 4 π c R ∑ s = e,i q$_{s}$n$_{s}$ [Ω$_{s}$ ( ψ ) R + Δ$_{ϕs}$] , (35) ∇ × B self = 4 π c ∑ s = i,e q$_{s}$n$_{s}$ [ V$_{s}$ + Δ U$_{s}$ ] , (34) η$_{e}$ Zη$_{i}$ = 1 , (33) where B self is as defined in Eq.(4) and the expression for Δ U$_{s}$ is given by Eq.(24). The toroidal component of this equation gives the generalized Grad-Shafranov equation for the poloidal flux function ψ$_{p}$ :