Kinetic Equilibria of Collisionless Tori

• The paper establishes invariant-based kinetic distribution functions using generalized Maxwellians for off-equatorial collisionless tori.
• It demonstrates that the balance among gravitational, centrifugal, and magnetic forces produces levitating density maxima away from the equator.
• The study reveals kinetic corrections to the thermal pressure that modify the traditional fluid equation of state in collisionless regimes.

Searching arXiv for the cited papers and closely related work on collisionless toroidal kinetic equilibria. arXiv search query: "(Cremaschini et al., 2013) collisionless tori kinetic equilibria off-equatorial plasma torus" Kinetic equilibria of collisionless tori are stationary solutions of the Vlasov–Maxwell system for toroidal plasma configurations in which binary collisions are negligible and the phase-space distribution is constrained by single-particle invariants rather than by collisional thermodynamic closure. In the specific setting of off-equatorial plasma tori around compact objects, the problem is to determine whether levitating, non-equatorial density maxima can persist in the combined action of gravity and magnetic confinement when the plasma is sufficiently dilute that a fluid description becomes inappropriate. For non-relativistic, multi-species, axisymmetric plasmas subject to an external dominant spherical gravitational field and a dipolar magnetic field, explicit equilibrium kinetic distribution functions can be constructed in terms of generalized Maxwellian functions with isotropic temperature and non-uniform fluid fields; under general conditions these equilibria admit off-equatorial tori and exhibit kinetic corrections that modify the ordinary thermal equation of state (Cremaschini et al., 2013).

1. Definition and astrophysical setting

In this context, a collisionless torus is an axisymmetric toroidal plasma structure described by distribution functions $f_s(\mathbf r,\mathbf v,t)$ for each species $s$, with the electromagnetic and gravitational fields treated in the quasi-static limit. The problem addressed for off-equatorial tori is motivated by accretion-disc coronae, which are expected to arise in a highly diluted environment; this is precisely the regime in which a fluid or non-ideal MHD description may be inadequate and a kinetic treatment becomes necessary (Cremaschini et al., 2013).

The non-relativistic model of off-equatorial equilibria assumes an external Newtonian gravitational potential

$\Phi_G(r)=-\frac{G_N M_+}{r},$

together with a dipolar vacuum magnetic field written through the poloidal flux

$$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$

The plasma-generated self-fields are ordered to be small relative to the external fields,

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$

so that the Vlasov equation can be analytically decoupled from Maxwell’s equations to leading order (Cremaschini et al., 2013).

The astrophysical significance of these equilibria is that they supply a collisionless counterpart to off-equatorial toroidal structures previously discussed in non-ideal MHD. In the kinetic description, the torus is not postulated as a fluid body obeying a barotropic closure; instead, it emerges from the invariant structure of single-particle motion and from the corresponding phase-space moments. This distinguishes the kinetic torus problem from conventional disc or corona modelling and places it in continuity with invariant-based kinetic equilibria developed for accretion-disc plasmas and magnetized toroidal plasmas more broadly (Cremaschini et al., 2010).

2. Vlasov–Maxwell framework and particle invariants

The underlying kinetic equation is the collisionless Vlasov equation,

$$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$

supplemented by Maxwell’s equations in the quasi-static limit. Axisymmetry and quasi-stationarity imply that, to leading order, each particle conserves the canonical toroidal momentum

$$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$

with $\psi=\psi_{\rm ext}+\psi_{\rm self}$, and the total energy

$$E_s=\frac{M_s}{2}v^2+Z_s e\,\Phi_{\rm eff}(\mathbf r),\qquad \Phi_{\rm eff}\equiv \Phi+\frac{M_s}{Z_s e}\Phi_G.$$

These two invariants are the arguments of the equilibrium distribution function for the off-equatorial-torus construction (Cremaschini et al., 2013).

The invariant-based strategy is a recurrent feature of kinetic torus theory. In strongly magnetized axisymmetric accretion-disc plasmas, the canonical toroidal momentum, total energy, and magnetic moment provide the exact or adiabatic invariants underlying generalized bi-Maxwellian equilibria (Cremaschini et al., 2012). In axisymmetric Tokamak plasmas with anisotropic temperatures, the same invariant structure supports asymptotic kinetic equilibria with toroidal differential rotation, quasi-stationary poloidal flows, and temperature anisotropy (Cremaschini et al., 2011). In more general toroidal geometry, a non-perturbative gyrokinetic formulation yields the three invariants

$G\equiv \{E,p_\varphi,\mu\},$

which are used to construct spatially non-symmetric kinetic equilibria (Cremaschini et al., 2023).

A plausible implication is that the existence of a torus in collisionless theory is controlled less by a macroscopic force-balance ansatz than by the availability of a sufficiently rich invariant set. That interpretation is explicit in the relativistic extensions as well: in covariant treatments without requiring spatial symmetry, the canonical energy $s$0 and the relativistic magnetic moment $s$1 are used as equilibrium invariants (Cremaschini et al., 2023), while for neutral collisionless matter in Kerr spacetime the energy, azimuthal angular momentum, and Carter constant determine the functional form of the equilibrium distribution (Cremaschini et al., 2023).

3. Equilibrium distribution functions and kinetic regimes

For off-equatorial collisionless tori, the equilibrium distribution is sought in the form

$s$2

with structure functions $s$3, $s$4, and $s$5 that depend on the invariants. The admissible equilibria are generalized Maxwellian functions characterized by isotropic temperature and non-uniform fluid fields (Cremaschini et al., 2013). The equivalent shifted-Maxwellian representation introduces

$s$6

so that the leading-order equilibrium is a rotating drifted Maxwellian in velocity space.

The functional dependence of the structure functions distinguishes three kinetic regimes. In the gravitationally-bound regime,

$s$7

in the magnetized regime,

$s$8

and in the mixed regime,

$s$9

The off-equatorial torus is admitted under general conditions precisely when both gravitational and magnetic fields contribute to shaping the spatial profiles of the equilibrium plasma fluid fields (Cremaschini et al., 2013).

The leading-order Chapman–Enskog expansion is written as

$\Phi_G(r)=-\frac{G_N M_+}{r},$0

where

$\Phi_G(r)=-\frac{G_N M_+}{r},$1

The density associated with this leading-order Maxwellian is

$\Phi_G(r)=-\frac{G_N M_+}{r},$2

Thus the equilibrium torus is fully specified, at leading order, by the number density $\Phi_G(r)=-\frac{G_N M_+}{r},$3, the temperature $\Phi_G(r)=-\frac{G_N M_+}{r},$4, and the rotation frequency $\Phi_G(r)=-\frac{G_N M_+}{r},$5 (Cremaschini et al., 2013).

This construction is closely related to the generalized bi-Maxwellian equilibria used in other toroidal settings. In axisymmetric collisionless accretion-disc plasmas, generalized bi-Maxwellian distributions encode toroidal differential rotation and temperature anisotropy (Cremaschini et al., 2010), while in spatially non-symmetric toroidal equilibria the generalized bi-Maxwellian depends on $\Phi_G(r)=-\frac{G_N M_+}{r},$6 and yields a pressure tensor with $\Phi_G(r)=-\frac{G_N M_+}{r},$7 (Cremaschini et al., 2023). The off-equatorial-torus case is more specialized in that the 2013 equilibrium is explicitly isotropic in temperature at the generalized-Maxwellian level, with kinetic corrections entering through the asymptotic expansion rather than through a bi-Maxwellian zeroth order (Cremaschini et al., 2013).

4. Existence of off-equatorial density maxima

The defining feature of an off-equatorial torus is that the species density has a maximum away from the equatorial plane, that is, at $\Phi_G(r)=-\frac{G_N M_+}{r},$8. In the kinetic formulation, such a maximum occurs where $\Phi_G(r)=-\frac{G_N M_+}{r},$9 has a zero crossing off the equator. Differentiating the exponent of the density with respect to $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$0 gives the criterion

$$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$1

This expresses the balance, at the torus center, among centrifugal force, magnetic contribution, pressure-gradient force, and gravity in the polar direction (Cremaschini et al., 2013).

Two explicit constructions are given. In the first, $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$2 and $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$3 are taken constant, a desired density profile $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$4 with off-equatorial maxima is prescribed, and the density relation

$$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$5

is inverted to obtain $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$6; this yields a quadratic equation in $$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$7. In the second, one imposes

$$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$8

and requires the Maxwellian exponent to vanish, so that

$$\mathbf B_{\rm ext}=\nabla\psi_{\rm ext}\times\nabla\varphi,\qquad \psi_{\rm ext}(r,\theta)=\mathcal M_0\frac{\sin^2\theta}{r}.$$9

which has two real roots for $$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$0 (Cremaschini et al., 2013).

These constructions show that levitating off-equatorial systems are not exceptional solutions produced by a single fine-tuned rotation law. Rather, they can be realized by different choices of the structure functions, provided the gravitational and magnetic terms jointly shape the equilibrium profiles (Cremaschini et al., 2013). A plausible implication is that the off-equatorial torus is a structural possibility of the invariant-based kinetic closure itself, rather than a special artifact of one parametrization.

The broader literature on collisionless tori supports this emphasis on configurational conditions. In axisymmetric accretion-disc plasmas, the existence of kinetic equilibria requires gravitational binding, axisymmetry, and appropriate kinetic constraints on the structure functions (Cremaschini et al., 2012). In more general toroidal geometries, necessary and sufficient conditions include a valid gyrokinetic diffeomorphism, functional independence of the invariants, consistency with Poisson and Ampère constraints, and smooth nested magnetic surfaces (Cremaschini et al., 2023).

5. Fluid moments and deviations from the thermal equation of state

The fluid moments of the equilibrium distribution determine the macroscopic fields carried by the torus. For the off-equatorial solutions, the pressure tensor is isotropic at the level of the total tensor,

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$1

but the scalar pressure is not simply $$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$2. Instead,

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$3

The kinetic corrections are explicitly identified as diamagnetic, energy-correction, and electrostatic contributions (Cremaschini et al., 2013).

The diamagnetic or FLR-type terms $$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$4 enter current calculations but drop out of the scalar pressure because they are odd in $$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$5. The energy correction is

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$6

where $$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$7 is a linear combination of the thermodynamic-force gradients

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$8

The pure electrostatic correction is

$$|\mathbf B_{\rm self}|/|\mathbf B_{\rm ext}|\sim O(\lambda)\ll1,\qquad \frac{|Z_s e\,\Phi|}{M_s|\Phi_G|}\sim O(\lambda)\ll1,$$9

These terms imply non-vanishing deviations from the assumption of thermal pressure and therefore modify the equation of state relative to the collisional law $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$0 (Cremaschini et al., 2013).

This point is central to the kinetic theory of collisionless tori. In strongly magnetized accretion-disc plasmas, the equilibrium pressure tensor is anisotropic already at leading order,

$$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$1

with FLR corrections generating off-diagonal stresses (Cremaschini et al., 2012). In Tokamak plasmas, the leading-order pressure tensor is likewise gyrotropic with $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$2, and poloidal flows appear only if anisotropy is present or if the toroidal rotation differs from the $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$3 rotation (Cremaschini et al., 2011). The off-equatorial-torus solutions of 2013 are therefore unusual in that isotropic-temperature generalized Maxwellians still produce non-thermal pressure corrections once the kinetic expansion is carried to the relevant order (Cremaschini et al., 2013).

A common misconception is that a scalar pressure tensor implies an exactly thermal equation of state. In this class of collisionless torus equilibria, that implication does not hold: the tensor remains isotropic while the scalar pressure receives kinetic-origin corrections associated with energy drift and electrostatic ordering (Cremaschini et al., 2013).

6. Self-consistency, extensions, and stability

The off-equatorial torus is completed by checking Maxwell’s equations a posteriori. Poisson’s equation,

$$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$4

is solved to leading order by inserting the density $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$5 and neglecting $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$6 corrections. The resulting $$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$7 must satisfy the ordering

$$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$8

Ampère’s law for the self-poloidal flux,

$$\frac{d}{dt}f_s \equiv \partial_t f_s+\mathbf v\cdot\nabla f_s +\frac{Z_s e}{M_s}\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr)\cdot\frac{\partial f_s}{\partial\mathbf v}=0,$$9

yields a generalized Grad–Shafranov equation for $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$0, and its solution must remain $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$1 relative to the dipole flux (Cremaschini et al., 2013). In this sense, the equilibrium is kinetic and self-consistent but perturbatively dominated by the prescribed external gravitational and magnetic fields.

The same microscopic-to-macroscopic closure appears across the broader theory of collisionless tori. In quasi-stationary accretion-disc plasmas, Maxwell’s equations determine the poloidal flux $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$2 and toroidal-field function $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$3, while finite-Larmor-radius, diamagnetic, and energy-correction terms generate a quasi-stationary kinetic dynamo that self-generates poloidal and azimuthal magnetic fields (Cremaschini et al., 2012). In the earlier axisymmetric gravitational equilibrium theory, the possibility of self-generating toroidal magnetic field without net radial accretion flow was already identified as a “kinetic dynamo effect” (Cremaschini et al., 2010). In spatially non-symmetric toroidal equilibria, Poisson and Ampère constraints likewise determine whether the generalized bi-Maxwellian can sustain self-generated toroidal and poloidal fields (Cremaschini et al., 2023).

Stability results depend on geometry and ordering. For strongly magnetized, gravitationally bound, axisymmetric accretion-disc plasmas described by stationary kinetic solutions with non-uniform fluid fields, stationary accretion flows, and temperature anisotropies, no axisymmetric unstable perturbations exist on low-frequency, long-wavelength scales long compared with the Larmor time and radius; this rules out axisymmetric magneto-rotational or thermal instabilities of that type (Cremaschini et al., 2012). In a distinct solid-torus setting with a perfectly conducting boundary and specular particle reflection, toroidally symmetric equilibria of the relativistic Vlasov–Maxwell system satisfy a sharp spectral criterion: the self-adjoint operator $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$4 is non-negative if and only if there are no exponentially growing modes (Nguyen et al., 2013).

Recent extensions move beyond the non-relativistic, axisymmetric, off-equatorial setting. Covariant gyrokinetic theory allows spatially non-symmetric equilibria in strong gravity without requiring any ignorable spatial coordinate, using the invariants $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$5 and Gaussian-like equilibrium distributions (Cremaschini et al., 2023). For spatially symmetric relativistic plasmas with non-stationary electromagnetic fields, axisymmetric “toroidal” equilibria can still be built from the adiabatic invariants $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$6 and $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$7, even when particle energy is not conserved (Cremaschini et al., 2023). For neutral collisionless matter in Kerr spacetime, the Carter constant produces non-isotropic equilibrium distributions $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$8, with anisotropic stress-energy tensors and toroidal isodensity surfaces localized near minima of an effective potential (Cremaschini et al., 2023). In a perturbative Einstein–Vlasov treatment around a Kerr black hole, a collisionless torus with a Gaussian-like distribution $$P_{\varphi s}=M_s R v_\varphi+\frac{Z_s e}{c}\psi(\mathbf r),$$9 generates a self-metric of Kerr type and a stress-energy tensor with pressure and temperature anisotropies and nonzero $\psi=\psi_{\rm ext}+\psi_{\rm self}$0 (Cremaschini et al., 2023).

Taken together, these results define kinetic equilibria of collisionless tori as a family of invariant-based toroidal solutions whose geometry, flows, and pressure structure are fixed by the conserved quantities of single-particle dynamics and by the a posteriori Maxwell or Einstein constraints. In the off-equatorial plasma-torus problem, the essential conclusion is that levitating collisionless tori are admitted under general conditions, and that their equilibrium thermodynamics is intrinsically kinetic rather than reducible to the ordinary thermal pressure law (Cremaschini et al., 2013).