Particle confinement, kinetic stability, and transport

Ideal-MHD stability addresses large-scale fluid perturbations. This parallel development addresses orbit loss, bootstrap current, neoclassical heat loss, and microinstabilities. References and source caveats are in 09-literature.md.

Contents

• Full-orbit dynamics
• Guiding-centre dynamics
• Symmetry, omnigeneity, and finite-time bounds
• Drift-kinetic transport
• Gyrokinetic stability and turbulence
• Profile evolution and reactor power
• Theorem backlog

Full-orbit dynamics

For a nonrelativistic particle of mass m>0 in specified fields:

For a static magnetic field and E=-∇Φ, prove conservation of

For q≠0, E=0 and uniform nonzero B, prove the helix solution and Larmor-radius formula. Include the straight-line q=0 or B=0 limit separately. These are useful nonvacuity and sign tests.

Define confinement as x(t) ∈ Ω_safe for all t ∈ [0,T] and all initial conditions in a declared set K. An invariant magnetic surface constrains field lines; full particle motion includes perpendicular velocity and does not automatically remain on that surface.

Possible first certificates are:

• interval-enclosed orbit integration for a compact initial set and finite time;
• an invariant energy or Lyapunov barrier separating initial states from a loss boundary;
• a reduced-orbit certificate with a proved full-orbit error smaller than the spatial margin.

Treat a probability of loss separately: define an initial probability measure and prove a measure bound. Tracking a sampled ensemble yields a numerical estimate, not a universal or measure theorem by itself.

Guiding-centre dynamics

Use Littlejohn's variational formulation. For static fields, a chosen leading-order guiding-centre model uses (X,v∥,μ), where μ=m v⊥²/(2|B|) and b=B/|B|. One convention is

Require m>0, q≠0, |B|≥B_*>0, and |B∥*|≥b_*>0 on the certified phase-space set. Derive the equations from the chosen action/Poisson structure. With a time-independent, single-valued scalar potential E=-∇Φ, prove conservation of H_gc=m v∥²/2+μ|B|+qΦ. Static fields alone do not ensure a global scalar potential on a multiply connected domain. Time-dependent fields add terms and must have a separate interface.

Exact invariance of μ belongs to this reduced model. In full Lorentz motion it is an adiabatic approximation. A model-reduction theorem must include initial-data matching, regularity, the small-gyroradius regime, and a time-dependent error bound.

Symmetry, omnigeneity, and finite-time bounds

For a genuine symmetry of the guiding-centre action, prove the corresponding Noether invariant. Extract the exact relation between Boozer field-strength quasisymmetry and this symmetry from Burby–Kallinikos–MacKay. A spatial symmetry of |B| alone should not be used as a theorem about an arbitrary kinetic system.

For trapped reduced orbits define the bounce action

at fixed energy and magnetic moment, specifying turning points, sign conventions, and regularity near the trapped–passing boundary. Omnigeneity concerns vanishing bounce-averaged radial drift; in an appropriate formulation it is expressed by independence of the bounce action from field-line label. Cary–Shasharina provides a starting point.

Keep quasisymmetry, omnigeneity, and quasi-isodynamic geometry as distinct predicates. None alone supplies all-mode MHD stability, all-time full-orbit confinement, or absence of turbulent heat transport.

An elementary approximate-invariant theorem is useful for imperfect designs. Require K along each trajectory to be continuously differentiable (or absolutely continuous with the bound almost everywhere):

Take ε≥0 and forward times t≥0 in a connected interval on which the trajectory exists. Require initial states inside the safe region and a continuous trajectory so loss requires a first crossing of its boundary. Combine the bound with a proved separation condition: initial states satisfy K≤k₀, while every loss-boundary state in the relevant energy/phase-space set satisfies K≥k_loss>k₀. For ε>0, loss is impossible before (k_loss-k₀)/ε within the declared existence interval. For ε=0, it is impossible throughout that interval. Establish the separation, continuity, phase-space enclosure, and derivative bounds in the actual dynamics; the derivative bound must hold up to any possible first crossing.

Study tolerances in derivative-sensitive norms. A small Fourier or volume-averaged quasisymmetry residual does not automatically bound every orbit's drift, especially near resonances and separatrices.

Drift-kinetic transport

Adopt a specified distribution perturbation g_s and a drift-kinetic equation schematically

where T includes the selected streaming, mirror, and drift terms and C is a specified linearized collision operator. Record boundary conditions, particle/momentum/energy conservation, nullspace constraints, and the thermodynamic drives in S.

Landreman–Smith–Mollén–Helander/SFINCS is a source for model choices, not a proof oracle.

Define radial particle and heat fluxes and parallel current through velocity integrals and flux-surface averages, using the chosen heat-flux convention. Prove:

• transport is zero for the correct source-free equilibria;
• conservation and entropy-production/dissipation identities;
• solvability conditions against collision invariants;
• uniqueness on an appropriate constrained quotient;
• an a posteriori bound ‖g-g_N‖ ≤ C‖residual‖ if the required operator stability estimate is proved;
• resulting flux/current errors through continuous functional bounds.

The C estimate is not automatic for a kinetic operator. Small collision frequency, trapped particles, and resonant precession can make it difficult.

For ambipolarity define A(E_r)=Σ_s q_s Γ_s(E_r). If a theorem proves continuity, endpoint signs on an interval, and A'≥a_*>0 there, then prove existence and uniqueness of an ambipolar root. If |δA|≤η, the shifted root has a displacement bound under the corresponding monotonicity/domain hypotheses. Do not assume one ambipolar root exists or is unique for every stellarator.

Feed the resulting bootstrap-current and electric-field envelopes back into equilibrium and stability. This closes a physics loop that a zero-current vacuum model omits.

Gyrokinetic stability and turbulence

Select the gyrokinetic ordering, equilibrium, species, electrostatic/electromagnetic approximation, collision model, and field equations. Derive a free-energy balance of the form

with exact definitions. Nonnegativity, nullspaces, and source signs must be proved for that model. Brizard–Hahm gives variational/energy foundations.

Formalize a restricted maximum-J trapped-particle stability theorem with its actual ordering and gradient conditions. Proll et al. and Helander et al. are useful sources; follow-up ion-driven modes demonstrate why this is not universal microstability.

Subsequent research targets:

1. Rigorous growth-rate bounds for a prescribed linear kinetic model including truncation errors.
2. A nonlinear free-energy estimate giving an absorbing bound or a finite-time bound under prescribed forcing.
3. A bound on time-averaged radial heat flux as a functional of free energy, with explicit averaging and initial-data dependence.
4. A useful coupling of orbit optimization, equilibrium, and turbulence suppression.

Do not equate linear microstability with a proved nonlinear turbulent transport law. A reactor can tolerate some turbulence if a sufficiently strong heat-loss bound is established.

Profile evolution and reactor power

For a simplified no-flow, prescribed-density model, use a radial volume label r with declared metric/area convention:

A phenomenological closure such as Q_s^r=-n_sχ_s ∂_rΘ_s needs a supplied interval law for χ_s and its regime. Add particle evolution, convection, interspecies exchange, and compression terms when moving beyond this restricted model. Prove conservation with boundary flux and comparison/positivity properties.

For distinct D and T species,

Use a verified enclosure of a sourced reactivity fit in its documented temperature domain, including the Bosch–Hale corrigendum. A fit is empirical input, not a first-principles theorem. Track alpha deposition, escaping energetic particles, neutron power, radiation, conduction, auxiliary heating, cryogenics, and pumping.

Distinguish plasma gain Q=P_fusion/P_aux from net electrical output. Reactor feasibility requires the coupled thermal/power/tritium/component contracts in 07.

Theorem backlog