Classical and neoclassical plasma transport

A random-walk estimate multiplies squared step length by decorrelation rate, up to coefficients determined by the detailed kinetic problem. Along a field, a thermal particle travels approximately one mean free path between scattering events, producing an inverse-collision-frequency diffusivity. Across a strong uniform field, collisions shift the guiding center by an orbit-scale distance and give a diffusivity proportional to collision frequency. More collisions can therefore reduce parallel transport while increasing perpendicular transport within the same ordered regime.

These estimates do not establish a valid local transport closure by themselves. A classical parallel Fourier-like closure needs the mean free path small compared with the parallel gradient length, with sufficiently slow evolution and a near-Maxwellian state. Strong magnetization requires cyclotron frequency much greater than collision frequency, and perpendicular gradients should be slow across an orbit. If the mean free path exceeds the system, extending the inverse-frequency formula indefinitely predicts an unphysical local diffusion process where streaming and boundaries actually dominate.

In a large-aspect-ratio axisymmetric tokamak, field strength varies along a field line and traps a fraction of particles between mirror points. Their guiding centers form banana-shaped orbits in a poloidal cross-section. The characteristic radial width contains the safety factor and inverse aspect ratio, and can greatly exceed a gyroradius while remaining small compared with equilibrium profile scales. Collisions scatter particles between trapped and passing regions, decorrelating those wider orbits.

A useful banana-regime scaling combines trapped fraction, squared banana width, and the enhanced rate at which pitch-angle scattering crosses the narrow trapped region. Multiplying these factors gives the familiar geometric enhancement of perpendicular diffusion. It is an order estimate for a specified tokamak regime, not an exact coefficient for every species, field geometry, or radial electric field. Stellarators and other nonaxisymmetric systems can have qualitatively different orbit losses and ambipolarity constraints.

The banana regime requires scattering out of the trapped region to be slow compared with bounce motion. A dimensionless collisionality compares the effective scattering rate with the bounce frequency. Increasing it moves through intermediate plateau behavior and toward a more collisional regime; the low-collisionality scaling should not be extrapolated across all of them. Even at low collisionality, a radially local neoclassical calculation needs orbit width small relative to the profile scale.

A collisionless closed orbit does not by itself produce a secular random walk across flux surfaces. Collisions, fluctuating fields, asymmetries, or boundaries supply the mechanism that breaks repeated orbit return. Turbulent fluxes can exceed a classical or neoclassical estimate and depend on correlations not present in an equilibrium orbit calculation. Comparing measured transport with a baseline is useful only after checking geometry, collisionality, orbit width, gradients, and the definition of the measured particle or heat flux.