Lagrangian formulation of transport theory: Like particle collisional transport and variational principle

Abstract

A formulation of transport theory, specifically treating like-particle collisions for the lowest collisionality regime, is developed for nonsymmetric geometries, using the Lagrangian picture. This theory provides a particularly simple method for calculating transport coefficients which is useful for devices such as stellarators and tandem mirrors. It is shown that the explicit piece of the transport coefficient, processes such as test particle diffusion 〈ΔψΔψ/Δt〉, provides a rigorous upper bound to the full diagonal transport coefficient. The more complex compound, implicit processes (involving perturbed distribution functions and not accessible from a simple Monte Carlo procedure), are always inward (for normal gradients) and can only reduce the full flux from its test particle value. The implicit fluxes have an associated variational principle, maximizing this inward flux. This variational principle differs from the usual minimum principle for entropy production, although the two principles are equivalent, as is shown. An application to systems with weak asymmetry, such as stellarators, bumpy tori, and tandem mirrors is considered.