Transport theory in the collisionless limit

Traditional transport theory provides a closure of fluid equations that is valid in the collisional, short mean-free-path limit. The possibility of extending an analogous closure to long mean-free path is examined here. An appropriate kinetic equation, using a model collision operator, is solved rigorously for arbitrary collisionality but weak, Maxwellian source terms. The corresponding particle and heat flows are then expressed in terms of the density and temperature profiles. The transport matrix is found to be symmetric even at vanishing collision frequency; in the collisionless limit it takes the form of nonlocal operators. The operator corresponding to thermal conductivity agrees with one found previously by Hammett and Perkins. However particle diffusion, which turns out to satisfy a local Fick's law for any finite collision frequency, becomes singular at vanishing collisionality, where the pressure gradient vanishes. We conclude that the fluxes can generally be expressed in terms of particle and energy sources, but not always in terms of pressure and temperature profiles