Stochastic Methods in Transport Theory

The methods of modern theory of stochastic processes appear to provide a useful tool for transport theory in magnetized plasmas. The Langevin equation formalism provides important, but limited information about diffusive processes. A quite promising new approach to modelling complex situations, such as transport in incompletely destroyed magnetic surfaces, is provided by the theory of Continuous Time Random Walks (CTRW), which is presented in some detail. A test problem is discussed in detail: transport of particles in a fluctuating magnetic field, in the limit of infinite perpendicular correlation length. The well-known subdiffusive behavior of the Mean Square Displacement (MSD), proportional to t^1/2, is recovered by a CTRW, but the complete density profile is only recovered under some additional conditions. The quasilinear approximation of the kinetic equation has the form of a non-markovian diffusion equation and can thus be generated by a CTRW. Finally, a new iterative map, called “tokamap” is presented and its relation to transport and CTRW is displayed.