Application of the Ito theory for Monte Carlo simulation of plasma diffusion

Abstract

In this paper the full set of stochastic differential equations (SDEs) are presented describing the guiding centre motion of test charged particles in a plasma with an arbitrary inhomogeneous magnetic field, when the drift approximation is applicable. The derivation is based on the Ito formula which is used to determine stochastic differentials of functions of the non-gyro-averaged velocity diffusion in strict correspondence with the general kinetic equations involving Coulomb collision operators. The drift SDEs are obtained by calculating the Ito stochastic integrals within time intervals admitting the gyro-averaging procedure. The proposed SDEs reproduce the well-known Monte Carlo operators for orbital invariants, however additionally accounting for the spatial drift caused by the cross-field diffusion process with a classical diffusion coefficient. All SDE coefficients are explicitly expressed in terms of the Rosenbluth potentials in a gyro-tropic or isotropic background plasma. The SDEs are presented in particular for the case of an axisymmetric toroidal magnetic configuration to describe the spatial two-dimensional poloidal diffusion process providing a detailed description of neoclassical orbital effects.