A Stochastic Model for Plasma Dynamics

Abstract

The dynamics of charged particles in magnetic structures, consisting of numerous small islands (microturbulence due to a great number of excited magnetic modes), may be viewed, in the collisional case, as the dynamics of stochastic, non isotropic diffusion processes, verifying, in the three dimensional physical space R ³, a stochastic differential equation of the form: (see (1) – (2))

$${\text{d}}{{\text{X}}_{\text{t}}} = {\beta _ + }({{\text{X}}_{\text{t}}},{\text{t}}){\text{ dt + D d}}{{\text{W}}_{\text{t}}}$$

(I.1)

where W_t is the standard brownian motion, β₊ is a drift term, witch will be determined by a stochastic dynamical asumption (see (3) – (4) – (5)), D is a diffusion tensor: D = α σ, where σ is a constant symmetric matrix:

$$\sigma = \left( {\begin{array}{*{20}{c}} {{\sigma _x}}{{\sigma _*}}0 \\ {{\sigma _*}}{{\sigma _y}}0 \\ 0 0{{\sigma _z}} \end{array}} \right) $$

(I.2)

${\alpha ^2} = \frac{{KT}}{\mu }\tau$, is the Einstein’s diffusion coefficient of brownian diffusion, τ is a characteristic time of the diffusion.