Diffusion of charged particles in a stochastic force-free magnetic field

Abstract

We study diffusion of charged particles in stationary stochastic magnetic field ${\bf B}$ with zero mean, $\langle {\bf B} \rangle = 0 $. In the case when electric current is carried by electrons, the field is force-free, $\mathrm{curl} \,{\bf B} = \alpha{\bf B} $, where $\alpha({\bf r})$ is an arbitrary scalar function. In a small region where the function $\alpha $ and the field magnitude $|{\bf B}|$ are approximately constant, the equations of motion of charged particles are integrated and reduced to the equation of mathematical pendulum. The transition from trapped to untrapped particles is continuously traced. Averaging over the magnetic field spectrum gives the spatial diffusion coefficient $D$ of particles as a function of the Larmor radius $r_L$ in the large-scale magnetic fields ($B_{LS}$) and magnetic field correlation length $L_0$. The diffusion coefficient turns out to be proportional to the Larmor radius, $D\propto r_L $, for $r_L <L_0 / 2\pi $, and to the Larmor radius squared, $ D \propto r_L^2 $, for $ r_L> L_0 /2\pi $. We apply obtained results to the diffusion of cosmic rays in the Galaxy, which contains a large number of independent regions with parameters $L_0$ and $B_{LS}$ varying in wide range. We average over $B_{LS}$ with the Kolmogorov spectrum and over $L_0$ with the distribution function $f(L_0)\propto L_0^{- 1+ \sigma}$. For the practically flat spectrum $\sigma = 1/15$, we have $ D\propto r_m^{0.7}$, which is consistent with observations.