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Let us consider a plasma in a strong uniform magnetic field $ B_0 $ and a weak random field $ B_0 \gg \delta B $. At each moment the particle rotates with a perturbed Larmor radius. The unperturbed component of the radius is, assuming the particle is ultra - relativistic

$ r_l \approx \frac{\varepsilon}{q B} $

where $ \varepsilon $ is the energy of the particle and $ q $ is its charge. The amplitude of the perturbation to the Larmor radius is

$ \delta r_l \approx \frac{\varepsilon}{q B_0^2} \delta B $

How long will it take the net displacement to be comparable to the unperturbed Larmor radius $ r_l $ ?

$ c \tau \left(\frac{\delta r_l}{r_l} \right)^2 \approx r_l \Rightarrow \tau \approx \frac{r_l}{c} \left(\frac{B}{\delta B}\right)^2 $

In a realistic situation, the random field has a certain spectrum $ \delta B^2 \propto k^{m-1} $, where $ m<1 $ is a constant. Magnetic fields at wavelengths larger than $ r_l $ vary the orbit in an almost coherent way, so their contribution to scattering is small. Magnetic fields on wavelengths much smaller than $ r_l $ are very weak, so their contribution to scattering is also small. Hence, the range of wavelengths that contributes the most to scattering is the same order of magnitude as $ r_l $. Substituting $ k \approx r_l^{-1} $ yields $ \tau \propto r_l^m $. The diffusion coefficient is

$ D \approx c^2 \tau \propto r_l^m \propto \varepsilon^m $.

For magnetic fields in a Kolmogorov spectrum

$ \delta B^2 \propto \delta v^2 \propto k^{-2/3} \Rightarrow m=\frac{1}{3} $.