Let us consider a plasma in a strong uniform magnetic field and a weak random field . At each moment the particle rotates with a perturbed Larmor radius. The unperturbed component of the radius is, assuming the particle is ultra - relativistic
where is the energy of the particle and is its charge. The amplitude of the perturbation to the Larmor radius is
How long will it take the net displacement to be comparable to the unperturbed Larmor radius ?
In a realistic situation, the random field has a certain spectrum , where is a constant. Magnetic fields at wavelengths larger than vary the orbit in an almost coherent way, so their contribution to scattering is small. Magnetic fields on wavelengths much smaller than 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 . Substituting yields . The diffusion coefficient is
For magnetic fields in a Kolmogorov spectrum