The problem of random motion of charged particles in an external magnetic field is studied under the assumption that the Langevin sources produce anisotropic diffusion in velocity space and the friction force is dependent on the direction of particle motion. It is shown that in the case under consideration, the kinetic equation describing particle transitions in phase space is reduced to the equation with a Fokker-Planck collision term in the general form (non-isotropic friction coefficient and nonzero off-diagonal elements of the diffusion tensor in the velocity space). The solution of such an equation has been obtained and the explicit form of the transition probability is found. Using the obtained transition probability, the mean-square particle displacements in configuration and velocity space were calculated and compared with the results of numerical simulations, showing good agreement. The obtained results are used to generalize the theory of large-scale fluctuations in plasmas to the case of anisotropic diffusion across an external magnetic field. Such diffusion is expected to be observed in the case of an anisotropic k spectrum of fluctuations generating random particle motion (for example, in the case of drift-wave turbulence).
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