The Diffusion Coefficient with Displacement Variance of Energetic Particles Caused by Adiabatic Focusing

The equation κzz = d 2/(2dt) describing the relation of the parallel diffusion coefficient κzz with the displacement variance 2 (hereafter DCDV) is a well-known formula. In this study, we find that DCDV is only applicable to two kinds of transport equations of the isotropic distribution function, one without cross-terms and the other without a convection term. Here, by employing the more general transport equation, i.e., the variable coefficient differential equation derived from the Fokker-Planck equation, a new equation of κzz as a function of 2 is obtained. We find that DCDV is the special case of the new equation. In addition, another equation of κzz as a function of 2 corresponding to the telegraph equation is also investigated preliminarily.