Diffusion coefficient with displacement variance of energetic particles with adiabatic focusing

The equation $\kappa_{zz}=d\sigma^2/(2dt)$ (hereafter DCDV) is a well-known formula of energetic particles describing the relation of parallel diffusion coefficient $\kappa_{zz}$ with the parallel displacement variance $\sigma^2$. In this study, we find that DCDV is only applicable to two kinds of transport equations of isotropic distribution function, one is without cross terms, the other is without 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 $\kappa_{zz}$ as a function of $\sigma^2$ is obtained. We find that DCDV is the special case of the new equation. In addition, another equation of $\kappa_{zz}$ as a function of $\sigma^2$ corresponding to the telegraph equation is also investigated preliminarily.