The Fokker-Planck (FP) equation ${\partial }_{t}f+\mu {\partial }_{x}f={\partial }_{\mu }(1-{\mu }^2){\partial }_{\mu }f$ is solved analytically. Foremost among its applications, this equation describes the propagation of energetic particles through a scattering medium (in $x$- direction, with $\mu$ being the $x$- projection of particle velocity). The solution is found in terms of an infinite series of mixed moments of particle distribution, $⟨{\mu }^j{x}^k⟩$. The second moment $⟨x^2⟩$ ($j=0$, $k=2$) was obtained by G. I. Taylor (1920) in his classical study of random walk: $⟨x^2⟩={⟨x^2⟩}_0+t/3+[\exp (-2t)-1]/6$ (where $t$ is given in units of an average time between collisions). It characterizes a spatial dispersion of a particle cloud released at $t=0$, with $\sqrt{{⟨x^2⟩}_0}$ being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, $⟨x^2⟩-{⟨x^2⟩}_0\approx t^2/3$ to a time-asymptotic, diffusive phase, $⟨x^2⟩-{⟨x^2⟩}_0\approx t/3$. The present paper provides all the higher moments by a recurrence formula. The full set of moments is equivalent to the full solution of the FP equation, expressed in form of an infinite series in moments $⟨{\mu }^j{x}^k⟩$. An explicit, easy-to-use approximation for a point source spreading of a pitch-angle averaged distribution $f_0(x,t)$ (starting from $f_0(x,0)=\delta (x)$, i.e., Green’s function), is also presented and verified by a numerical integration of the FP equation.

• Received 6 October 2016

DOI:https://doi.org/10.1103/PhysRevD.95.023007