Exact solution of the Fokker-Planck equation for isotropic scattering

The Fokker-Planck (FP) equation ∂tf+μ∂xf=∂μ(1−μ2)∂μf is solved analytically. Foremost among its applications, this equation describes the propagation of energetic particles through a scattering medium (in x- direction, with μ being the x- projection of particle velocity). The solution is found in terms of an infinite series of mixed moments of particle distribution, ⟨μjxk⟩. The second moment ⟨x2⟩ (j=0, k=2) was obtained by G. I. Taylor (1920) in his classical study of random walk: ⟨x2⟩=⟨x2⟩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 ⟨x2⟩0 being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, ⟨x2⟩−⟨x2⟩0≈t2/3 to a time-asymptotic, diffusive phase, ⟨x2⟩−⟨x2⟩0≈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 ⟨μjxk⟩. An explicit, easy-to-use approximation for a point source spreading of a pitch-angle averaged distribution f0(x,t) (starting from f0(x,0)=δ(x), i.e., Green’s function), is also presented and verified by a numerical integration of the FP equation.