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, ⟨μ^jx^k⟩. 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 (-2 t ) -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 ⟨μ^jx^k⟩. 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.