Exact solution of the Fokker-Planck equation for isotropic scattering

The Fokker-Planck (FP) equation partial derivative(t)f + mu partial derivative(x)f = partial derivative(mu)(1 - mu(2))partial derivative(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 root < x(2)>(0) being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, < x(2)> - < x(2)>(0) approximate to t(2)/3 to a time-asymptotic, diffusive phase, < x(2)> - < x(2)>(0) approximate to 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.