Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 75 / 3 / 2 / Solution

For isotropic initial statistics, isotropic noise preserves $P=P(B,t)$. Isotropy of the noise alone does not turn an arbitrary anisotropic initial vector distribution into an isotropic one; alternatively take the angularly averaged magnitude distribution. The radial gradient satisfies ${\partial }_{j}P=P_B{B}_j/B$, so the diffusion-vector contraction in the preceding equation is $B{B}_i{P}_B/2$. Its divergence givesNow $F=4\pi B^{2}P$ is the magnitude probability density function, normalized by ${\int }_0^{\infty }FdB=1$. Sincewe obtain the radial magnetic-field Fokker-Planck equationThe outward Fokker-Planck probability current is $J=-D(B^2{F}_B-2BF)$. Zero probability flux at zero and infinity preserves normalization. For positive initial magnitude the solution below remains on $B>0$; an atom initially at $B=0$ instead remains an absorbing zero-field component.