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Set $x=\ln (B/B_0)$ and $f=BF$ in the radial magnetic-field Fokker-Planck equation. Then $f_t=D{f}_{xx}-3D{f}_x$, so $f$ is a Gaussian distribution drifting at speed $3D$. For initial magnitude $B_0>0$, $F=[B\sqrt{4\pi Dt}{]}^{-1}\exp [-(\ln (B/B_0)-3Dt{)}^2/(4Dt)]$. This lognormal distribution has moments $B_0^q{e}^{Dq(q+3)t}$; the width of its logarithm grows as $\sqrt{2Dt}$.

Twice integrating the radial magnetic-field Fokker-Planck equation by parts gives the displayed law when moments and endpoint terms are controlled. In particular $⟨B^2⟩$ grows at rate $10D=2\gamma$, with $\gamma =5{\kappa }_2/4$. The corresponding magnetic energy density is $⟨B^2⟩/(8\pi )$ in Gaussian units.

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.