Lognormal magnetic-field amplification ($\ln (B/B_0)\sim N(3Dt,2Dt)$)

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 $\sqrt{2Dt}$.