Lognormal magnetic-field amplification
= Lognormal magnetic-field amplification
{title2=$\ln(B/B_0)\sim N(3Dt,2Dt)$}
Set $x=\ln(B/B_0)$ and $f=BF$ in the <radial magnetic-field Fokker-Planck equation>. Then $f_t=Df_{xx}-3Df_x$, 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 $\sqrt{2Dt}$.