Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 75 / 3 / 4 / Solution

For $B_0>0$, change to logarithmic magnitude and its correctly transformed probability density function:Using ${\partial }_B=B^{-1}{\partial }_x$, the radial equation becomesThis is diffusion with constant drift. Set $y=x-3Dt$ and $g(y,t)=f(y+3Dt,t)$; then $g_t=D{g}_{yy}$. The initial Dirac delta transforms to $f(x,0)=\delta (x)$, so the heat-kernel solution givesDividing by $B$ yields the lognormal magnetic-field amplification lawExpanding the square and using $B=B_0{e}^x$ gives the equivalent requested formThe logarithm is a Gaussian random variable with mean $3Dt$ and variance $2Dt$, which proves normalization and spreading directly. The exponential moments of that Gaussian distribution give $⟨B^q⟩=B_0^q{e}^{Dq(q+3)t}$, independently checking the energy growth. The limit at $t\downarrow 0$ is the initial Dirac delta distribution, not a pointwise finite function.