  We study fermion number non-conservation (or chirality breaking) in Abelian
gauge theories at finite temperature. We consider the presence of a chemical
potential $\mu$ for the fermionic charge, and monitor its evolution with
real-time classical lattice simulations. This method accounts for short-scale
fluctuations not included in the usual effective magneto-hydrodynamics (MHD)
treatment. We observe a self-similar decay of the chemical potential,
accompanied by an inverse cascade process in the gauge field that leads to a
production of long-range helical magnetic fields. We also study the chiral
charge dynamics in the presence of an external magnetic field $B$, and extract
its decay rate $\Gamma_5 \equiv -{d\log \mu\over dt}$. We provide in this way a
new determination of the gauge coupling and magnetic field dependence of the
chiral rate, which exhibits a best fit scaling as $\Gamma_5 \propto
e^{11/2}B^2$. We confirm numerically the fluctuation-dissipation relation
between $\Gamma_5$ and $\Gamma_{\rm diff}$, the Chern-Simons diffusion rate,
which was obtained in a previous study. Remarkably, even though we are outside
the MHD range of validity, the dynamics observed are in qualitative agreement
with MHD predictions. The magnitude of the chiral/diffusion rate is however a
factor $\sim 10$ times larger than expected in MHD, signaling that we are in
reality exploring a different regime accounting for short scale fluctuations.
This discrepancy calls for a revision of the implications of fermion number and
chirality non-conservation in finite temperature Abelian gauge theories, though
not definite conclusion can be made at this point until hard-thermal-loops
(HTL) are included in the lattice simulations.
