The chiral anomaly present in the standard model can have important phenomenological consequences, especially in cosmology and heavy-ion physics. In this talk, I will focus on the contribution from the Abelian gauge fields. Despite an absence of topologically distinct sectors, they have a surprisingly rich vacuum dynamics, partly because of the chiral anomaly. I will present results obtained from real-time classical lattice simulations of a U(1) gauge field in the presence of a chiral chemical potential. This simulations account for short distance fluctuations, contrary to effective descriptions such as Magneto-Hydrodynamics (MHD). I will discuss various phenomena, such as inverse magnetic cascade, which occur in this system. In particular, in the presence of a background magnetic field, the chemical potential exponentially decays. The associated chiral decay rate is related to the diffusion of the Abelian Chern-Simons number in a magnetic background, in the absence of chemical potential. The value obtained from the simulations is an order of magnitude larger than the one predicted by MHD. If this result is shown to be robust under corrections such as Hard Thermal Loops, it will call for a revision of the implications of fermion number and chiral number non-conservation in Abelian theory at finite temperature.