  We study the long term evolution of magnetic fields generated by an initially
unmagnetized collisionless relativistic $e^+e^-$ shock. Our 2D particle-in-cell
numerical simulations show that downstream of such a Weibel-mediated shock,
particle distributions are approximately isotropic, relativistic Maxwellians,
and the magnetic turbulence is highly intermittent spatially, nonpropagating,
and decaying. Using linear kinetic theory, we find a simple analytic form for
these damping rates. Our theory predicts that overall magnetic energy decays
like $(\omega_p t)^{-q}$ with $q \sim 1$, which compares favorably with
simulations, but predicts overly rapid damping of short wavelength modes.
Magnetic trapping of particles within the magnetic structures may be the origin
of this discrepancy. We conclude that initially unmagnetized relativistic
shocks in electron-positron plasmas are unable to form persistent downstream
magnetic fields. These results put interesting constraints on synchrotron
models for the prompt and afterglow emission from GRBs.
