We study the long term evolution of magnetic fields generated by a collisionless relativistic e ^ { + } e ^ { - } shock which is initially unmagnetized .
Our 2D particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock , particle distributions are close to isotropic , relativistic Maxwellians , and the magnetic turbulence is highly intermittent spatially .
The non-propagating magnetic fields in the turbulence form relatively isolated regions with transverse dimension \sim 10 - 20 skin depths .
These structures decay in amplitude , with little sign of downstream merging .
The fields start with magnetic energy density \sim ( 0.1 - 0.2 ) of the upstream kinetic energy within the shock transition , but rapid downstream decay drives the fields to much smaller values , below 10 ^ { -3 } of equipartition after \sim 10 ^ { 3 } skin depths .

In an attempt to construct a theory that follows field decay to these smaller values , we explore the hypothesis that the observed damping is a variant of Landau damping in an unmagnetized plasma .
The model is based on the small value of the downstream magnetic energy density , which suggests that particle orbits are only weakly perturbed from straight line motion , if the turbulence is homogeneous .
Using linear kinetic theory applied to electromagnetic fields in an isotropic , relativistic Maxwellian plasma , we find a simple analytic form for the damping rates , \gamma _ { k } , in two and three dimensions for small amplitude , subluminous electromagnetic fields .
We find that magnetic energy does damp due to phase mixing of current carrying particles as ( \omega _ { p } t ) ^ { - q } with q \sim 1 .
This overall decay compares well to that found in simulations , since it depends primarily on the longest wavelength modes , kc / \omega _ { p } \ll 1 .
However , the theory predicts overly rapid damping of short wavelength modes .
We speculate that magnetic trapping of a substantial fraction of the particles within the highly spatially intermittent downstream magnetic structures may be the origin of this discrepancy .
In addition , trapping may form the basis for MHD-like behavior , permitting a small fraction of the initial magnetic energy to persist for times much greater than have been followed in the simulations .

We briefly speculate on other physical processes , which depend on the presence of suprathermal particles , that may cause the generation of longer wavelength magnetic fields that create a magnetized plasma ( kr _ { Larmor } \ll 1 ) , in which the damping is not as fast .
However , absent such additional physical processes , 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 .
We also comment on the relevance of these results for relativistic shocks in electron-ion plasmas .