  We assess the impact of non-thermally shock-accelerated particles on the
magnetohydrodynamic (MHD) jump conditions of relativistic shocks. The adiabatic
constant is calculated directly from first principle particle-in-cell
simulation data, enabling a semi-kinetic approach to improve the standard fluid
model and allowing for an identification of the key parameters that define the
shock structure. We find that the evolving upstream parameters have a stronger
impact than the corrections due to non-thermal particles. We find that the
decrease of the upstream bulk speed yields deviations from the standard MHD
model up to 10%. Furthermore, we obtain a quantitative definition of the shock
transition region from our analysis. For Weibel-mediated shocks the inclusion
of a magnetic field in the MHD conservation equations is addressed for the
first time.
