Relating $\hat{q}$, $\eta/s$ and $\Delta E$ in an expanding Quark-Gluon Plasma

We use linear viscous hydrodynamics to describe the energy and momentum deposited by a fast moving parton in a quark gluon plasma. This energy-momentum is in turn used to compute the probability density for the production of soft partons by means of the Cooper-Frye formula. We use this probability density to render manifest a relation between the average transverse momentum given to the fast moving parton from the medium $\hat{q}$, the entropy density to shear viscosity ratio $\eta/s$ and the energy lost by the fast moving parton $\Delta E$ in an expanding medium under similar conditions to those generated in nucleus-nucleus collisions at the LHC. We find that $\hat{q}$ increases linearly with $\Delta E$ for both trigger and away side partons that have been produced throughout the medium. On the other hand, $\eta/s$ is more stable with $\Delta E$. We also study how these transport coefficients vary with the geometrical location of the hard scattering that produces the fast moving partons. The behavior of $\hat{q}$, with $\Delta E$ is understood as arising from the length of medium the parton traverses from the point where it is produced. However, since $\eta/s$ is proportional to the ratio of the length of medium traversed by the fast parton and the average number of scatterings it experiences, it has a milder dependence on the energy it loses. This study represents a tool to obtain a direct connection between transport coefficients and the description of in-medium energy loss within a linear viscous hydrodynamical evolution of the bulk.