  We present the first holographic simulations of non-equilibrium steady state
formation in strongly coupled $\mathcal{N}=4$ SYM theory in 3+1 dimensions. We
initially join together two thermal baths at different temperatures and
chemical potentials and compare the subsequent evolution of the combined system
to analytic solutions of the corresponding Riemann problem and to numeric
solutions of ideal and viscous hydrodynamics. The time evolution of the energy
density that we obtain holographically is consistent with the combination of a
shock and a rarefaction wave: A shock wave moves towards the cold bath, and a
smooth broadening wave towards the hot bath. Between the two waves emerges a
steady state with constant temperature and flow velocity, both of which are
accurately described by a shock+rarefaction wave solution of the Riemann
problem. In the steady state region, a smooth crossover develops between two
regions of different charge density. This is reminiscent of a contact
discontinuity in the Riemann problem. We also obtain results for the
entanglement entropy of regions crossed by shock and rarefaction waves and find
both of them to closely follow the evolution of the energy density.

High Energy Physics - Theory;Strongly Correlated Electrons