Abstract
We present the first holographic simulations of non-equilibrium steady state formation in strongly coupled \mathcal{N}=4𝒩=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 analytical solutions of the corresponding Riemann problem and to numerical 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.
Highlights
One might ask if these Non-Equilibrium Steady State (NESS) are a curiosity of integrable CFT2 or if they exist in more general theories and dimensions higher than two? This question was addressed in several studies by constructing solutions of the Riemann problem in relativistic hydrodynamics
[15, 19,20,21,22], in holographic CFT3 [23], in theories with gravity duals in the limit of large number of dimensions [24] and in non-relativistic theories with Lifshitz scale symmetry [25]. This led to the insight that the formation of NESSs does not rely on conformal symmetry or integrability, but rather is a universal feature of the hydrodynamic description of any fluid, independent of the underlying equation of state
Two regions with constant but different charge densities emerge inside the NESS region, which indicates the formation of a contact discontinuity
Summary
Describing the far-from-equilibrium dynamics of strongly coupled quantum systems is extremely challenging. This question was addressed in several studies by constructing solutions of the Riemann problem in relativistic hydrodynamics [15, 19,20,21,22], in holographic CFT3 [23], in theories with gravity duals in the limit of large number of dimensions [24] and in non-relativistic theories with Lifshitz scale symmetry [25] This led to the insight that the formation of NESSs does not rely on conformal symmetry or integrability, but rather is a universal feature of the hydrodynamic description of any fluid, independent of the underlying equation of state. In two appendices we derive the Rankine-Hugoniot jump conditions and provide numerical evidence that our results are independent on how we approximate the initial interface of the Riemann problem on the gravity side
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