# Non-equilibrium steady state formation in 3+1 dimensions

### Submission summary

 As Contributors: Christian Ecker Preprint link: scipost_202107_00020v2 Date accepted: 2021-08-16 Date submitted: 2021-08-02 11:20 Submitted by: Ecker, Christian Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Computational Fluid Dynamics High-Energy Physics - Theory Approach: Theoretical

### Abstract

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.

Published as SciPost Phys. 11, 047 (2021)

Dear editor and referees,

We would like to thank the referees for suggesting the publication of our work and for pointing us to some additional references, which we were happy to add to our manuscript.

Best regards,
Christian Ecker, Johanna Erdmenger, Wilke van der Schee

### List of changes

Added a remark and three of the suggested references in the second bracket of the following sentence in the paragraph above Fig.2 on p.7:

"Recently, [29] (based on [30–34]) managed to prove the local second law of thermodynamics (see also [35–37] for a statistical physics point of view) under the very general assumptions of unitarity ..."