Non-equilibrium steady state formation in 3+1 dimensions

Dear editor and referees,

We would like to thank the editor for suggesting a minor revision and the referees for the excellent and detailed comments on our manuscript.
In our opinion, all points raised are valid and we are happy to address each of them which will significantly improve our manuscript.

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

The following is a list of the points raised by the referees (e.g. r3.1 is request 1 by referee 3 etc.) and how we address them (marked by *) in the revised manuscript.

r3.1) add general references on Riemann problem for hyperbolic equations, and some explanations for choosing the correct physical solution
*) We added a reference to the book chapter by Bressan on page 4.
We also expanded the discussion of the shock-rarefaction wave solution on page 8, emphasizing that this solution is everywhere consistent with the second law of thermodynamics.
We added further references that noted this already some time ago in the context of relativistic Rankine-Hugoniot equations.
(See also point r3.7 below.)

r3.2) add discussion of viscous hydro in early sections, in the context of hydrodynamics, and connect with that made in the context of holography and in the analysis of the results
*) Our original presentation of viscous corrections was indeed very schematic.
We now added the constituent equations when referring to the relevant figure (now figure 11) and hope this clarifies.
The viscous term referred on the extremal surface equation has nothing to do with viscosity and we removed this reference for clarity.

r3.3) discuss how solution to Riemann problem presented in Sect 2 compares with previous literature and general textbook material
*) This is addressed under point r3.1 above.

r3.4) Clarify Figs 6 and Fig 8
*) As a clarification, we added in the caption of Fig.6 the times for which the hydro curves are shown (at t = 20, 40, and 60).
The ideal hydrodynamic results do depend on the initial conditions and are in that sense merely an illustration.
It is also for this reason that we choose not to show these in Fig. 8.
The shock-shock is effectively shown by the dashed lines already, since this solution is a simple (two-)step function.

r3.5) Clarify the "ideal hydro" solution and its relation with the holographic calculation
*) We realise that our presentation of how we construction our ideal hydrodynamic solutions was not entirely clear;
we now added " such that the evolution resembles the holographic result at $t=20$" to clarify that we did not choose initial conditions equivalent to the holographic calculation (otherwise we would not have spelled out eqn (67)), since these would form a true shock too quickly to be shown.
In that sense, these curves are a bit arbitrary, but we still think they are important enough to show how (simple) ideal hydrodynamics behaves in comparison.

r3.6) It may also be worth emphasizing that the differences between shock+rarefaction and holographic is mainly due to the different (smooth) initial condition in the holographic numerics. Clarify early enough in the discussion the origin of the bulk of the difference between shock+rarefaction and holographic solution
*) Indeed the question if the holographic solution at later times depends on the details of the initial profile of the interface initial is very important, but since this is not the case (verified in Appendix B) we decided not to put this in the main text.
After presenting the solution we discuss in detail the differences between the steady-state energy densities of the shock-rarefaction wave and the holographic solution at intermediate and late times.
It would be interesting if we could say more about the t to infinity limit and if the holographic solution would limit to the shock-rarefaction solution, but unfortunately, our numerics do not allow such a statement at the moment.

r2.1) I think that when the second law of thermodynamics is discussed and its violations, it is important to state that this is the local version of the second law. The closest thing to a proof of this statement exists in 1612.07705, which is based on the Schwinger-Keldysh effective field theory of hydrodynamics developed in 1305.3670, 1405.3967, 1511.03646, 1511.07809, 1701.07436 and numerous other papers. My suggestion for the authors is to extend the discussion of this and comment further on the meaning of this breaking of the local second law in some cases and how this is restored in the full UV-complete simulation. This is naturally a very interesting aspect of their work and shows how important it is go beyond hydrodynamics. It would be nice to discuss and stress this further and well as discuss in greater depth what this law really means and why it is sometimes violated.
*) For the hydrodynamics case, we refer to our answer to point r3.1 above.
We also cited the paper 1612.07705, and the other papers it is based on and that were mentioned by the referee, below equation (16), and clarified the local nature of the violation of the positivity theorem for the entropy current.
We discuss in Fig.14 the local entropy production from the holographic computation (see also Fig.15), and we now added an extra sentence "As expected from 1612.07705 the divergence of the local entropy current is however everywhere positive." to bring this more into context.
As far as the full solution is concerned, we added a paragraph on page 18 emphasizing the consistency of our holographic result with the second law of thermodynamics.

r2.2) Regarding the second law, in the introduction, it says that the authors ‘confirm that the double-shock solution violates the second law of thermodynamics also in 3+1 dimensions’.
What does it mean that they are confirming it?
Was it predicted somewhere?
In that case, they could add a reference.
I suppose they are confirming a general expectation that similar things should happen in all dimensions.
*) Indeed, the original wording was misleading.
We rephrased the corresponding statement and added references that discuss the issues with the double-shock solution.

r2.3) The authors use words ‘analytic’ and ‘numeric’ several times.
Should it not be ‘analytical’ and ‘numerical’, especially when it comes to the latter?
*) We changed all appearances of ‘analytic’ and ‘numeric’ by ‘analytical’ and ‘numerical’ wherever this was grammatically correct.

r2.4) Above eq. (17) I think that it should be not ‘monotonous’ but ‘monotonic’ .
*) We changed ‘monotonous’ to ‘monotonic’.

r2.5) Below eq. (27), what does it mean that a ‘similar expression’ can be derived from the Rankine-Hugoniot condition.
Why similar and not the same?
*) What we mean is that expressions (27) and (29) do not have the same form, but must, of course, evaluate to the same value.
Demanding equality of (27) and (27) fixes in this case the flow velocity in the NESS region.
We changed ‘similar expression’ to ‘analogous expression’ and hope the sentence is sufficiently clear now.

r1.1) Do the authors have any idea or comment about the cause of the difference between the contact discontinuity for the charge density in the Riemann problem and the smooth interpolating configuration ("crossover region") they find in the gravity solution?
Of course one cannot expect to see a sharp discontinuity in a numerical analysis starting from a smooth initial condition, but the behavior of the interpolating configuration does not seem to evolve to a discontinuity-like form, even at late time.
*) This is an excellent point that can be understood using the old Appendix B.
In response to the point, we moved the diffusion part to the main text (now Fig 9) and slightly adapted the text.
This should make it clear that in rescaled coordinates the full solution truly becomes a contact discontinuity on a timescale of the square root of t.

r1.2) Would an analysis as the one in figure 12, but for the "crossover region", have been helpful in understanding its behavior?
Maybe it would have suggested a possible analytic form.
*) This is another excellent point, and we think this analytical form can be seen from the Fig 9 as introduced in the previous point.

r1.3) Actually, in the original coordinates also the "shock wave" seems to be different from the one of the Riemann problem, since it expands with time.
It is only in the z/t coordinates that it seems to evolve to a shock wave at late times.
Why should we expect to see this behavior only in the z/t coordinates?
*) Fig.13 provides numerical evidence that in the holographic calculation the width of the wave that moves towards the cold side saturates to some finite value.
The holographic calculation can be seen as a resummation of all higher-order gradient corrections of an effective hydrodynamic description of the problem.
It therefore also includes viscous effects that prevent the formation of sharp shocks.
The reason the holographic evolution converges to a sharp shock in the z/t coordinate is that the approximately constant width divided by time approaches zero at late times.

r1.4) The backreaction of the charge on the metric has been ignored.
Is it correct to state that this approximation does not allow for a check that the two regions of different charge density within the NESS do not correspond to two different energy densities?
A-priori the metric could be influenced by these two charges and the energy density could be different.
*) The backreaction of the gauge field to the metric was neglected to simplify the numerical analysis.
When the backreaction is included, we indeed expect that the charge density will leave a nontrivial imprint on the energy density in the NESS region.
We plan to study this in future work.

r1.5) At page 6, 3 lines after formula (12), it is stated that "For larger values of $\chi$ the charge density becomes non-monotonic:", but from figure 1 the non-monotonicity seems to appear for *smaller* values of $\chi$, doesn't it?
*) This was indeed a mistake.
We changed "larger" to "smaller".

r1.6) In the formulas (38) and (42) for the temperature I believe there is a sign mistake, otherwise the chargeless limit of the solution would result in a negative temperature.
*) We are very grateful that the referee brought these mistakes to our attention.
There were actually several errors in (38) and (42) which we have corrected in the revised version.

r1.7) I would have liked to see a citation to some literature in appendix A.
*) Several references were added.

r1.8) In the last line of the caption of figure 2 "hot" should be written as "Hot".
*) Replaced "hot" by "Hot".

r1.9) I would have liked to read the last sentence of the paragraph after formula (43) somewhere around formula (41): it would have been immediately useful.
*) We moved this sentence directly after (41).

r1.10) In the sentence starting 4 lines after formula (51) there is a "are" too much.
*) Removed superfluous "are".

r1.11) I believe the "red" in the 5th line of the caption of figure 9 should be a "blue".
*) We changed "red" to "blue" in the caption of Fig.10, previously Fig.9.

r1.12) In formula (73) the is a dot missing.
*) We replaced the comma by a dot in (76), previous (73).

r1.13) Formula (74) is the inverse of the wordy definition in the 2 lines above it, right?
*) We thank the referee for finding this, we now changed "ratio of the renormalised entanglement entropy and the total energy inside the entangling region" to "ratio of the total energy and the renormalised entanglement entropy inside the entangling region"

r1.14) Formula (77) is lacking a "dx" on the r.h.s.
*) We added "dx" in (80), previously (77).