Transport through interacting defects and lack of thermalisation

Dear Editor,

we thank you for your time in considering our submitted manuscript "Transport through interacting defects and lack of thermalisation" for publication in Scipost Physics. In her/his report, the referee raised some points which helped us in further clarifying our exposition: we are thankful for the useful input. Hereafter we address in detail the questions of the referee and the consequent improvements on the manuscript. We hope the revised version of our work can satisfy the Referee's request and meet the publication criteria of Scipost Physics.

Detailed comments:

1) Referee: "This is a nice and original work on a classical 1-d model written as if it were a paper on a quantum model, which makes reading it quite confusing. It considers a classical system of quadratic oscillators with a non-linear potential confined in a region of size L."

1) Answer: We apologize for the misunderstanding, of course it was not our intention to confuse the reader. As stated already in the previous version of the manuscript, we focus on the classical setup to have access to large-scale numerical simulations which are beyond the reach of quantum numerical methods. Nonetheless, the microscopic mechanism at the origin of the boundary generalized resistance is explained in terms of simple kinetic arguments which are quantitatively captured by the Boltzmann approach (which can be formulated within the quantum framework as well). Since we look at the classical model just as a convenient tool to check our general picture and the same mechanism is expected to hold in the quantum scenario, we did not sufficiently advertise the use of the classical setup. In the revised version of the manuscript, we repair this mistake through the following changes

• In the abstract, we replace "quasiparticles"-> "carriers". While we think the concept of quasiparticle can be used in the classical scenario as well, we agree the reader could be induced to believe we are focusing mostly on quantum models. For the same reason, in the abstract we now explicitly added "Focusing on classical systems, we study..."

-We explicitly state in the introduction that we focus on the classical scenario. Nevertheless, let us point us that classical physics has been an irreplaceable tool to assess the validity of non-equilibrium scenarios also in connection with integrability, as shown for example by Refs. [54-57] of the revised manuscript.

-We changed the title of Section 2 to make the use of a classical model explicit.

• In Eq.(1), \psi^\dagger -> \psi^. The use of \psi^\dagger instead of \psi^ was a typo, indeed it never appeared again in the manuscript.

• Below Eq. 3, we removed the explicit mention to the Wigner distribution. Even though the local phase-space distribution of the classical model is defined similarly to the quantum Wigner distribution, we agree it could be confusing for the reader.

2) Referee: "Another point: while in the quantum transport set-up driven-dissipative boundaries (which lead to the Lindblad equation) are used to mimic the leads attached to a mesoscopic object (in the markovian limit etc.), in classical mechanics I am not sure this can be easily justified nor I have seen it used before (no reference is quoted here, so I assume this is an invention of the authors). The authors should justify this set-up more convincingly."

2) Answer: To the best of our knowledge, we are not aware of the use of this strategy within the classical scenario and we could not find a proper reference. This is also the main motivation behind appendix A, which provides a detailed account of how the strategy works. In the revised version of the manuscript, we summarize in a few words the general idea before digging into the details: essentially, outgoing carriers flowing out of the defect are removed from the system thanks to the dissipation. On the other hand, the tunable drive allows injecting an excitation distribution of our choice. The effect of the boundaries is then captured by Eq. 16: numerically solving for the stationary solution \partial_t n=0 allows for a determination of the injected mode density as a function of the tunable drive s(k). This allows for an exact characterization of the injected mode density in terms of the boundary noise: for example, in Fig. 3 the injected mode density (k>0) of the microscopic simulations (symbols) is exactly tuned to the target value (solid curve) obtained by solving Eq. 16.