Activity driven transport in harmonic chains

In this paper, the authors study a harmonic chain coupled to
an active bath. More precisely, the system is subjected at its two extremities,
i.e. at site $j=1$ and site $j=N$, to a noise which has a non-zero persistence
time, respectively $\tau_1$ and $\tau_N$ (and thus does not satisfy the fluctuation-dissipation
theorem), in addition to a Gaussian white noise (which satisfies the fluctuation-dissipation theorem).
There is no other source of noise in the system. They show analytically
that the system reaches a nonequilibrium steady state, with a nonzero energy current. The main
focus of the paper is on the characterisation of this current, which exhibits two interesting features:
(i) a non-monotonic behavior as a function of the persistence times $\tau_1$ and $\tau_N$ (which they call
``negative differential conductivity'') and (ii) a change of sign again as $\tau_1$ and $\tau_N$ are varied (which
they call ``current reversal''). These two effects do not have any counterpart in the case where the
two baths are passive, i.e., when the noise at the two extremities is purely thermal, a case that was studied
a long time ago by Rieder, Lieb and Lebowitz.

The paper is physically sound and presents one rare example where the non-equilibrum dynamics
of an extended system coupled to an active bath can be carried out analytically. As far as I know, these
results are new. Most of them are also carefully compared to numerical simulations, showing a very good agreement.
The organisation of the paper as well as the presentation of the results is rather clear -- although the English wording still requires some work
(see below) -- while technical details have been relegated to three Appendices.

Therefore, I would like to recommend the publication of the present manuscript to SciPost. The authors might
however consider the following comments:

i) Below Eq. (2): please indicate that the indices $j$ and $l$ take only the values $j,l =1,N$.

ii) In Eq. (5) the authors mention both the thermal and active currents, but eventually give only the active one. I think
that they should at least refer to Eq. (11) where the thermal current is given.

iii) In the caption of Fig. 2, the authors should describe what $\tau_m$ is (and maybe define this notation also in subsection 3.1.1)

iv) Above Eq. (13) the authors mention the tri-diagonal structure of the matrix $G(\omega)$. They should at least refer to Eq. (35) in
Appendix A. In fact, since this matrix plays an important role in the computation, I would suggest to give it in the text earlier -- maybe around
Eq. (8).

v) In Eq. (13), recall that $j=1$ or $j=N$.

vi) In the caption of Fig. 4, as well as in Section 3.1.2, the notation $(\bar{\tau}, \bar{\tau})$ is a bit strange: maybe choose a better one?

vii) On p. 8, below Eq. (15): what do the authors mean by "... numerical simulations performed with Eq. (3) in Fig. 5(a)"? They should clarify
their statement here.

viii) In Appendix A, below Eq. (23) the authors use the fact that the matrix $G(\omega)$ is symmetric but this matrix is given only later in the paper, in Eq. (35).
As mentioned above, I think that this matrix should be given much earlier (and in the text, not only in the Appendix).

ix) In Eq. (25) there is a misprint: $H_R$ should be $S_R$.

x) Below Eq. (31): "The fist term on the second line..." --> "The first term in the integrand on the second line"... Without this precision, the sentence sounds a bit odd.

xi) In Appendix A, below Eq. (41): "given by" has nothing to do here. It should be deleted.

xii) Above Eq. (44) the authors could give a hint or a reference on how to evaluate this integral.

xiii) At the end of Appendix C, the authors write that the linear response is purely "frenetic". They should at least explain a bit what is meant
by "frenetic" here since here this comes completely out of the blue.

Minor points
-------------------

As mentioned above, the paper needs some polishing. Here is a list (surely non-exhaustive) of instances where the wording needs
to be corrected:

1) p .2: "on study of simple..." --> "on THE study of simple..."

2) p. 2: "introducing disorders" --> "introducing disorder"

3) p. 2: "lies with the Lorentzian.." --> "lies IN the Lorentzian.."

4) p. 2: "a uniform value at the bulk" --> "a uniform value IN the bulk"

5) p. 3, below Eq. (3): "as such exponential correlations" --> "SINCE exponential correlations"

6) On top of p. 4: "the average is over the NESS" --> "the average is taken in the NESS"

7) Below Eq. (6): "This suggests the possibility of interpreting..." --> "This would suggest..." since eventually, this interpretation is seemingly wrong.

8) Below Eq. (6): "different than thermal ones" --> "different FROM thermal ones". This mistake also appears in some other places in the paper.

9) On p. 5 below Eq. (11): "remains same" --> "remains THE same"

10) On p. 5: "is obtained exploiting" --> "is obtained BY exploiting"

11) On p. 8: above Eq. (15): "the the kinetic" --> "the kinetic"

12) On p. 8: "remain different than" --> "remainS different FROM"

13) On p. 8: "different than the energy" --> "different FROM the energy"

14) On p. 9, the expression for $J_{\rm act}$ in the passive limit does not have any number, please add one.

15) Below that un-numbered equation: "which is same" --> "which is THE same"

16) On p. 9, in the conclusion: "leads to an NESS" --> "leads to A NESS"

17) In Appendix A, Eq. (22) please replace the full stop by a coma.

18) Similarly, in Appendix A, Eq. (25) please replace the full stop by a coma.

19) In Appendix B, the first pair of equations is numbered as (51) and (52) while in many other places, they used (33a) and (33b) or (36a) and (36b). Pease
uniformise the notations.

20) Below Eq. (52): "at the bulk" --> "IN the bulk"

- novel model of reservoirs
- clearly written

The manuscript deals with a model of transport in a linear
chain driven by nonequilibrium (thermal and
nonthermal) energy baths. The main novelty of the work is the introduction
of active reservoirs, an issue that has not been considered before in
this field. The authors demonstrate two effects
(i) negative differential conductivity (NDC), and (ii) current direction
reversal at some finite value of the activity drive.
Although (i) has been reported in many examples with standard thermal
baths, (ii) is a novel effect characteristic of the non-equilibrium
nature of the reservoirs. I think the work may open a novel
research subfield (for instance the effect of acive baths in
nonlinear systems).

The results are obtained via the well-known approach of nonequilibrium
Green functions, a method that, with little extension of the calculation
(namely addition of the active forces in eq. (9)) allows the authors to
derive expression of the main observables (energy current and temperature
field).

The main idea of the paper is interesting and original. The methods
are sound. The presentation
is very accessible and self-contained (the Authors wisely condensed the technical
details in the appendix, leaving only the results in the main text).

I would recommend publication after the following issues have been
considered and the text revised accordingly.

1- I do not quite agree with the sentence "Equation (12) is a Landauer-like
formula where the transmission coefficient depends on ...".
The transmission coefficient is an intrisic property of the chain (and the
boundary conditions). What is different here ar just the source terms
g(tau,omega). As a related remark, after eq. (12): upon comparison with the Landauer formalua,
"phonon spectrum $\omega^2 G_{1N}(\omega)$" should rather be
"transmission coefficient $\omega^2 |G_{1N}(\omega)|^2$" (notice the square missing)

2- same in the caption of Figure 3: I would not call the curves
a plot of the "phonon band" but the "transmission coefficient".

3- to demonstrate NDC Authors use an approach based on eq.(14) and
comment that "when the number ... is positively correlated with the
current an NDC emerges". However this is not very explicative: how can
we understand when this happens in (14)? this point need some
clarification (perhaps upon extending the comments at the end of
appendix C?).