Review Reports

Round 1

Reviewer 1 Report

This work presents a proposal of a measure for the violation of time-reversal-symmetry, and a procedure in which this measure is used to control time-reversal-asymmetry. The procedure may imitate the effects of Lévy noise. Simple corrections on the Fluctuation-Dissipation Relation are obtained.

A remarkably clear and nice work, I strongly recommend this paper to be published.

Author Response

The Reviewer is full of praise for our work. The take his words as an affirmation that we did a good & thorough job scientifically and that we wrote an interesting & intelligible manuscript.

Reviewer 2 Report

I have now read the submitted manuscript, I find it interesting and certainly has potential. There are a number of possible improvements that may be made first, these are listed below.

1. Is there a particular reason that in the relation \sigma and \delta in Figure 3 is not starting at 0? What is the source of this. The claim is basically that this relation works? This would be only working in some "average" sense?
2. The number of steps ascending and descending should be discussed further, in the Levy case since this would generate a mismatch?
3. Why noty investigate the case of Cauchy-distribution where a closed form of the PDF can be found, and compute more analytical relations in particular then using the parabolic well.
4. The expressions are for \Delta x and \Delta E however the arc length deviates from these expressions possibly on the same scale as the delta? How do we know? In some sense how close to equilibrium is needed?
5.  

Author Response

Point 1). The points are due to stochastic simulations. That the depicted best-fitting straight line does not exactly hit the (0,1) point is to be expected. Ultimately, the points follow the theoretical prediction very well.

Points 2) and 3). The Reviewer refers to the "mismatch" between ascending and descending steps. This is the "time-reversal asymmetry" that is in the title of Section 3. This asymmetry is also displayed in Figure 1b. The subject is kind of novel. We published thorough articles about it in Physical Review E in 2021 (Reference 23 of this manuscript) and in Europhysics Letters in 2016 (Reference 22 of the manuscript). We have added a sentence in Section 3 referring to the Cauchy distribution. The Reviewer mentions the Cauchy distribution in his 3rd point and suggests a more detailed consideration of this case. The Cauchy distribution is indeed in many ways the analytically easiest α-stable distribution. For a Cauchy distribution a more explicit derivation is possible and we have a section about that in our 2016 Europhysics Letters publication. We now mention this in Section 3 in the revised current manuscript.

4). The arc length along the parabola is not really relevant in our context. We consider a system (cf. Fig. 1a and Fig. 2) where the deterministic force that is driving the particle to x=0 is proportional to the |x|, i.e. the distance to x=0. This leads to a potential V(x)=A x². The particle does, of course, not move on a parabolic surface. It moves on the x-axis and V(x) represents a potential energy that can be associated with the particle's dynamics.

Round 2

Reviewer 2 Report

I think that most questioned have been answered however some more clarity to avoid misunderstanding would be appreciated.