Spectral Properties of Effective Dynamics from Conditional Expectations

Round 1

Reviewer 1 Report

The authors have fully addressed my concerns, and the article is now in my opinion significantly improved

Reviewer 2 Report

The authors have strongly modified their manuscripts with respect to the one I reviewed earlier in 2020. This revised version of the manuscript has answered to my previous comments.

This paper is still hard to understand for applied physicists or chemists but I think that it is unavoidable for such theoretical work, but the topic of reduction of dimensionality is very important and needs to be adressed in such a way.

I am not able to judge the exactness of the mathematical developments and I hope another referee will be able to comment on these developments.

Reviewer 3 Report

Though in my view, the paper is still addressed to mathematicians rather than to a general readership, the authors have done their best to improve the manuscript. On the other hand, since improvement is a never-ending process, I believe it is high time to terminate it and to publish the paper as it stands. 

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The manuscript analyses an approach for approximating the dynamics of slowly varying modes in stochastic dynamics.  The material is rather interesting, but rather heterogeneous. The first part presents formal derivations of the properties of the reduced dynamics, with a language which would be appropriate for an applied math journal. I don't have the expertise to judge the formal correctness of the derivations, but the most important result (eq. 20 and 21, an error bound of the dominant generator eigenvalues), seem to me novel and in principle useful.

The second part of the manuscript describes numerical experiments aimed at verifying if a reduced dynamics whose coefficients are estimated using the Kramers–Moyal formulae, eq 22-23, is able to reproduce the implicit time scale of the system "Conjecture 1". The  answer to this question seems to be "no, except if the system is extremely simple": panel B of fig 5 and 6  show that for small lag parameter s the estimated time scales are pretty far off, and seem to approach the correct value only for large s. This is not by itself a surprise, since the Psi angle is not a good CV for the example of fig 5, and the system in fig 6 is intrinsically  high-dimensional. Moreover, estimating the coefficient by the KL formulas on a grid seems to me a rather naif approach compared to what other groups do (see for example https://aip.scitation.org/doi/10.1063/1.3058436). But, beside the quality of the results, what I found disappointing is that the examples  to do not seem much related with the first part of the manuscript (and are instead focused on conjecture 1, which seem to me of pretty limited practical and conceptual power, as shown by the examples). Why are the error bond in eq 20  and 21 not discussed, verified and demonstrated in practical examples? This in my opinion would have made a great manuscript.  

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, the authors explore the conservation of timescales of slow motions through the process of reduction of the dimension of the problem. They advocate the use of Kramers-Moyal (K-M) formulae to determine the drift and diffusion parameters for the low-dimensional simulator of the full model. They use numerical examples to provide evidences to their conjecture.

 

The subject of this paper is very interesting because retaining the dynamics of the more important slow motions in molecular systems is of great concern when one wants to lower the dimensionnality of the problem. I don’t have the background to follow all the mathematical developments of the authors (I hope another referee does) and will only comment on the numerical examples. The quality of the paper seems quite good to me and, because of the relevance of the subject, I think this work deserves publication in “Entropy” but needs major revisions to clarify the two points.

 

The authors present 4 numerical examples in their paper. The first one (Lemon slice potential) seems to indicate that the offset s in the K-M formulae should be taken small and that the results are much degraded for high values of s, as for example can be seen in Fig 2D. But in the 3 other examples, it appears that choosing a small value for s leads to worse results than choosing a large value. This seems to be especially true for the molecular simulation examples (Fig 5B and 5C or Fig 6B). I don’t understand why this is so. I think the authors should comment on that.

 

My other concern is for the last example for which the conclusion is not so clear to me. Indeed, looking at Fig 6B that presents implied time scales in the system, it seems that the highest s, the better the results. Values of s greater than 100 seem to be a good choice. But when we look at Fig 6D, we can see that the stationary distributions are better reproduced at small s, or at least that they begin to be not so good when s > 100. This seems contradictory to me. Maybe it is not and maybe it just points to the fact that we have to chose between retaining dynamics or stationary distribution when performing reduction in the number of dimensions. Anyway, I really think the authors should comment on that.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Multi-dimensional diffusion in external potential is an interesting and important topic to study. It is well-known that in the bounded domains the drift-and-diffusion operator has a discrete spectrum. Inverse eigenvalues of this operator have the meaning of the relaxation time of the system. In many important cases, there is a gap in the spectrum separating fast and slow modes.

It is often important - both for technical simplicity and for a better intuitive understanding of the problem in question - to map an initial multidimensional problem into space of macroscopic (collective, reaction, etc) variable with a smaller dimension. It is very important to understand whether, under what conditions, and to which extent such mapping preserves the edge of the relaxation spectrum, i.e., the set of largest relaxation times of the problem.

The manuscript by F. Nueske et al. tries to address this interesting topic. However, it has several important drawbacks.

First, the exact formulation of the problem the authors address is missing (as opposed to the statement that they are generally trying to elucidate the problem outlined above). Even in the conclusion, the authors formulate the results using imperfective verbs (we discussed, we analyzed...). The only hard result the authors mention is "relative error bound for dominant eigenvalues in terms of the H1m-approximation error of the corresponding eigenfunctions was proved" (as far as I understand, it is proposition 1 that they refer to, although it is not exactly clear). I am not sure I understand the practical importance of this new bound. I hoped the numerical examples would elucidate its use, but they did not.

Second, it seems that the authors made almost deliberate effort to make their text impenetrable for non-mathematicians. Judging by the authors' choice of notation and terminology (explaining the meaning of 'potential' and 'diffusion' but using sophisticated terminology of the functional analysis without any explanation) the authors aimed this paper to the audience of pure mathematicians. Meanwhile, to the best of my understanding Entropy is a multi-disciplinary journal. Its topics lie in physics, information theory, machine learning, chemistry, biology, economics, etc., but not in pure mathematics. I expect that a huge majority of its readers have no idea what Frobennius inner product or Sobolev spaces are, and it includes many people who can understand perfectly well the topic of multi-dimensional diffusion and might find the authors' results interesting and important.

Third, on top of that, there are numerous inconsistencies in notation, and other murky places, which make the manuscript extremely hard to follow and understand, for example:
- Q in the rhs of (17) is never defined, so it is difficult to understand what Galerkin approach actually does (not to mention how it does that)
- the authors keep mentioning the notion of "good set of reactive coordinates in the sense of Proposition 1", while Proposition 1 does not define any notion of a good reactive coordinate. It says "the relative error of eigenvalues is bound by this expression", without mentioning any classification of reactive coordinates into good and bad in terms of this bound or any other measure;
- the potentials used for examples are written in a very strange way: (i) what is the purpose of +0.05 in (30)?, (ii) why potential in (31) is centered at (2,2), not (0,0)?, (iii) why write -9 (x-2)^2 + 0.5 (x-2)^2 instead of simply -8.5 (x-2)^2?, (iv) what is exactly the potential the authors call alanine dipeptide is not at all clear - there is neither formula nor a picture nor a reasonably complete explanation in the text, ditto for deca alanine.  
- etc

Based on that I think the paper cannot be published as it stands. If the authors want to write for the multidisciplinary audience, they should fundamentally revise the presentation.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf