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Nonequilibrium Probability Currents in OpticallyDriven Colloidal Suspensions
by Samudrajit Thapa, Daniel Zaretzky, Ron Vatash, Grzegorz Gradziuk, Chase Broedersz, Yair Shokef, Yael Roichman
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Submission summary
Authors (as registered SciPost users):  Yael Roichman · Yair Shokef · Samudrajit Thapa 
Submission information  

Preprint Link:  https://arxiv.org/abs/2310.12718v1 (pdf) 
Date submitted:  20231020 12:00 
Submitted by:  Thapa, Samudrajit 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Experimental, Computational 
Abstract
In the absence of directional motion it is often hard to recognize athermal fluctuations. Probability currents provide such a measure in terms of the rate at which they enclose area in phase space. We measure this area enclosing rate for trapped colloidal particles, where only one particle is driven. By combining experiment, theory, and simulation, we single out the effect of the different time scales in the system on the measured probability currents. In this controlled experimental setup, particles interact hydrodynamically. These interactions lead to a strong spatial dependence of the probability currents and to a local influence of athermal agitation. In a multipleparticle system, we show that even when the driving acts only on one particle, probability currents occur between other, nondriven particles. This may have significant implications for the interpretation of fluctuations in biological systems containing elastic networks in addition to a suspending fluid.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2024111 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2310.12718v1, delivered 20240111, doi: 10.21468/SciPost.Report.8396
Report
The authors study two optically trapped colloidal particles that are hydrodynamically coupled due to the solvent. One particle is driven by switching the trap position. Considering the projected positions $x_i$ of both particles, the authors address the question of estimating probability currents through measuring the enclosed area of the trajectory, and from this the rate (AER) of area changes, in the plane spanned by both particle positions. This idea first appeared in a review (Ref. 4) and the authors now elaborate on this concept through experiments and simple analytical calculations.
Inferring entropy production or at least lower bounds to the entropy production is currently intensively studied in order to assess complex (living) systems driven away from equilibrium. While the authors admit that the “connection between the underlying activity in the system and its manifestation in the AER is not fully understood”, I’m still missing a discussion on the relation between AER and entropy production. The AER vanishes in equilibrium but is it supposed to be a bound? Does it equal the entropy production rate in some limit? What do I learn from quantifying the AER except that it is nonzero if the system is driven? Otherwise I feel this manuscript is sufficiently interesting for publication.
Further points:
 In the abstract, please reword “phase space”. Phase space denotes the space spanned by positions and momenta.
 On page 5, the authors write “with 30 nm spatial resolution”. Is that correct? That would be far below the optical resolution. I understand that the particles are µm and above the resolution threshold but still. What is the field of view and what is the pixel size? The scale bar in Fig. 1b is 50 nm, that would be about the error bar so that essentially the whole trajectory is of the order of the measurement error. It is hard to see how the authors estimate an error of 2 nm^2/ms for the AER?
 In Sec. V, the authors determine expressions for the AER, which is possible since they consider simple linear stochastic equations of motion. They then introduce hydrodynamic coupling through Eq. (15). However, it seems the coupled stochastic equations are solved for fixed $J$ [Eq. (18)] so that the positions fluctuate but the quantity $r$ is fixed (I suppose to the mean average?) and independent of the $\delta x$. This is a drastic simplification. The authors should at least attempt a Taylor expansion of the mobility tensor around $r$ and discuss whether the limitations of that approximation are consistent with the experiments.
Report #2 by Anonymous (Referee 2) on 20231224 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2310.12718v1, delivered 20231224, doi: 10.21468/SciPost.Report.8340
Strengths
1) insightful minimal model
2) combines experiments and theory
3) well organised
4) wider applicability
Weaknesses
1)some potential issues with the derivation of the out of equilibrium hydrodynamicallycoupled model
2) some parts of the model need to be clarified
3) parts of the text need to be specified better (see specific issues raised in the requested changes)
Report
The manuscript reports on the validation of the areaencolingrate (AER) as a measure of outofequilibrium state in stochastic systems. The authors start with experiments focussing on the behaviour of a pair of colloidal particles held at distance “r” by separate optical tweezers (OTs). When particle 1 is actively driven by regularly displacing the position of its OT (typical size b0), the driving is felt also by particle 2 due to interparticle hydrodynamic coupling. The authors then observe the dynamics of the coupled system along the two dimensional projection of its phase space given by the coordinates (x1,x2) of the colloids along the line joining their average position. In the (x1,x2) space, the system’s position displays a net circulation around the point (avg(x1),avg(x2)) which can be quantified in terms of a AER. The experimental AER appears to be largely compatible with the results from simulations of the same system, showing a scaling with b0^2/r.
After analysing the experimental results, the authors lay down the general theoretical framework to analyse similar systems, with particles coupled either through hydrodynamics (dissipative coupling) or elastic coupling (conservative coupling).
Overall, I find the paper interesting and well organised. It provides good insights into the outofequilibrium behaviour of a simple model system, which can be of guidance for more complex cases. At the same time there are in my opinion a few issues that the authors need to address. Once these have been resolved satisfactorily, I will be happy to recommend the manuscript for publication.
Requested changes
The comments are ordered as the related text appears in the draft manuscript.
1) Pg.4. “[..] suspended in double distilled ionized water […]”.
I suspect there is a typo here.
2)Sec. Experimental Design. The authors should quantify the experimental error they expect for the parameters /tau and b0 of the driven optical tweezer. Amongst other things, this is important for example when discussing the plateauing observed at small b0 in the experimental points of Fig.4.
3)Sec. Experimental Design. As far as I can tell, the value of the parameter r/d is only found in the caption to Fig.1. Please make sure that this is explicitly stated in the main text of the paper.
4) Sec. Numerical Simulations. I am slightly puzzled by the choice of the RotnePrager approximation, rather than just the Oseen tensor, for the interparticle hydrodynamic interactions. Of course I agree that this is a better approximation in general. However, the correction term over the Oseen tensor is of the order (1/6)(d/r)^2, which is approximately 0.01 here (d being the diameter of the colloids). It does not hurt, but it also does not seem needed to me.
5) Sec. IV. When discussing the experimental values vs. the numerical values obtained either with the parabolic or the Gaussian approximations, it is clear that the agreement is better with the latter. However, later in the manuscript the authors also show that a finitesamplingtime can lead to an underestimate of the AER. Could this not be at play here as well? In other words, the *real* experimental AER could be closer to the parabolic traps’ case, with the difference only due to finite sampling time effects.
6) Sec. IV. Regarding the scaling of the AER with b0^2/r, perhaps one could already expect that as a simple consequence of hydrodynamic coupling. Given a typical displacement of particle 1 of “b0”, the typical displacement of particle 2 resulting from hydrodynamic coupling should be “b0 d/r” (d is the colloid radius). The area, then, will go like the product of the two, i.e. (b0^2)(d/r).
7)Sec. V, pg. 10. “governed by a Langevin equation with arbitrary coefficients”. It seems to me that Eq. (1) is actually a specific class of Langevin equations, rather than the most generic one.
8) Pg.11. “Equation 1 is quite general…”. Please be more specific.
9) Pg. 11. “A symmetric matrix V […] detailed balance”. It seems to me that requiring that the matrices V and D are mutually diagonalisable would be sufficient and more general than what is currently written in the text.
10) Pg. 13. “Note that the AER is independent of the distance […] motion”. I am not sure why this would be surprising. It appears to be a direct consequence of the assumption that all the couplings here are perfect Hookean springs. Under this assumption, the forces bear no dependence on the actual distance between the colloids, but only on the variation of this distance. Then it is not surprising that also the AER is independent of the interparticle separation.
11) Pg. 14, Eq.15. In reference also to point number 4) above, the tensor used here is the Oseen tensor. I am not sure why the authors introduce the RotnePrager approximation.
12) Pg.15, Eqs.17,18. I agree with the authors’ previous formulae on the elastically coupled colloids. However, I am not sure I agree with their approach in this case. I will try to explain myself. Please bear with me as it is a bit long to write it down. I hope it will be clear.
The system leading to Eq. 18 has
particle 1, temperature T+\DeltaT
particle 2, temperature T
Let us call this system, system 0.
The current analysis should be valid independently of whether \DeltaT is positive or negative. In particular, if instead of \DeltaT we take \DeltaT, and have the following system (system 1)
particle 1, temperature T\DeltaT
particle 2, temperature T
then the diagonal terms of the tensor D should be equal to those in Eq.18, with the sign of \DeltaT inverted:
D11= T \DeltaT
D22= TJ^2\DeltaT
Now let’s go back and consider the original system (system 0), but define (T+\DeltaT) as T*:
T*=T+\DeltaT.
Then particle 1 is at temperature T* and particle 2 at temperature (T*\DeltaT). Let’s now relabel particle 2 as particle 1*, and particle 1 as particle 2*. Then the complete “star” system (system 0*) is
particle 1*, temperature T*\DeltaT
particle 2*, temperature T*.
According to the prescription above, the tensor D for this system should have diagonal terms
D1*1* = T* \DeltaT
D2*2* = T*J^2\DeltaT
However, the systems 0 and 0* are actually the same system, implying that D1*1* should be equal to D22 and D2*2* should be equal to D11. This is clearly not the case.
In other words, having particle 1 at \DeltaT with respect to particle 2, should be equal to having particle 2 at \DeltaT with respect to particle 1. This does not seem to be the case here.
In my opinion, this problem arises from Eq.17. I do not think that Eq. 17 is correct. The correct form should be such that, if we consider T to be the average temperature and have particle 1 at temperatures T+0.5\DeltaT and particle 2 at temperatures T0.5\DeltaT, then the two particles are treated in an equivalent way.
In other words, the system is not out of equilibrium because particle 1 is at temperature \DeltaT above particle 2, but because the two particles do now have the same temperature.
This problem persists in later parts of the manuscript and should be addressed there as well.
13) Pg.15. Given the tensor D, the tensor F is chosen by the authors through a Cholesky decomposition. However, what we know of the two tensors is simply that D=0.5FF^T. The particular choice of the Cholesky decomposition of D is but one of infinitely many choices one can make for the tensor F. For example, for any rotation matrix O, (FO) is also a perfectly fine choice… but it will give rise to a different “microscopic” dynamics.
This of course is due to the fact that, whilst there is one FokkerPlanck equation that can be derived from any Langevin equation, the opposite is not true. There are infinitely many Langevin equations that give rise to the same FokkerPlanck equation. The authors should be more careful when presenting this part in their paper. The prescription they have used is valid, but is not unique. (Of course the AER is independent of the actual choice of F as long as 0.5FF^T=D)
14) The authors have looked at systems of 2 and 3 particles, and thanks to their analysis it is reasonably straightforward now to generalise the results to more particles (although the calculations might be tedious). However, I was wondering whether the authors could comment about the case of a single particle. Is there a simple way to test its potential out of equilibrium state using an AERlike measurement?
Author: Samudrajit Thapa on 20240715 [id 4621]
(in reply to Report 2 on 20231224)
Requested changes: Referee: The comments are ordered as the related text appears in the draft manuscript.
1) Pg.4. “[..] suspended in double distilled ionized water […]”. I suspect there is a typo here.
Reply: We fixed this typo to ‘deionized water’.
2) Sec. Experimental Design. The authors should quantify the experimental error they expect for the parameters tau and b0 of the driven optical tweezer. Amongst other things, this is important for example when discussing the plateauing observed at small b0 in the experimental points of Fig.4.
Reply: We have mentioned at the end of the first paragraph on page 6 that the error on the repositioning rate is about 1 Hz. In the experimental description section we state that our trap positioning precision is 10 nm. This is the main source of error in b0. In the revised manuscript we add the error in b0 (10 nm) in all relevant locations.
3) Sec. Experimental Design. As far as I can tell, the value of the parameter r/d is only found in the caption to Fig.1. Please make sure that this is explicitly stated in the main text of the paper.
Reply: It is now explicitly stated in the revised manuscript.
4) Sec. Numerical Simulations. I am slightly puzzled by the choice of the RotnePrager approximation, rather than just the Oseen tensor, for the interparticle hydrodynamic interactions. Of course I agree that this is a better approximation in general. However, the correction term over the Oseen tensor is of the order (1/6)(d/r)^2, which is approximately 0.01 here (d being the diameter of the colloids). It does not hurt, but it also does not seem needed to me.
Reply: We agree with the referee that considering the Oseen tensor would suffice. We verify this with simulations and also the agreement between simulations with RotnePrager and analytical results with the Oseen tensor shows this. However, we decided to keep the RotnePrager tensor because it ensures momentum conservation. Following the referee’s comments we have clarified this in section VI in the revised manuscript (below Eq. 14, page 14).
5) Sec. IV. When discussing the experimental values vs. the numerical values obtained either with the parabolic or the Gaussian approximations, it is clear that the agreement is better with the latter. However, later in the manuscript the authors also show that a finitesamplingtime can lead to an underestimate of the AER. Could this not be at play here as well? In other words, the real experimental AER could be closer to the parabolic traps’ case, with the difference only due to finite sampling time effects.
Reply: In Figs. 4 and 5 we choose a sampling rate of 120 fps for both the simulations and the experiments. Thus the effect of sampling rate rate is ruled out in the discrepancy between simulations with parabolic traps and the experiments. We discuss the effect of finite sampling rate only in section VIIB, and in the revised manuscript we have added Fig. 12 in Appendix A which shows how the AER varies with the imaging rate.
However, following the referee reports, after thorough investigation, we identified a numerical artifact in the behavior of our simulations, and have established that choosing a Gaussian trap does not lead to significant changes in the AER in contrast to choosing parabolic traps. We have therefore removed the parts with Gaussian traps in the revised manuscript.
6) Sec. IV. Regarding the scaling of the AER with b0^2/r, perhaps one could already expect that as a simple consequence of hydrodynamic coupling. Given a typical displacement of particle 1 of “b0”, the typical displacement of particle 2 resulting from hydrodynamic coupling should be “b0 d/r” (d is the colloid radius). The area, then, will go like the product of the two, i.e. (b0^2)(d/r).
Reply: We thank the referee for this important insight. We have now added this in the revised manuscript.
7) Sec. V, pg. 10. “governed by a Langevin equation with arbitrary coefficients”. It seems to me that Eq. (1) is actually a specific class of Langevin equations, rather than the most generic one.
Reply: In the revised manuscript we have removed the expression “with arbitrary coefficients”.
8) Pg.11. “Equation 1 is quite general…”. Please be more specific.
Reply: We have changed this to “In Eq. 1 we may consider…”.
9) Pg. 11. “A symmetric matrix V […] detailed balance”. It seems to me that requiring that the matrices V and D are mutually diagonalisable would be sufficient and more general than what is currently written in the text.
Reply: We wanted to relate to the standard equilibrium dynamics for which D is diagonal with identical elements (all particles have the same temperature). Then in the simplest case when V is symmetric, it is apparent that B=0. We have changed the sentence in the revised manuscript to better explain this.
10) Pg. 13. “Note that the AER is independent of the distance […] motion”. I am not sure why this would be surprising. It appears to be a direct consequence of the assumption that all the couplings here are perfect Hookean springs. Under this assumption, the forces bear no dependence on the actual distance between the colloids, but only on the variation of this distance. Then it is not surprising that also the AER is independent of the interparticle separation.
Reply: We agree that it is not surprising, but we want to highlight that the massspring model cannot be used to explain the experimental system we have, because in our system we observe distance dependence of the AER. Following the referee’s comment, we have added a sentence (page 13) in the revised manuscript clarifying that the context of this remark is to emphasize the inability of the theoretical results for a massspring system to explain our experimental results.
11) Pg. 14, Eq.15. In reference also to point number 4) above, the tensor used here is the Oseen tensor. I am not sure why the authors introduce the RotnePrager approximation.
Reply: Considering the Oseen tensor in the simulations suffices. We however kept the RotnePrager tensor in the simulations because it ensures momentum conservation. We have clarified this in section VI in the revised manuscript (below Eq. 14, page 14).
12) Pg.15, Eqs.17,18. I agree with the authors’ previous formulae on the elastically coupled colloids. However, I am not sure I agree with their approach in this case. I will try to explain myself. Please bear with me as it is a bit long to write it down. I hope it will be clear.
The system leading to Eq. 18 has
particle 1, temperature T+\DeltaT particle 2, temperature T
Let us call this system, system 0. The current analysis should be valid independently of whether \DeltaT is positive or negative. In particular, if instead of \DeltaT we take \DeltaT, and have the following system (system 1)
particle 1, temperature T\DeltaT particle 2, temperature T
then the diagonal terms of the tensor D should be equal to those in Eq.18, with the sign of \DeltaT inverted:
D11= T \DeltaT D22= TJ^2\DeltaT
Now let’s go back and consider the original system (system 0), but define (T+\DeltaT) as T*:
T*=T+\DeltaT.
Then particle 1 is at temperature T and particle 2 at temperature (T\DeltaT). Let’s now relabel particle 2 as particle 1, and particle 1 as particle 2. Then the complete “star” system (system 0*) is
particle 1, temperature T\DeltaT particle 2, temperature T.
According to the prescription above, the tensor D for this system should have diagonal terms
D11 = T \DeltaT D22 = TJ^2\DeltaT
However, the systems 0 and 0 are actually the same system, implying that D11 should be equal to D22 and D22* should be equal to D11. This is clearly not the case.
In other words, having particle 1 at \DeltaT with respect to particle 2, should be equal to having particle 2 at \DeltaT with respect to particle 1. This does not seem to be the case here. In my opinion, this problem arises from Eq.17. I do not think that Eq. 17 is correct. The correct form should be such that, if we consider T to be the average temperature and have particle 1 at temperatures T+0.5\DeltaT and particle 2 at temperatures T0.5\DeltaT, then the two particles are treated in an equivalent way. In other words, the system is not out of equilibrium because particle 1 is at temperature \DeltaT above particle 2, but because the two particles do now have the same temperature.
This problem persists in later parts of the manuscript and should be addressed there as well.
Reply: We thank the referee for this detailed comment. There is some ambiguity on how to write the diffusion matrix in the presence of two temperatures and hydrodynamic interactions. We present the case with different heat baths only as an example and the diffusion matrix is a choice for the scheme presented in Fig. 8a. We have clarified this below Eq. 16 in the revised manuscript. Following the referee’s comments we have now moved this case to Appendices C and D. We note that the system with different heat baths is different from our main focus in this article, namely, a system of optically trapped particles where one of them is driven by repositioning one of the traps. For this system we don’t have any ambiguity because in this case there is no temperature difference, just a single ambient temperature (which is irrelevant since its fluctuations are uncorrelated with the driving) and driving by trap repositioning.
13) Pg.15. Given the tensor D, the tensor F is chosen by the authors through a Cholesky decomposition. However, what we know of the two tensors is simply that D=0.5FF^T. The particular choice of the Cholesky decomposition of D is but one of infinitely many choices one can make for the tensor F. For example, for any rotation matrix O, (FO) is also a perfectly fine choice… but it will give rise to a different “microscopic” dynamics. This of course is due to the fact that, whilst there is one FokkerPlanck equation that can be derived from any Langevin equation, the opposite is not true. There are infinitely many Langevin equations that give rise to the same FokkerPlanck equation. The authors should be more careful when presenting this part in their paper. The prescription they have used is valid, but is not unique. (Of course the AER is independent of the actual choice of F as long as 0.5FF^T=D)
Reply: We agree with the referee and have now added an explanation on why the Cholesky decomposition is usually used. In the revised manuscript, in section V after Eq. 10, we note that there are several ways to decompose D but the final result in terms of AER does not depend on which decomposition is used. The Cholesky decomposition is widely used because of the important property that the existence of Cholesky decomposition of a matrix means that the matrix is positive definite, which ensures that the eigenvalues are positive. In the case of the diffusion matrix, this ensures that the diffusion coefficients are nonnegative.
14) The authors have looked at systems of 2 and 3 particles, and thanks to their analysis it is reasonably straightforward now to generalise the results to more particles (although the calculations might be tedious). However, I was wondering whether the authors could comment about the case of a single particle. Is there a simple way to test its potential out of equilibrium state using an AERlike measurement?
Reply: In case of a single particle, if it is driven such that it exhibits directional motion (say if the trap driving it moves in circles, rather than along a line) then there would clearly be an observable current in physical space, and we won’t need to search for probability currents in phase space (as we do, for instance, with the space spanned by x1 and x2 for two particles). However, for singleparticle systems, if currents are noisy and hard to observe, in principle, one can still detect and quantify them with the AER in physical space. Indeed, Ref. [17] considered probability fluxes to quantify nonequilibrium motion of a beating flagellum of Chlamydomonas reinhardtii by decomposing its motion into different modes. In the case of a colloidal sphere, for which we track only its position, there is not much sense in looking at AER for a single particle. We discuss this in section IX of the revised manuscript.
Report #1 by Anonymous (Referee 1) on 20231222 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2310.12718v1, delivered 20231222, doi: 10.21468/SciPost.Report.8306
Strengths
1 This is an interesting paper on a central issue of stochastic thermodynamics: the investigation of probability currents in outofequilibrium systems, with the goal of quantitatively distinguishing these systems from their equilibrium counterparts.
2 The authors combine experiments, numerical simulations and theory to make their points.
3 The readability of the paper is very good. It is clear what the authors did (and did not do), and results obtained from previous papers are well referenced.
Weaknesses
1 The structure of the paper is suboptimal. Each part is dedicated to a different topic and the links between them are minimal. The overall consistency could be improved.
2 The paper often deviates from its main message and the link between the different theoretical parts (two temperatures, colored noise, three particles) and the experiments is sometimes hard to understand.
3 The scalings from the experimental data in Figs. 4 and 5 are not so convincing.
3 Some numerical observations are left unexplained (strong influence of the shape of the potentiel and the acquisition rate) even at a qualitative level.
4 The importance of looking at the AER, vs other observables, is not explained enough.
Report
Outofequilibrium systems exhibit probability currents that may be challenging to measure, especially in experiments. In this article, the authors put forward (i) a model experimental system: beads trapped by optical tweezers with a stochastic repositining of one trap at equally spaced times and (ii) a measure of probability currents: the area enclosing rate (AER). They additionally perform numerical simulations (Stockesian dynamics) and theoretical computations of the AER in simple cases.
I nevertheless find this article not entirely convincing since the experimental scalings are unclear, the most puzzling numerical observations are not explained so well, and the numerous theoretical models could be better connected to the experiments. In short, I am currently not in favor of the publication of this article in SciPost Physics as it is now. But I do believe it has a good potential and I encourage the authors to improve it and resubmit.
I list several questions and comments below. Some other points are listed in "Weaknesses" and "Requested changes" and will not be repeated here.
1 One point that puzzles me in the article is that on the one hand the theory (sections VI and VII) is done at the linear order in displacements ; while on the other hand the simulations show that nonlinear effects (Gaussian traps) change the AER by one order of magnitude. Do the authors have any idea why the nonlinear effects are so strong, and how they can be explained, at least qualitatively.
2 In the discussion, "We also demonstrate that the AER peaks when the driving time scale $\tau$ is comparable to the relaxation time scale $\gamma/\tau$". I'm sorry but I do not see what part of the main text is dedicated to this issue.
3 Why did the authors choose a frequency $1/\tau=36$ Hz? How do the results depend on this frequency?
4 Fig. 3b gives the fealing that the plateau is almost at zero. Maybe the $y$ range can be made smaller (or error bars included).
5 What would be a derivation for the potential $U(x)$ used at the top of page 9 for a Gaussian beam (reference?)? Also, shall I understand from the values that $w\sim \lambda/2$?
6 What is the status of section V? Is it a reminder of known results? A warmup before the following section? Some new computations related to AER?
7 Is there is qualitative link between the twotemperature problem and the colored noise problem? For instance an approximate mapping? Or are they two separate outofequilibrium issues?
8 Is there a reason to use the Cholesky decomposition, as opposed to the (symmetric) square root of a symmetric matrix? Do both lead to the same results?
9 Eq. (14), maybe say that the $1/r^3$ term will be neglected in the following, since it is never used.
10 I think the rational for section VIII could be made clearer, since it is not (directly) connected with the experiments.
Let me end this report by saying that I shall be supportive if the authors experience difficulties to resubmit due to the current situation in their country.
Requested changes
1 Add an outline at the end of the introduction to introduce and show the consistency between the different parts.
2 Refactor the whole article so that the message is clear throughout and that the sections are better linked with one another. This may involve moving some of the theory into appendices.
3 Give more details earlier in the text about what the AER is, for which pair of variables it may be computed, why it is important and what alternative observables could have been considered. Giving some basic theoretical results (such as Eq 6) earlier may help.
4 Be clearer (and more honest) about what can be deduced from the experimental data. Why do the authors think that the scalings from the simulations are also seen in experiments (Figs 4, 5)? To which degree of certainty? Maybe explain why the noise level and the error bars are quite large. If needed, additional experimental data may be shown in appendices.
5 Better comment the results of the simulations even when analytical results are not available. End of IV, is there any insight why the difference is so large when the shape of the trap is changed? End of VII, what happens between $10^4$ and 120 fps, why is the difference so large, how does it depend on the framerate ?
6 Give more details about the numerical simulations. Which framework is used (homemade or standard one), how is the code implemented ? Alternatively the authors may consider opensourcing their code (github / Zenodo).
7 "s" is the standard SI symbol for "second", not "sec" [see https://www.bipm.org/en/measurementunits/sibaseunits]. This should be corrected.
Author: Samudrajit Thapa on 20240715 [id 4620]
(in reply to Report 1 on 20231222)
We thank the referee for listing the strengths of the paper. We are also grateful for listing the weaknesses, after addressing which we believe the revised manuscript is stronger. We reply below, describing how we addressed the weaknesses.
1) We have restructured the paper, explained more explicitly the logical connections between the different parts, and also moved some of the theoretical sections to the appendices. We hope that the referee will deem the revised version to be more consistent.
2) In the revised manuscript, we focus on the main messages: i.e. the AER for a system of optically trapped, hydrodynamically interacting particles where one of them is driven optically. Consequently, we have moved the theoretical results for a system of particles in contact with heat baths at different temperatures to the Appendix. We now also better connect the sections by providing an outline of the paper in the introduction section and by adding connecting sentences at the beginnings and ends of sections.
3) One of our primary motivations for this work was to extend prior tests of the AER by exploring a system where components are linked through hydrodynamic rather than elastic interactions. Hydrodynamic interactions are both distancedependent and longranged. In Fig. 5, our investigation demonstrated a scaling of the AER with distance, with a noticeable decay in its strength over distance. The amplitude of the AER decayed below measurement error already at intermediate distances. Consequently, measuring the complete scaling of the AER decay experimentally posed a significant challenge. Nevertheless, the scaling in our experimental findings is consistent with numerical computations, which, in turn, corroborate our theoretical analysis. Moreover, Fig. 4 shows the scaling of experimental AER with the driving amplitude, which agrees with the scaling in the numerical simulations and that expected from our subsequent theoretical analysis. In the revised manuscript, we address this point.
We thank the referee for asking about the difference between using Gaussian or parabolic traps in the simulations. After a thorough investigation of this issue, we identified that the coupling between thermal fluctuations and Gaussian traps caused particles to escape their traps occasionally, and this led to significantly lower values of AER in those simulations. Since we expect (and have verified for parabolic traps) that thermal fluctuations should not affect the AER, we reran the Gaussian trap simulations without thermal fluctuations, and this eliminated the escapes and yielded AER values that were very close to those obtained with parabolic traps. Therefore, we conclude that the Gaussian shape of the traps cannot explain the deviation between experiments and (parabolic traps) simulations. We have updated the paper accordingly.
We also thank the referee for asking how the estimated AER depends on the acquisition rate. Following this question, we have added a new Figure 12 in Appendix A that highlights that at low frame rates, the estimated AER is lower than the steady state value, and only at high enough frame rates do the estimated AER values converge.
4) We thank the referee for pointing this out. We added a paragraph in the discussion section recalling the connection of the AER to the cycling frequency and entropy production rate. We also highlighted that the AER and the cycling frequency have the advantage of being relatively simple to compute directly from the measured particle trajectories.
Report: Referee: Outofequilibrium systems exhibit probability currents that may be challenging to measure, especially in experiments. In this article, the authors put forward (i) a model experimental system: beads trapped by optical tweezers with a stochastic repositioning of one trap at equally spaced times and (ii) a measure of probability currents: the area enclosing rate (AER). They additionally perform numerical simulations (Stockesian dynamics) and theoretical computations of the AER in simple cases.
I nevertheless find this article not entirely convincing since the experimental scalings are unclear, the most puzzling numerical observations are not explained so well, and the numerous theoretical models could be better connected to the experiments. In short, I am currently not in favor of the publication of this article in SciPost Physics as it is now. But I do believe it has a good potential and I encourage the authors to improve it and resubmit.
Reply: Following the referee’s comments, we have revised the manuscript adding explanations of the issues raised by the referee and also restructuring the manuscript to connect the sections better. We thoroughly investigated the issue alluded to by the referee regarding the simulations considering different shapes of the potential, and we found that using Gaussian traps does not lead to significant difference in the AER values as compared to using parabolic traps, and therefore this cannot explain the observed discrepancy between results from simulations and experiments. We modified section IV accordingly, and we remark on possible reasons for this discrepancy. Our investigations revealed that taking into account the size of the particles does not resolve this discrepancy. However, the particles getting closer to the confining walls might result in lower values of the AER in the experiments. The experiments were performed at a distance of a few microns (~2 microns) from the bottom wall, while the height of the sample cell was ~20 microns, and the distances between the spheres were 38 microns. Under these conditions, momentum is absorbed by both the bottom and top glass walls (see Ref. [34]). This leads to a weaker hydrodynamic interaction between the particles (see Ref. [35]) and, consequently, a lower AER.
In the theoretical analyses, we present three cases: (i) a massspring system where the particles are in contact with heat baths at different temperature but there is no hydrodynamic interactions (ii) particles interacting hydrodynamically and in contact with different heat baths, and (iii) particles interacting hydrodynamically and in contact with the same heat bath but the system is driven out of equilibrium by repositioning the trap position of one of the particles periodically. Case (i) highlights that without hydrodynamic interactions the AER does not depend on interparticle distance dependence. Case (ii) is presented to compare the massspring system with that of optically trapped particles and shows that including hydrodynamic interactions results in the distance dependence of the AER as observed in the experiments. Finally, case (iii) mimics the experimental setup where the particles are optically trapped, interact hydrodynamically and the system is driven out of equilibrium by repositioning the trap position of one of the particles periodically. We explain this above Eq. 16 in section VI.
Referee: I list several questions and comments below. Some other points are listed in "Weaknesses" and "Requested changes" and will not be repeated here.
1 One point that puzzles me in the article is that on the one hand the theory (sections VI and VII) is done at the linear order in displacements ; while on the other hand the simulations show that nonlinear effects (Gaussian traps) change the AER by one order of magnitude. Do the authors have any idea why the nonlinear effects are so strong, and how they can be explained, at least qualitatively.
Reply: We thank the referee for drawing our attention to this issue. After thorough investigation of the simulations, we identified a numerical artifact in simulations with Gaussian traps. Specifically, the coupling between thermal fluctuations and Gaussian traps caused particles to occasionally escape their traps, and this led to significantly lower averaged values of AER in those simulations. Since we expect (and have verified for parabolic traps) that thermal fluctuations should not affect the AER, we reran the Gaussian trap simulations without thermal fluctuations, and this eliminated the escapes and yielded AER values that were very close to those obtained with parabolic traps. Therefore, we conclude that the Gaussian shape of the traps cannot explain the deviation between experiments and (parabolic traps) simulations.
2 In the discussion, "We also demonstrate that the AER peaks when the driving time scale tau is comparable to the relaxation time scale gamma/tau". I'm sorry but I do not see what part of the main text is dedicated to this issue.
Reply: We showed this in the description of Fig. 9. To make it clearer, following the referee’s comments, we have added the description in the caption of Fig. 9 in the revised manuscript, and also emphasized this in the last paragraph of section VIIB in the revised manuscript.
3 Why did the authors choose a frequency 1/tau=36 Hz? How do the results depend on this frequency?
Reply: This value is the trap repositioning frequency in our experimental setup, which is determined by the highest reliable switching rate of our optical traps in our experimental setup. In the revised manuscript we have added a sentence stating this in section VIIA. Figure 9 describes how the AER depends on this frequency. We have highlighted this both in the caption and in the description of Fig. 9 in the revised manuscript.
4 Fig. 3b gives the fealing that the plateau is almost at zero. Maybe the y range can be made smaller (or error bars included).
Reply: Indeed, it is hard to infer the plateau value from the main panel of Fig. 3b. Therefore, the figure includes an inset which zooms in on the longtime behavior, and clearly shows that the plateau value is nonzero. In the revised manuscript we have highlighted this both in the description in the last paragraph of section II and in the caption of Fig. 3.
5 What would be a derivation for the potential U(x) used at the top of page 9 for a Gaussian beam (reference?)? Also, shall I understand from the values that w∼λ/2?
Reply: Thanks to the referee’s comments on the effect of shape of the traps, we found that Gaussian traps do not lead to significant difference in AER values as compared to using parabolic traps. We have therefore removed the parts with Gaussian traps in section IV in the revised manuscript.
6 What is the status of section V? Is it a reminder of known results? A warmup before the following section? Some new computations related to AER?
Reply: In section V we recall previous results on how to compute the AER starting with a Langevin equation driven by white noise. This sets up the framework that we subsequently use to obtain analytical results of the AER for hydrodynamically coupled colloidal particles. We also recall previous results on the AER for a massspring model and highlight that the theoretical results for this system are incapable of explaining the experimental results presented in section III. We have added this description to the beginning of section V in the revised manuscript.
7 Is there is qualitative link between the twotemperature problem and the colored noise problem? For instance an approximate mapping? Or are they two separate outofequilibrium issues?
Reply: The twotemperature and the colored noise are two separate outofequilibrium situations. We present the twotemperature case as an example to compare between the massspring system and the system of optically trapped particles. The colored noise case, on the other hand, mimics the experimental system where the trap position of one of the particles is repositioned periodically thereby driving the system out of equilibrium. We have added this clarification in section VI of the revised manuscript (page 16).
8 Is there a reason to use the Cholesky decomposition, as opposed to the (symmetric) square root of a symmetric matrix? Do both lead to the same results?
Reply: The positive square root can also be used, which would lead to the same AER. In the expression for AER, only the diffusion matrix appears, and therefore there is some flexibility on the choice of F. However, the Cholesky decomposition has important properties. The existence of the Cholesky decomposition of a matrix means that the matrix is positive definite. This means that the eigenvalues are positive. In the case of the diffusion matrix, this ensures that the diffusion coefficients are nonnegative. This is why the Cholesky decomposition is widely used, and we preferred using it. However, we reiterate that the final results are not affected by the decomposition used.
9 Eq. (14), maybe say that the 1/r^3 term will be neglected in the following, since it is never used.
Reply: We have mentioned this below Eq. (14) in the revised manuscript.
10 I think the rational for section VIII could be made clearer, since it is not (directly) connected with the experiments.
Reply: We have clarified in the revised manuscript that the question we address in section VIII is whether the AER computed from nondriven particles can help detect nonequilibrium signatures of a system where there may be untracked driven particles. This is crucial to biological systems where it is not possible to track all particles.
Requested changes: 1 Add an outline at the end of the introduction to introduce and show the consistency between the different parts.
Reply: We have added an outline in the revised manuscript.
2 Refactor the whole article so that the message is clear throughout and that the sections are better linked with one another. This may involve moving some of the theory into appendices.
Reply: We have restructured the revised manuscript by moving some of the theory to the appendix and have also added explanations at several places.
3 Give more details earlier in the text about what the AER is, for which pair of variables it may be computed, why it is important and what alternative observables could have been considered. Giving some basic theoretical results (such as Eq 6) earlier may help.
Reply: The AER can be computed between any two degrees of freedom. Following also another referee’s suggestion, we add descriptions of the cycling frequency as an alternate observable and also present the connection between AER and the entropy production rate in section IX of the revised manuscript . We also highlight that the AER and the cycling frequency can be computed directly from the raw trajectories obtained in single particle tracking. While in order to maintain the continuity of the paper we decided to not move the basic theoretical results earlier, we provide an outline in the introduction which points to the relevant section V for the theoretical details.
4 Be clearer (and more honest) about what can be deduced from the experimental data. Why do the authors think that the scalings from the simulations are also seen in experiments (Figs 4, 5)? To which degree of certainty? Maybe explain why the noise level and the error bars are quite large. If needed, additional experimental data may be shown in appendices.
Reply: Hydrodynamic interactions decay fast (~1/r). For this reason, the signaltonoise ratio in experiments decreases rapidly with the distance between the traps, and quickly reaches the experimental noise level. These distancedependent interactions are fundamentally different from the discrete elastic systems that were used previously. However, they are prevalent in active and biological systems, which is why we chose to study them. At short distances and large driving strength the scaling of AER with the driving strength and distance between the particles in experiments agree with simulated data and analytical predictions. See for instance the experimental datapoints corresponding to b0 > 60 nm in Fig. 4 or the experimental datapoints corresponding to r/d <=3 in Fig. 5b.
5 Better comment the results of the simulations even when analytical results are not available. End of IV, is there any insight why the difference is so large when the shape of the trap is changed? End of VII, what happens between 10^4 and 120 fps, why is the difference so large, how does it depend on the framerate ?
Reply: After thorough investigation, we found that choosing a Gaussian trap does not lead to significant changes in the AER in contrast to choosing parabolic traps. We have therefore removed the parts with Gaussian traps in the revised manuscript. Figure 9 presents AER vs. k1 * tau/gamma but with k1 and gamma fixed. Therefore this figure shows how the AER varies with the repositioning time tau, and highlights that the peak is when k1 * tau/gamma ≈1, i.e. the repositioning time is comparable to the relaxation time gamma/k1. Figure 12 in Appendix A shows how the AER changes with the image acquisition rate. It highlights that at low imaging rates the estimated AER values are lower than the steady state value and only at high imaging rates do the estimated AER values converge. This is because a lower imaging rate corresponds to temporal coarsegraining in phase space, thereby reducing the measured area.
6 Give more details about the numerical simulations. Which framework is used (homemade or standard one), how is the code implemented ? Alternatively the authors may consider opensourcing their code (github / Zenodo).
Reply: We have clarified in section III of the revised manuscript that the codes are home made and have provided a link to github where we made them publicly available.
7 "s" is the standard SI symbol for "second", not "sec" [see https://www.bipm.org/en/measurementunits/sibaseunits]. This should be corrected.
Reply: We have corrected this in the revised manuscript.
Author: Samudrajit Thapa on 20240715 [id 4622]
(in reply to Report 3 on 20240111)Referee: The authors study two optically trapped colloidal particles that are hydrodynamically coupled due to the solvent. One particle is driven by switching the trap position. Considering the projected positions xi of both particles, the authors address the question of estimating probability currents through measuring the enclosed area of the trajectory, and from this the rate (AER) of area changes, in the plane spanned by both particle positions. This idea first appeared in a review (Ref. 4) and the authors now elaborate on this concept through experiments and simple analytical calculations.
Inferring entropy production or at least lower bounds to the entropy production is currently intensively studied in order to assess complex (living) systems driven away from equilibrium. While the authors admit that the “connection between the underlying activity in the system and its manifestation in the AER is not fully understood”, I’m still missing a discussion on the relation between AER and entropy production. The AER vanishes in equilibrium but is it supposed to be a bound? Does it equal the entropy production rate in some limit? What do I learn from quantifying the AER except that it is nonzero if the system is driven? Otherwise I feel this manuscript is sufficiently interesting for publication.
Reply: We thank the referee for the positive evaluation of our work. Following the comments, we have now added a discussion, in section IX, on other observables, namely the cycling frequency and the entropy production rate. The AER matrix provides the rate at which phasespace trajectories enclose area and is given by the difference of the product of the drift matrix and the Covariance matrix from its transpose. The cycling frequency, on the other hand, is defined as the rate at which a trajectory revolves in coordinate space and differs from the AER only by the normalization factor given by the determinant of the Covariance matrix. The entropy production rate (EPR) for a linear system is given by the trace of the matrix computed as a product of the AER, the inverse Covariance matrix and the inverse Diffusion matrix (see Ref. [15] and Eq. 37 in the revised manuscript). The advantage that the AER and the cycling frequency have over the EPR is in the fact that they can be computed directly from the raw single particle tracking data. Moreover they can be computed for any two degrees of freedom. Indeed, the AER can be leveraged, in the case of multidimensional systems, to perform a dissipative component analysis to identify the components which contribute the most to EPR and provide lower bounds on the EPR (see Ref. [15]).
Referee: Further points:
Reply: The concept of AER has to do with looking at currents in phase space, and in principle one could also consider momentum degrees of freedom. When we say that we quantify currents in phase space by measuring AER, we mean a reduced phase space, since AER is defined only in terms of projections to 2D subspaces. Therefore following the referee’s comment, we have rephrased “phase space” in the abstract to “reduced phase space”.
Reply: We thank the referee for this comment. We have corrected it to “with 20 nm spatial resolution”. The diffraction limit is indeed approximately 250 nm, which is the distance two objects need to be apart to distinguish them apart. However, the localization accuracy of the center of mass of a single particle is only limited by the signaltonoise ratio of its measured intensity profile, 20 nm is the conventional estimation using 100x objectives in bright field illumination. The full field of view in these conditions is approximately 100 um^2 and the pixel size is 0.58 nm. Note that the localization is usually of subpixel values. The AER is a time averaged property in which thermal noise and part of other measurement noise are canceled, which is how we obtain a 10 nm^2/ms error, not a 2 nm^2/ms.
Reply: We thank the referee for this careful observation. Indeed in the analytical derivations we assume that J is fixed. This assumption is valid only when r>>sqrt(<(Delta x)^2>). This is always true in the cases we consider because sqrt(<(Delta x)^2>) ≈ b0. The largest driving strength we consider, i.e. b0=110 nm is much smaller than the shortest distance between the trap positions, i.e. r=2d=3 um.