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Harmonic chain far from equilibrium: singlefile diffusion, longrange order, and hyperuniformity
by Harukuni Ikeda
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Harukuni Ikeda 
Submission information  

Preprint Link:  https://arxiv.org/abs/2309.03155v2 (pdf) 
Date submitted:  20231109 20:17 
Submitted by:  Ikeda, Harukuni 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In one dimension, particles can not bypass each other. As a consequence, the meansquared displacement (MSD) in equilibrium shows subdiffusion ${\rm MSD}(t)\sim t^{1/2}$, instead of normal diffusion ${\rm MSD}(t)\sim t$. This phenomenon is the socalled singlefile diffusion. Here, we investigate how the above equilibrium behaviors are modified far from equilibrium. In particular, we want to uncover what kind of nonequilibrium driving force can suppress diffusion and achieve the longrange crystalline order in one dimension, which is prohibited by the MerminWagner theorem in equilibrium. For that purpose, we investigate the harmonic chain driven by the following four types of driving forces that do not satisfy the detailed balance: (i) temporally correlated noise with the noise spectrum $D(\omega)\sim \omega^{2\theta}$, (ii) centerofmass conserving noise, (iii) periodic driving force, and (iv) periodic deformations of particles. For the driving force (i) with $\theta>1/4$, we observe ${\rm MSD}(t)\sim t^{1/2+2\theta}$ for large $t$. On the other hand, for the driving forces (i) with $\theta<1/4$ and (ii)(iv), MSD remains finite. As a consequence, the harmonic chain exhibits the crystalline order even in one dimension. Furthermore, the density fluctuations of the model are highly suppressed in a large scale for the driving forces (i) with $\theta<0$ and (ii)(iv). This phenomenon is known as hyperuniformity. We discuss that hyperuniformity of the noise fluctuations themselves is the relevant mechanism to stabilize the longrange crystalline order in one dimension and yield hyperuniformity of the density fluctuations.
Current status:
Reports on this Submission
Strengths
1gives a broad assessment of the effects of nonequilibrium noise inspired by a range of physical situations
2calculations are straightforward but set out clearly
3interesting concluding discussion about relation to model A/B dynamics
Weaknesses
1in most scenarios considered here, hyperuniformity  one of the key aspects being studied  is "inherited" more or less directly from properties of the assumed noise
2treatment of centerofmass mode and implicit steady steady assumptions not very clear
Report
The paper studies the effects of various types of nonequilibrium noise on the dynamics of the harmonic chain. The scenarios considered are inspired in part by models considered elsewhere in the literature, e.g. active particles with periodically varying sizes. Due to the linearity of the problem, most quantities of interest can be calculated in closed form, given the power spectral density $D_q(\omega)$ of the Fourier modes of the noise. The author studies in particular the structure factor $S(q)$ and its behaviour for $q\to 0$, in order to detect hyperuniformity, the meansquare displacement (MSD), and its longtime limit as a probe of crystalline order.
Of the acceptance criteria for SciPost Physics, the (only) one I can see being met is "Provide a novel and synergetic link between different research areas" as the paper does draw together a range of nonequilibrium situations considered in the literature.
Requested changes
1The treatment of the centreofmass mode $q=0$ is not very clear and this mode will generically show diffusive motion, breaking many of the statements on finite MSD etc at least for long times and in systems of finite size, and preventing the system from reaching a steady state as the author implicitly assumes. I would ask the author to make these points explicit and work in the centerofmass frame, i.e. set $\tilde{u}_{q=0}(t=0)=0$ and $D_{q=0}(\omega)=0$.
2I found the noise in Sec. 4 not as well motivated as the others  clearly there are many ways to have centerofmass conserving noise other than the gradienttype noise considered here; knocking out the centerofmass noise mode (see above, equivalent to subtracting the average of the $\xi_j$ from each $\xi_i$) would be an obvious one. This should be discussed.
3The language is generally intelligible but there are a few places where the errors are conspicuous and these should be fixed, including:
On the ground state > In the ground state
Gaussian color_ed_ noise
hyperniformiy, hyperuniformiy > hyperuniformity
powerlow > powerlaw
deriving forces > driving forces
symmetry braking > symmetry breaking
4In Fig. 1 the solid line for $\theta=0.4$ looks like it is slightly decreasing, which physically it shouldn’t  please check
Report
This work theoretically studied the vibrational dynamics of 1D harmonic chains with several active noises. In particular, the author focused on the stability of the crystalline phase and the largescale density fluctuations. The author treated four types of active noises (temporally correlated noise, centerofmass conserving noise, periodic driving, and periodic deformation) and derived the conditions under which the longrange order survives even in 1D and the hyperuniformity of the density emerges. In the obtained formula, the hyperuniformity exponent and the lower critical dimension are expressed using the exponents in the noise spectrum.
I feel that the paper is beneficial for future theoretical, simulation, and experimental research on dense active particles. The author treated different types of models in a unified and transparent manner, enabling readers to follow the calculations easily. Despite the simplicity of the calculations, the results and the claims are general and essential. Based on this significance, I basically recommend its publication in SciPost Physics. However, before the final recommendation, I ask the author to clarify the following point.
Requested changes
1 In the Fourier transformation Eq.(3) and (6), the zerowave number case $q=0$ is omitted. This means that the global translation mode for the displacements and noises is omitted from the analysis. On the other hand, in Sec. 4.1, the author wrote.
“To preserve the center of mass, the noise should satisfy $\sum_{j=1}^N \xi_j = 0$.“
This seems strange as the author excluded the $q=0$ mode from the beginning. I would like to ask the author to describe more about the reason why the q=0 mode can be omitted in the analysis, and to motivate the centerofmass conserving noise in a selfconsistent manner.
Author: Harukuni Ikeda on 20240712 [id 4615]
(in reply to Report 3 on 20231224)
Dear Referee, Thank you very much for reviewing my paper. I am very happy to see that the Referee has a high opinion of my work. I have carefully studied your reports and have revised the paper. I believe that I have fully answered your questions. In what follows, I will reply to your comments separately and explain major changes made in the revision. All the changes in the revision are highlighted in the file diff.pdf. I hope and believe that you will find the revised version acceptable for publication in SciPost. Sincerely yours,
Harukuni Ikeda
This work theoretically studied the vibrational dynamics of 1D harmonic chains with several active noises. In particular, the author focused on the stability of the crystalline phase and the largescale density fluctuations. The author treated four types of active noises (temporally correlated noise, centerofmass conserving noise, periodic driving, and periodic deformation) and derived the conditions under which the longrange order survives even in 1D and the hyperuniformity of the density emerges. In the obtained formula, the hyperuniformity exponent and the lower critical dimension are expressed using the exponents in the noise spectrum.
I feel that the paper is beneficial for future theoretical, simulation, and experimental research on dense active particles. The author treated different types of models in a unified and transparent manner, enabling readers to follow the calculations easily. Despite the simplicity of the calculations, the results and the claims are general and essential. Based on this significance, I basically recommend its publication in SciPost Physics. However, before the final recommendation, I ask the author to clarify the following point.
Thank you very much for your careful reading and positive evaluations of our work.
1 In the Fourier transformation Eq.(3) and (6), the zerowave number case q=0
is omitted. This means that the global translation mode for the displacements and noises is omitted from the analysis. On the other hand, in Sec. 4.1, the author wrote.
“To preserve the center of mass, the noise should satisfy $\sum_{j=1}^N\xi_j=0$.”
This seems strange as the author excluded the $q=0$ mode from the beginning. I would like to ask the author to describe more about the reason why the $q=0$ mode can be omitted in the analysis, and to motivate the centerofmass conserving noise in a selfconsistent manner.
Thank you for the suggestion. The discussion in Sec. 4.1. was indeed misleading. The crucial point of Hexner and Levine’s argument is not the conservation of the center of mass but rather that the noise in the continuum limit reduces to a conserving noise $\xi(x)=\partial_x\eta(x)$.
To clarify this point, we decided to use “conserving noise” instead of “centerofmass conserving noise” or “centerofmass conserving dynamics” throughout the revised manuscript.
We also modified the sentence above equation (29) as follows:
“To preserve the center of mass $X\equiv \sum_{j=1}^N x_j$, the noise should satisfy $\sum_{j=1}^N \xi_j=0$.”
→
“A conserving noise in the continuum limit $\xi(x)$ is written as $\xi(x)=\partial_x \eta(x)$, where $\eta(x)$ denotes another white noise.”
Attachment:
Strengths
This paper describes three situations in which a onedimensional system of particles unable to overcome one another develop either long range order, or longrange correlations with hyperuniformity.
The article is very well written, very thorough and interesting.
It may have connections with the behavior of active particle systems in confined spaces.
Weaknesses
Phase transitions out of equilibrium in one dimensional space are quite frequent, so this `violation' of the Mermin Wagner result is not in itself the maun point of the paper.
Report
yes
Author: Harukuni Ikeda on 20240712 [id 4614]
(in reply to Report 2 on 20231222)
Dear Referee, Thank you very much for reviewing my paper. I am very happy to see that the Referee has a high opinion of my work. I have carefully studied your reports and have revised the paper. I believe that I have fully answered your questions. In what follows, I will reply to your comments separately and explain major changes made in the revision. All the changes in the revision are highlighted in the file diff.pdf. I hope and believe that you will find the revised version acceptable for publication in SciPost. Sincerely yours,
Harukuni Ikeda
This paper describes three situations in which a onedimensional system of particles unable to overcome one another develop either long range order, or longrange correlations with hyperuniformity.
The article is very well written, very thorough and interesting.
It may have connections with the behavior of active particle systems in confined spaces.
Thank you very much for your careful reading and positive evaluations of our work.
Phase transitions out of equilibrium in one dimensional space are quite frequent, so this `violation' of the Mermin Wagner result is not in itself the maun point of the paper.
We are sorry that our motivation has not been explained clearly in the previous version of the manuscript.
As mentioned by the referee, several nonequilibrium onedimensional systems are known to exhibit phase transition. However, to the best of our knowledge, there are still no known examples of continuous symmetry breaking in one dimension. So, we believe that the violation of the Mermin Wagner result in one dimension is sufficiently novel.
To clarify this point, we added the following sentence at the beginning of section 7.6:
“Several nonequilibrium systems are known to exhibit phase transition in one dimension. However, to the best of our knowledge, there are still no known examples of continuous symmetry breaking in one dimension. This work provides several promising candidates.”
Attachment:
Report
This manuscript describes some interesting theoretical results for a onedimensional system of particles, interacting via harmonic potentials, and subjected to different kinds of driving or noise forces. The main questions to be addressed are whether longranged positional order survives the introduction of noise, and what is the nature of largescale density fluctuations in the resulting steady state.
Several different types of driving/noise forces are considered, which allows some general trends to be identified. Overall, I think the manuscript should be suitable for sciPost after revision. However, there are some aspects of the presentation that need more precision before the manuscript can be accepted.
The most important point is that some aspects of the setup (Section 2.1) need to be more precise. The author does not give any initial conditions for the particles, to supplement equation (1). Related to this, I believe that equation (10) is valid only if the system is already in its steady state at time $t=0$. From this it seems that the angle brackets in (10) etc should represent a steadystate dynamical average. Is this correct? Full details are needed here because there are some subtleties with these kinds of analysis, see the following two points.
The author also writes that the "equilibrium position" of particle $j$ is $R_j=ja$ but since the boundaries are periodic, taking $R_j=(jc)a$ would be an equally appropriate choice for any $c$ between $0$ and $1$. If the system supports longranged positional order, this means that the particles will converge to "equilibrium" positions that depend on the initialization of the system, hence $c$ also depends on the initial condition. So it is not acceptable to omit the details of initialization. (If $c$ is not zero, this also gives a phase to the order parameter $R$ in (12).)
In addition, the author takes $q=2\pi k/(Na)$ with $k=1,2,\dots,N$ so that $q$ is always positive, but then equations like (7) include $\delta_{q,q'}$, which only makes sense if they allow $q<0$, for example by indexing $k$ symmetrically from $N/2$ to $N/21$. Whichever choice is made, there is an underlying physical question about the centerofmass mode (either $k=N$ in the author's convention or $k=0$ in the symmetric convention ). Since this mode has no restoring force ($\lambda_q=0$), it may be that the system never converges to any steady state, in which case the steady state averages mentioned above are not welldefined. A related issue is that for the simplest possible case of equilibrium dynamics with finite $N$, the scaling of the MSD as $t^{1/2}$ will cross over at large enough times to diffusive scaling, MSD $\sim t/N$, because the center of mass of the particles will undergo free diffusion. I think that several of these problems might be avoided by working in the centerofmass frame, which should be equivalent to setting $D_q=0$ for the centerofmass mode, from the start. There would be other options too.
If these problems are fixed then whole study will have a solid mathematical foundation. This is needed in order to properly evaluate some of the later analysis (see for example point 2 in the numbered list below).
Requested changes
1 Clarify the setup of the problem, as described in the report above.
2 What is the size of the terms that are neglected in the first approximate equality in equation (13)? Under what circumstances is it consistent to truncate this expansion? (The required assumption seems to be that $qu_j$ is small compared to unity. Even if $q$ is small, could it be that $u_j$ is very large so the expansion breaks? This step may be ok in systems with longranged positional order it should be justified carefully.)
3 at the end of Sec 3.1 the discussion of "hyperuniform in time" is not very clear. I would suggest to compare with a very simple system which is a single particle with position $X_j$, moving as
\[
\dot X_j = \xi_j
\]
where $\xi_j$ has noise spectrum (15). I assume that one also gets anomalous scaling of the MSD in this case, depending on $\omega$. This may allow the author to connect their results with previous work on subdiffusion or fractional diffusion.
4 in eq(15), surely the sec in the denominator would be more appropriate as a cosine in the numerator.
5 please give a reference (or more detailed justification) for the inequality in (45).
6 it is unfortunate to use $R_j$ for the "equilibrium" position of particle $j$ and then later to use $R$ for the order parameter in (12).
7 In the Introduction. Please clarify the following points. First: the MerminWagner theorem only applies if interactions are shortranged. Second: the fact that particles cannot bypass one another is only true if the interactions are strong enough. Third: it is not necessarily true that fluctuating hydrodynamics is limited to low densities, see for example the macroscopic fluctuation theory of Bertini et al, which is valid at all densities in lattice models.
Author: Harukuni Ikeda on 20240712 [id 4613]
(in reply to Report 1 on 20231222)
Dear Referee, Thank you very much for reviewing my paper. I am very happy to see that the Referee has a high opinion of my work. I have carefully studied your reports and have revised the paper. I believe that I have fully answered your questions. In what follows, I will reply to your comments separately and explain major changes made in the revision. All the changes in the revision are highlighted in the file diff.pdf. I hope and believe that you will find the revised version acceptable for publication in SciPost. Sincerely yours,
Harukuni Ikeda
This manuscript describes some interesting theoretical results for a onedimensional system of particles, interacting via harmonic potentials, and subjected to different kinds of driving or noise forces. The main questions to be addressed are whether longranged positional order survives the introduction of noise, and what is the nature of largescale density fluctuations in the resulting steady state.
Several different types of driving/noise forces are considered, which allows some general trends to be identified. Overall, I think the manuscript should be suitable for sciPost after revision.
Thank you very much for your careful reading and positive evaluations of our work.
However, there are some aspects of the presentation that need more precision before the manuscript can be accepted.
The most important point is that some aspects of the setup (Section 2.1) need to be more precise. The author does not give any initial conditions for the particles, to supplement equation (1). Related to this, I believe that equation (10) is valid only if the system is already in its steady state at time t=0. From this it seems that the angle brackets in (10) etc should represent a steadystate dynamical average. Is this correct? Full details are needed here because there are some subtleties with these kinds of analysis, see the following two points.
Thank you for the question. As mentioned by the referee, the boundary conditions were not explicitly stated in the previous manuscript.
In this study, we set the boundary condition $\tilde{u}_q(\pm\infty)=0$ to simplify the Fourier transformation. We also assume that at finite time, $\tilde{u}_q(t)$ reaches a steady state independent from the boundary condition. To clarify this point, we have added the following sentence to the paragraph above equation (8) in the revised manuscript.
"We impose the boundary condition $\tilde{u}_q(\pm \infty)$ and assume that $\tilde{u}_q(t)$ reaches a steady state independent from the boundary condition at finite $t$.”
Also, we added the following sentence below equation (6).
“Note that $\tilde{\xi}_{q=0}(t)=0$ in the centerofmass frame.”
The author also writes that the "equilibrium position" of particle j is $R_j=ja$ but since the boundaries are periodic, taking $R_j=(jc)a$ would be an equally appropriate choice for any $c$ between 0 and 1. If the system supports longranged positional order, this means that the particles will converge to "equilibrium" positions that depend on the initialization of the system, hence $c$ also depends on the initial condition. So it is not acceptable to omit the details of initialization. (If $c$ is not zero, this also gives a phase to the order parameter $R$ in (12).)
Thank you for pointing that out. To clarify the initial condition, we modify the sentence above equation (2) as follows:
“Let $a$ be the lattice constant, $R_j=ja$ be the equilibrium position of the $j$th particle, and $u_j=x_jR_j$ be the displacement from the equilibrium position. The dynamical equation for $u_j$is then written as”
→
“Let $a$ be the lattice constant. Without loss of generality, we can assume that the equilibrium position of the $j$th particle is given by $R_j=ja$. The dynamical equation for the displacement $u_j=x_jR_j$ is then written as”
In addition, the author takes $q=2\pi k/(Na)$ with $k=1,2,\dots,N$ so that q is always positive, but then equations like (7) include $\delta_{q,q'}$, which only makes sense if they allow $q<0$, for example by indexing $k$ symmetrically from $N/2$ to $N/21$.
Thank you for pointing that out. Following the suggestion, we modified the sentence below equation (3) as follows:
" where $q\in \left{\frac{2\pi k}{Na} \right}_{k=1,\cdots,N}$.”
→
“where $q\in \left{\frac{2\pi k}{Na} \right}_{k=N/2,\cdots, N/21}$ if $N$ is even, and $q\in \left{\frac{2\pi k}{Na} \right}_{k=(N1)/2,\cdots, (N1)/2}$ if $N$ is odd.”
We also modified the sentence below equation (11) as follows:
“where we have replaced the summation for $q\in \left{\frac{2\pi k}{Na}\right}_{k=1,\cdots, N}$ with an integral for $q\in (0,2\pi/a]$.”
→
“where we have replaced the summation for $q$ with an integral for $q\in (\pi/a,\pi/a)$.”
Accordingly, we modified the integration range in Eqs. (11), (16), (17), (23), and (33) from $\int_0^{2\pi/a}$ to $\int_{\pi/a}^{\pi/a}$.
Whichever choice is made, there is an underlying physical question about the centerofmass mode (either $k=N$ in the author's convention or $k=0$ in the symmetric convention ). Since this mode has no restoring force ($\lambda_q=0$), it may be that the system never converges to any steady state, in which case the steady state averages mentioned above are not welldefined. A related issue is that for the simplest possible case of equilibrium dynamics with finite $N$, the scaling of the MSD as $t^{1/2}$ will cross over at large enough times to diffusive scaling, MSD $\sim t/N$, because the center of mass of the particles will undergo free diffusion. I think that several of these problems might be avoided by working in the centerofmass frame, which should be equivalent to setting $D_q=0$ for the centerofmass mode, from the start. There would be other options too.
Thank you for pointing that out. Following the suggestion, we decided to use the centerofmass frame. To clarify this point, we have added the following sentence to the paragraph below equation (1) in the revised manuscript.
“We investigate the model in the centerofmass frame, which is, in practice, equivalent to replacing the noise in (1) as $\xi_j\to \xi_j\sum_{k=1}^N \xi_k/N$.”
1 Clarify the setup of the problem, as described in the report above.
We clarified the setup in the revised manuscript as detailed above.
2 What is the size of the terms that are neglected in the first approximate equality in equation (13)? Under what circumstances is it consistent to truncate this expansion? (The required assumption seems to be that $qu_j$ is small compared to unity. Even if q is small, could it be that $u_j$ is very large so the expansion breaks? This step may be ok in systems with longranged positional order it should be justified carefully.)
Thank you for the suggestion. As mentioned by the referee, the expansion by $qu_j$ may fail in the fluid phase.
To clarify this point, we added the following sentence below equation (13):
“Note that the expansion (13) is verified only in the crystal phase. In the fluid phase, even for small $q$, $u_j$ can be very large so the expansion by $qu_j$ breaks.”
I am aware that the expansion (13) can reproduce the correct result for the classical harmonic chain in equilibrium, implying that more rigorous treatments may justify the expansion even in the fluid phase; see, for instance, Appendix. B in [J. Kim and S. Torquato, PRB 97, 054105 (2018)], which demands a bit more cumbersome calculation. Since this article mainly focuses on the connection between hyperuniformity and translational order in the crystal phase, we simply removed the discussions of $S(q)$ in the fluid phase throughout the revised manuscript. The modifications are summarized in “diff.pdf”.
3 at the end of Sec 3.1 the discussion of "hyperuniform in time" is not very clear. I would suggest to compare with a very simple system which is a single particle with position $X_j$, moving as $\dot{X}_j=\xi_j$
Thank you for the suggestion. The suggested model has been studied in a previous work in the context of anomalous diffusion. A simple scaling argument leads to ${\rm MSD}\sim t^{1+2\theta}$. To clarify this point, we added the following sentence at the end of Sec. 3.1:
“The powerlow spectrum (15) has been often used in the context of anomalous diffusion[38]. A single free particle driven by the noise, $\dot{x}=\xi(t)$, exhibits ${\rm MSD}\propto t^{1+2\theta}$ for large time $t$.”
4 in eq(15), surely the sec in the denominator would be more appropriate as a cosine in the numerator.
Thank you for pointing that out. We modify the equation accordingly.
5 please give a reference (or more detailed justification) for the inequality in (45).
Thank you for the suggestion. In the revised manuscript, we added the following footnote to justify equation(45):
“To prove the inequality $\cos(x)\geq 1x^2/2$, it is convenient to introduce an auxiliary function $f(x)=\cos(x)(1x^2/2)$. Since $f(x)$ is an even function, it is sufficient to show $f(x)\geq 0$ for $x\geq 0$, which follows from $f(0)=0$ and $f'(x)=\sin(x)+x\geq 0$ for $x\geq 0$.
6 it is unfortunate to use $R_j$ for the "equilibrium" position of particle j and then later to use R for the order parameter in (12).
Thank you for the suggestion. We decided to use $O$ for the order parameter in the revised manuscript.
7 In the Introduction. Please clarify the following points. First: the MerminWagner theorem only applies if interactions are shortranged.
We modified the corresponding sentence in the introduction as follows:
“However, as proved by Mermin and Wagner, the longrange order cannot exist in one and two dimensions in equilibrium”
→
“However, as proved by Mermin and Wagner, the longrange order cannot exist in one and two dimensions in equilibrium if interactions are shortranged”
Second: the fact that particles cannot bypass one another is only true if the interactions are strong enough.
We modified the first sentence in the introduction as follows:
“In onedimensional manyparticle systems, the particles cannot bypass one another.”
→
“In onedimensional manyparticle systems, the particles cannot bypass one another if the interactions are strong enough.”
Third: it is not necessarily true that fluctuating hydrodynamics is limited to low densities, see for example the macroscopic fluctuation theory of Bertini et al, which is valid at all densities in lattice models.
Thank you for the suggestion. We modified the corresponding sentence as follows:
“So far the most of theoretical studies of hyperuniformity far from equilibrium are based on the fluctuating hydrodynamics, which can be justified only at sufficiently low densities and can not be applied in the crystal phase[…]”
→
“So far, most of the theoretical studies of hyperuniformity far from equilibrium have been conducted on low densities much below the crystal phase[…]”
Author: Harukuni Ikeda on 20240712 [id 4616]
(in reply to Report 4 on 20240105)Dear Referee, Thank you very much for reviewing my paper. I am very happy to see that the Referee has a high opinion of my work. I have carefully studied your reports and have revised the paper. I believe that I have fully answered Referee’s questions. In what follows, I will reply to your comments separately and explain major changes made in the revision. All the changes in the revision are highlighted in the file diff.pdf. I hope and believe that you will find the revised version acceptable for publication in SciPost. Sincerely yours,
Harukuni Ikeda
Thank you very much for your careful reading and positive evaluations of our work.
Thank you for pointing that out. Following the suggestion, we decided to use the centerofmass frame. To clarify this point, we have added the following sentence to the paragraph below equation (1) in the revised manuscript.
“We investigate the model in the centerofmass frame, which is, in practice, equivalent to replacing the noise in (1) as $\xi_j\to \xi_j\sum_{k=1}^N \xi_k/N$.”
To simplify the Fourier transform, we impose the boundary condition $\tilde{u}_q(t=\pm \infty)=0$, instead of the initial condition $\tilde{u}_q(t=0)=0$.
Thank you for the suggestion. The discussion in Sec. 4.1. was indeed misleading. The crucial point of Hexner and Levine’s argument is not the conservation of the center of mass but rather that the noise in the continuum limit reduces to a conserving noise $\xi(x)=\partial_x\eta(x)$.
To clarify this point, we decided to use “conserving noise” instead of “centerofmass conserving noise” throughout the revised manuscript.
We also modified the sentence above equation (29) as follows:
“To preserve the center of mass $X\equiv \sum_{j=1}^N x_j$, the noise should satisfy $\sum_{j=1}^N \xi_j=0$.”
→
“A conserving noise in the continuum limit $\xi(x)$ is written as $\xi(x)=\partial_x \eta(x)$, where $\eta(x)$ denotes another white noise.”
Thank you for the suggestions. We modified the typos.
The solid line for $\theta=0.4$ denotes a constant rather than a decreasing line. The appearance of a decreasing trend may be an optical illusion.
Attachment:
diff_enz9YFf.pdf