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Harmonic chain far from equilibrium: single-file diffusion, long-range order, and hyperuniformity

by Harukuni Ikeda

Submission summary

Authors (as registered SciPost users): Harukuni Ikeda
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-11-09 20:17
Submitted by: Ikeda, Harukuni
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


In one dimension, particles can not bypass each other. As a consequence, the mean-squared displacement (MSD) in equilibrium shows sub-diffusion ${\rm MSD}(t)\sim t^{1/2}$, instead of normal diffusion ${\rm MSD}(t)\sim t$. This phenomenon is the so-called single-file diffusion. Here, we investigate how the above equilibrium behaviors are modified far from equilibrium. In particular, we want to uncover what kind of non-equilibrium driving force can suppress diffusion and achieve the long-range crystalline order in one dimension, which is prohibited by the Mermin-Wagner 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) center-of-mass 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 long-range crystalline order in one dimension and yield hyperuniformity of the density fluctuations.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 4 on 2024-1-5 (Invited Report)


1-gives a broad assessment of the effects of non-equilibrium noise inspired by a range of physical situations
2-calculations are straightforward but set out clearly
3-interesting concluding discussion about relation to model A/B dynamics


1-in most scenarios considered here, hyperuniformity - one of the key aspects being studied - is "inherited" more or less directly from properties of the assumed noise
2-treatment of center-of-mass mode and implicit steady steady assumptions not very clear


The paper studies the effects of various types of non-equilibrium 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 mean-square displacement (MSD), and its long-time 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 non-equilibrium situations considered in the literature.

Requested changes

1-The treatment of the centre-of-mass 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 center-of-mass frame, i.e. set $\tilde{u}_{q=0}(t=0)=0$ and $D_{q=0}(\omega)=0$.
2-I found the noise in Sec. 4 not as well motivated as the others - clearly there are many ways to have center-of-mass conserving noise other than the gradient-type noise considered here; knocking out the center-of-mass 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.
3-The 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
power-low -> power-law
deriving forces -> driving forces
symmetry braking -> symmetry breaking
4-In Fig. 1 the solid line for $\theta=-0.4$ looks like it is slightly decreasing, which physically it shouldn’t - please check

  • validity: high
  • significance: good
  • originality: good
  • clarity: top
  • formatting: perfect
  • grammar: good

Anonymous Report 3 on 2023-12-24 (Invited 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 large-scale density fluctuations. The author treated four types of active noises (temporally correlated noise, center-of-mass conserving noise, periodic driving, and periodic deformation) and derived the conditions under which the long-range 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 zero-wave 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 center-of-mass conserving noise in a self-consistent manner.

  • validity: good
  • significance: high
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 2 on 2023-12-22 (Invited Report)


This paper describes three situations in which a one-dimensional system of particles unable to overcome one another develop either long range order, or long-range 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.


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.



  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

Anonymous Report 1 on 2023-12-22 (Invited Report)


This manuscript describes some interesting theoretical results for a one-dimensional 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 long-ranged positional order survives the introduction of noise, and what is the nature of large-scale 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 steady-state 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 $R_j=(j-c)a$ would be an equally appropriate choice for any $c$ between $0$ and $1$. If the system supports long-ranged 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 $N/2-1$. Whichever choice is made, there is an underlying physical question about the center-of-mass 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 well-defined. 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 center-of-mass frame, which should be equivalent to setting $D_q=0$ for the center-of-mass 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 long-ranged 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 Mermin-Wagner theorem only applies if interactions are short-ranged. 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.

  • validity: ok
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: excellent

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