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Coexistence of coarsening and mean field relaxation in the long-range Ising chain

by Federico Corberi, Alessandro Iannone, Manoj Kumar, Eugenio Lippiello, Paolo Politi

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Submission summary

Authors (as registered SciPost users): Federico Corberi · Manoj Kumar · Paolo Politi
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-02-17 09:58
Submitted by: Corberi, Federico
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance $r$ decaying as $r^{-\alpha}$. For $\alpha =0$, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with $\alpha >1$ there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. $0<\alpha <1$, we show that the system shows both features, with probability $P_\alpha (N)$ of having the latter one, with the different limiting behaviours $\lim_{N\to \infty}P_\alpha (N)=0$ (at fixed $\alpha<1$) and $\lim_{\alpha \to 1}P_\alpha (N)=1$ (at fixed finite $N$). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry and we argue that the growth law of the size $L$ of the coarsening domains is $L(t)\sim t^{1/\alpha}$.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2021-3-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2102.08217v1, delivered 2021-03-26, doi: 10.21468/SciPost.Report.2732


1. This paper deals with a very interesting subject that seems not to have been addressed previously, namely, the dynamics of a long-range-interacting one-dimensional Ising model after a temperature quench, when the interactions are "strongly" long-ranged (i.e., the system is not additive).
2. The work reported in the paper is sound and the results are very interesting and in some cases also unexpected: in particular, the coexistence of coarsening and relaxation for the same system (in the sense carefully explained in the paper) is, to the best of my knowledge, a phenomenon never observed before.
3. The paper is well organized and contains all the information needed to understand the work done (and check the results).
4. The theoretical reasoning and the numerical simulations are clearly explained.
5. This paper may suggest further interesting work along the same direction.


1. While the more technical parts of the paper are very clear and I expect can be followed even by interested researchers not actively working in the specific field, the presentation of the model is not as clear as it could be, in my opinion (see Requested changes).
2. There is room to improve the quality of the presentation in the Introduction (see Requested changes).
3. References to the long-range-interacting systems in general and related issues might be expanded (maybe even more than what is explicitly suggested in the Requested changes).


In my opinion, this paper is a relevant addition to the field of nonequilibrium statistical physics. I recommend this paper for publication in SciPost, provided the minor issues listed above and below are taken into consideration by the authors.

Requested changes

1. (refers to item 1. of the Weaknesses section) At the beginning of Sec. II, first, a definition of $ J_{ij}$ is given inline in the text, definition that includes the Kac normalization factor (which maybe deserves a line of comment, absent here) and then Eqs. (1) and (2) are presented. However, in my opinion one should reverse the order of presentation, that is, present Eqs. (1) and (2) before and afterwards promote also the definition of the $J_{ij}$ to a numbered equation, for the sake of clarity.
2. The sentence "For intermediate values of $\alpha$, but having $ \alpha> d$, the spatial dimension, some new feature, depending on the specific system at hand, may be determined by the extended interaction, however the basic assumptions of statistical mechanics, which mostly rely on the additivity property, are retained." is not very clear and is very heavy to read: my suggestion is to reformulate it.
3. In addition to Ref. [1], the book "Physics of long-range interacting systems" by A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo (Oxford, 2014) should be cited.
4. The terminology "weak long range" and "strong long range" is explained at the beginning if the paper. However, being such a terminology not completely agreed upon, the authors should explicitly cite some references where their terminology is introduced and/or discussed.

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

Anonymous Report 2 on 2021-3-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2102.08217v1, delivered 2021-03-26, doi: 10.21468/SciPost.Report.2715


1) The work deals with a long range interacting spin system , a more relevant or general paradigm , compared to the prevalent short range systems, while understanding nonequilibrium kinetics of phase transitions. It would certainly add to the current impetus gained in this direction.

2) Additionally, it ventures into an important and general, albeit almost untouched, aspect of distinguishing the nonequilibrium dynamics of the mean field cases with the non-mean field ones.

3) The manuscript is clearly written without invoking any unnecessary jargon, and the analyses seem to be easily adaptable for such studies in other scenarios.


1) The manuscript lacks enough mention of the previously and currently published works in this directions. Given the importance of studying long range systems, highlighting such works would be helpful to expand the reach of their own work.

2) As pointed out by Referee1 there are few typos which should be taken care of.


This is a nice piece of work concerning an important problem ( at least in the broad field of theoretical and computational physics) in a comprehensive manner. Specially intriguing is the observation of the coexistence of mean-field and coarsening behavior.

Requested changes

1. The scaling of the characteristic length $L(t)$ of the coarsening is mentioned in the abstract but has never been discussed explicitly in the text. At least I do not find any numerical data supporting it. Of course, $\tau \sim N^z$ is shown with $z=\alpha$. However, one should keep in mind that $z=1/\alpha$ is not true for all coarsening systems, for example in fluids.

2. No unit of time is defined for the numerical results.

3. Eq. (9) is the heart of their theoretical finding. Can the authors comment on its validity if one fixes the initial condition to m(0)=0 ?

4. In Fig.2 and its caption the probability is written as $P(N,t)$ whereas in the text I see the order of $N$ and $t$ is changed as $P(t,N)$. It is better to stick to one.

5. The pre-asymptotic correction should be vanishing with increase of N. However, having a careful look at the data in Fig.3 (a) reveals the following:
With the reasonable assumption that the $N=10^5$ data in Fig. 3(a) is the closest one representing the master curve, one can see that the deviation of the data for $N< 10^5$ is not systematic. An explanation is required.

6. In the conclusion: Please provide references for the study of the nearest neighbor Ising model regarding universality of the growth exponent at different $T$.

7. The author should exclusively comment on the growth exponent for $T<T_c$ in $d=2$ ? They must be aware of the seminal analytical works by Bray and Rutenberg in this regard and also if there exists any numerical evidence supporting that.

8. What is the value of $T_c $ for $\alpha=0.7$ ? The authors claim that the data for $T=T_c/2$ presented in the inset of Fig. 3 is $N$ independent for large $N$ which is not so convincing. Is this simply finite-size effects ?

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

Anonymous Report 1 on 2021-3-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2102.08217v1, delivered 2021-03-16, doi: 10.21468/SciPost.Report.2709


1. The authors investigate non-equilibrium relaxation of the one-dimensional Ising chain in the strong long-range regime. The dynamical properties in this regime were not well understood previously.

2. This regime exhibits peculiar dynamic critical properties, having features of both the mean-field regime and the short-range regime.

3. Numerical simulations in finite systems allow to compute the time-dependent probability that a configuration contains domains. The size- and time-dependence of this probability are studied systematically and the validity of a postulated scaling form is verified.

4. The paper is well written, and compelling data are presented.


1. There are only a few typos that the authors should correct: "the WORLD domain does not refer to spin domains" (page 3); "Montecarlo" -> "Monte Carlo" (page 3)


This is an interesting paper that investigates relaxation processes for a hitherto poorly studied case, namely the Ising chain in the strong long-range regime. The obtained results are novel and provide insights into a non-equilibrium systems that displays a combination of features of well studied limiting regimes. I recommend publication of this paper in SciPost Physics.

Requested changes

1. Correct the few typos.

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

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