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Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system

by Anupam Kundu

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Anupam Kundu
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-22
Date submitted: 2023-04-24 08:16
Submitted by: Kundu, Anupam
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We study transport in a one-dimensional lattice system with two conserved quantities -- 'volume' and energy. Considering a slowly evolving local equilibrium state that is slightly deviated from an underlying global equilibrium, we estimate the correction to the local equilibrium distribution. This correction arises mainly through the space-time correlations of some local currents. In the continuum limit, we show that the local equilibrium distribution along with the correction yields drift-diffusion equation for the 'volume' and super-diffusion equation for the energy in the linear response regime as macroscopic hydrodynamics as one would obtain from non-linear fluctuating hydrodynamic theory. We find explicit expression of the super-diffusion equation. Further, we find diffusive correction to the super-diffusive evolution. Such a correction allows us to study a crossover from diffusive to anomalous transport. We demonstrate this crossover numerically through the spreading of an initially localized heat pulse in equilibrium as well as through the system size scaling of the stationary current in non-equilibrium steady state.

Published as SciPost Phys. 15, 038 (2023)

Author comments upon resubmission

Distinguished Editors,
Thank you for your email, sent March 31, 2023, regarding the submission titled “Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system.”. We also wish to thank the Reviewer for the report. We also thank the journal team for processing our paper and sending it to the referees.
The referees have made positive and constructive comments which have helped us improve the manuscript. In particular the Reviewer-1 has said the following “A microscopic derivation of superdiffusion equations, even in a simple model, is a challenging problem. In this manuscript, the author manages to make progress in the derivation using only a limited number of physically motivated assumptions and without invoking more ”heuristic” arguments up to Eq. (36). Furthermore, the treatment presented here also allows for the study of subleading diffusion corrections. Therefore I think the manuscript matches the acceptance criteria of SciPost Physics and I would recommend its publication after some minor revisions.”
The Reviewer-2 has made the following comment “It is an interesting cal- culation and it deserves publication.”.
The reviewer-3 has made the following comment '' - a nice and rather precise derivation of linearised hydrodynamic equations, showing diffusion and super-diffusion. I find the calculation, in this specific model, very illuminating and helpful to understand linearied hydrodynamics more generally. "
As advised I have resubmitted a revised version of scipost 202210 00087v1 that addresses all the points raised by the reviewers. Separate detailed responses to individual reviewers are also uploaded.

I hope the revised version will get accepted for publication soon.

Anupam Kundu

List of changes

List of Changes:
1. Clarified the meaning of the exchange term in Eq. (1)
2. Added more discussions on previous derivations of the super-diffusive equation (page2, 2nd paragraph).
3. Clarified the discussion on crossover scale in the introduction (page2, last paragraph along with new citations 20-24).
4. Clarified the precise approximations used for the derivation in section 3 (after Eq.~20 and sec.~3.2)
5. The figure 2 is updated with more convincing simulation data along with comparison
between diffusive and super-diffusive scaling collapse. Also the description of the figure and the
related discussions have been improved (2nd paragraph after Eq.~ 53).
6. Added more discussion on the advantage of our method in the context of the HCVE model and discussed
the connection with the NLFHD theory ( Discussion after Eq.~36).
7. Moved the discussion about the continuum limit to the beginning of sec. 3.3.
8. Other small changes related to typos and grammatical mistakes

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