Anomalous Stochastic Transport of Particles with Self-Reinforcement and Mittag–Leffler Distributed Rest Times
Round 1
Reviewer 1 Report
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Comments for author File: 
Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 2 Report
This manuscript proposes a model that takes into account the effects of self-reinforcement and power-law distribution of rest time, and the effects of them on anomalous transport phenomena are discussed. It is theoretically and numerically shown that sub-diffusion and super-diffusion appear depending on the setting parameters. However, I have some concerns and will wait for the authors' response to the remarks below before recommending the acceptance of this manuscript.
Major remarks:
(1) In this paper, it is claimed that sub-diffusion appears in the long-time limit. On the other hand, in a recent paper reported at PRL ["Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion" by David Hartich and Aljaž Godec: Phys. Rev. Lett. 127, 080601 (2021)] that sub-diffusion in the long time limit is forbidden by the thermodynamic uncertainty relation. Clarify how this contradiction should be understood to be consistent.
(2) I agree with the importance of the phenomenon that the rest time distribution follows a power law, but why does it have to be the Mittag-Leffler distribution? Although the authors cite Reference 2 in page 3 of the manuscript, but explain that can be understood by reading this manuscript alone.
(3) Explain the physical meaning of beta and tau_0 defined in the integral escape rate. These parameters are compared with different values in Figure 2-4, and are necessary to understand the results clearly.
(4) The authors propose eq.(4) for self-reinforcement. I think this formula is intuitive, as the authors say, but I wonder if there are any examples that correspond to actual phenomena. If you have such experimental systems, cite them.
(5) Is it possible to analytically show the super-diffusion in the intermediate time scale seen for small tau_0 from the model equations?
(6) Does the result that the model always leads to sub-diffusion in the long time limit hold for cases other than w_3 = 1/3?
Minor remarks:
・Why bacteria and dogs in Fig. 1? Can't it be one or the other, or just a particle?
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Report on the manuscript “Anomalous stochastic transport of particles with self-reinforcement and Mittag-Leffler distributed rest time” by D. Han, D.V. Alexandrov, A. Gavrilova, and S. Fedotov.
In the manuscript, there is considered a diffusion model in which the diffusing particle can be in three states: moving forward, backward, or resting. A molecule can stay in the first two states in the time that has an exponential distribution, and the time in the third state has a distribution expressed by the Mittag-Leffler (ML) function. Self-reinforcement of the molecule is defined as the favoring of the direction of the last molecule jump. This process is described by the hyperbolic fractional diffusion equation. The authors have derived this equation, and on its basis they have derived an equation describing the temporal evolution of the mean square displacement (MSD) of the molecule. Solutions to these equations have been made by means of the Monte Carlo method, analytical solutions have been obtained within a long time limit. The authors have concluded that in the considered process it is possible to move from superdiffusion to subdiffusion.
The assumptions for the model considered in the paper are well presented. The organization of the manuscript is good. The presented results are interesting and deserve to be published. However, some considerations are too brief. In order for the reader to have a chance to become acquainted with the details of the considerations, an additional description should be added at some points or, if possible, other articles in which similar considerations are conducted should be cited. These comments apply to the following issues:
- The authors wrote (page 8): “However, in this second example, the fractional diffusion limit exists, which means that the system can be approximated accurately by the fractional diffusion equation (…)“. What does "fractional diffusion limit" mean?
- How was Eq. (30) derived? Why could equation (30) be derived for the process described in Sec. 3.2, and could not be derived for the process considered in Sec. 3.1?
- The authors wrote (page 8, line 116-118) “(…) the first example is inherently different because there is no accurate fractional diffusion equation to approximate the system and so the second moment should not be interpreted in diffusive, subdiffusive or superdiffusive terms”. This sentence causes some confusion, as the considerations based on Figures 2 and 3 show that the identification of subdiffusion and superdiffusion is based on the temporal evolution of MSD. Why the process described in Sec. 3.1, for which the time evolution of MSD is expressed by equation (22), cannot be interpreted as anomalous diffusion? So what is the definition of anomalous diffusion adopted by the authors? Can this definition be related to the continuous time random walk model?
- The authors note that in the process under consideration there may be superdiffusion in the initial time. How does the presented model interpret this process, since the waiting time for the molecule to jump is described by the heavy-tailed ML function, which is typical for subdiffusion?
Minor remark: (page 3) the assumption that $w_3=1/3$ is not clearly explained.
In my opinion, the manuscript can be published if the above comments are included in the new version of the manuscript.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I believe that the authors have adequately addressed my comments and would like to recommend the publication of this manuscript on Fractal Fract.
I have just a minor comment:
For ref.59, It's better that PRL information is referred instead of arXiv.
