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Unifying description of the damping regimes of a stochastic particle in a periodic potential

by Antonio Piscitelli, Massimo Pica Ciamarra

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Submission summary

As Contributors: Massimo Pica Ciamarra · Antonio Piscitelli
Arxiv Link: (pdf)
Date submitted: 2017-06-05 02:00
Submitted by: Piscitelli, Antonio
Submitted to: SciPost Physics
Academic field: Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We introduce a physically motivated theoretical approach to investigate the stochastic dynamics of a particle confined in a periodic potential. The particle motion is described through the Il'in Khasminskii model, which is close to the usual Brownian motion and reduces to it in the overdamped limit. Our approach gives access to the transient and the asymptotic dynamics in all damping regimes, which are difficult to investigate in the usual Brownian model. We show that the crossover from the overdamped to the underdamped regime is associated with the loss of a typical time scale and of a typical length scale, as signaled by the divergence of the probability distribution of a certain dynamical event. In the underdamped regime, normal diffusion coexists with a non Gaussian displacement probability distribution for a long transient, as recently observed in a variety of different systems. We rationalize the microscopic physical processes leading to the non-Gaussian behavior, as well as the timescale to recover the Gaussian statistics. The theoretical results are supported by numerical calculations and are compared to those obtained for the Brownian model.

Ontology / Topics

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Brownian motion Damping Il'in-Khasminskii model Stochastic dynamics
Current status:
Has been resubmitted

Submission & Refereeing History

Reports on this Submission

Anonymous Report 1 on 2017-8-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1705.08083v1, delivered 2017-08-13, doi: 10.21468/SciPost.Report.208


1) The paper presents a reasonably clear and complete analysis of the motion of particle in a periodic potential according to a model proposed over 50 years ago by Il'in and Khas'minskii.


1) The model used in the paper is not defined properly. This is especially important since the original paper introducing this model is over 50 years old and may not be easily accessible to all interested researchers.
Specifically, in the ll'in-Khas'minskii model "instantaneous interaction with the heat bath occurs at a constant rate $t_c^{-1}$". Does it mean that the interaction events are equally spaced in time? Or, are they distributed according to a probability distribution? Also, for more mathematically minded readers it would be nice to have an equation of motion for the probability distribution within the ll'in-Khas'minskii model.
2) The standard model to describe stochastic motion (the Brownian motion model) can be derived from a more fundamental (more complete) description of the particle+bath system. While the derivation is approximate, it provides some physical understanding of the assumptions behind this model. It is not clear whether (and how) the ll'in-Khas'minskii model follows from the description of the particle+bath system in terms of the coordinates, positions and interaction of the particle and the particles constituting the bath.
3) The physical interpretation and thus the estimation of the parameters in the Langevin equation describing the standard model is reasonably clear. How could one estimate $t_c$, i.e. the characteristic time of the ll'in-Khas'minskii model?
4) The paper introduces and uses a variety of acronyms. Some of them are relatively non-standard (e.g. VhD). This makes the paper a bit difficult to follow.
5) Some figures (e.g. panels (d-f) of Fig. 7) are difficult to read. The inset in panel c of Fig. 7 is impossible to read in the standard size.


This paper discusses the application of an approach introduced by Il'in and Khas'minskii to the motion of a stochastic particle in a periodic potential. It provides additional details regarding previously described results. It discusses the transition from overdamped to underdamped motion. It shows that according to the Il'in-Khas'minskii model, diffusive mean-squared displacement can coexist with non-Gaussian probability distribution of displacements.
I recommend that the paper is accepted after the authors considered the (relatively minor) changes requested below.

Requested changes

1) I would appreciate some suggestions regarding the applicability of the present model and, in particular, how "the loss of a typical time ... length scale" could be accessed from the analysis of the trajectories. For example, if one had a trajectory, how would one decide whether to describe it using the present model or the Brownian motion model?
2) The model used in the paper should be properly defined.
3) The number of acronyms used should be reduced.
4) Readability of the figures should be improved.
Minor points:
a) The authors write in the Abstract that they "introduce a physically motivated theoretical approach". Rather, they use a previously introduced approach (by Il'in and Khas'minskii) to describe motion in a periodic potential.
b) According to the caption, the full line in Fig. 2a is an empirical fit rather than a prediction and thus should be labeled as such.
c) On p.7 the authors attribute the plateau to "the interaction with the thermal noise and confining barrier". I think the noise itself does not cause the plateau?
d) Some more editing (e.g. removing "Langeving" on p. 1) should be done.

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

Login to report


Antonio Piscitelli  on 2017-08-18  [id 162]

answer to question

We thank the referee for her/his helpful comments and remarks and have
modified the manuscript following her/his suggestions. We provide a
point-to-point response to all remarks in the file attached answer_referee.pdf.
Should the referee found it useful, a pdf version of our manuscript with
all changes highlighted in red is available at