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

by Antonio Piscitelli, Massimo Pica Ciamarra

Submission summary

As Contributors: Massimo Pica Ciamarra · Antonio Piscitelli
Arxiv Link: (pdf)
Date accepted: 2017-08-26
Date submitted: 2017-08-18 02:00
Submitted by: Piscitelli, Antonio
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We analyze the classical problem of the stochastic dynamics of a particle confined in a periodic potential, through the so called Il'in and Khasminskii model, with a novel semi-analytical approach. 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

Published as SciPost Phys. 3, 018 (2017)

List of changes

In the revised version of the manuscript :

-In Sec.2 we introduced the Il'in Kashminskii equation of
motion when reviewing previous results. We specified more
precisely the model

-We eliminated the acronym VhD for Van Hove, substituting
with the explicit expression, and reduced the number of acronyms to
improve clarity

- We eliminated the inset in panel c of Fig 7. We have also modified
panels (d-f) of Fig. 7 to improve their readability

-We modified the first sentence in the abstract and the first sentence of Sec. IIIb

-We corrected the label in Fig. 2a

-We corrected a sentence in Sec. 5a and introduced a comment at the end of Sec. 4

-We corrected Eq.20 and introduced Eq.19

-We corrected few typos

Submission & Refereeing History

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Resubmission 1705.08083v2 on 18 August 2017

Reports on this Submission

Anonymous Report 1 on 2017-8-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1705.08083v2, delivered 2017-08-18, doi: 10.21468/SciPost.Report.214


The authors made changes suggested in the first report.

  • validity: -
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