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Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

  • Matthew Chase
  • Tom J. McKetterick
  • Luca Giuggioli
  • V. M. Kenkre
Regular Article

Abstract

Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steady-state depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.

Keywords

Statistical and Nonlinear Physics 

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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Matthew Chase
    • 1
  • Tom J. McKetterick
    • 2
    • 3
  • Luca Giuggioli
    • 2
    • 3
    • 4
  • V. M. Kenkre
    • 1
  1. 1.Consortium of the Americas for Interdisciplinary Science and the Department of Physics and Astronomy, University of New MexicoAlbuquerqueUSA
  2. 2.Bristol Centre for Complexity Sciences, University of BristolBristolUK
  3. 3.Department of Engineering MathematicsUniversity of BristolBristolUK
  4. 4.School of Biological Sciences, University of BristolBristolUK

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