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

M Chase, Thomas John McKetterick, Luca Giuggioli, VM Kenkre

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
289 Downloads (Pure)


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-Leer 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-Markoan 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 Markoan counterparts at small (large) values of the difference between the two times.
Original languageEnglish
Article number87
Number of pages15
JournalEuropean Physical Journal B
Issue number4
Early online date6 Apr 2016
Publication statusPublished - Apr 2016


  • Statistical and Nonlinear Physics


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