Journal of Statistical Physics

, Volume 158, Issue 5, pp 1100–1125 | Cite as

Brownian Motion of a Rayleigh Particle Confined in a Channel: A Generalized Langevin Equation Approach

  • Changho Kim
  • George Em KarniadakisEmail author


We study confined Brownian motion by investigating the memory function of a \(d\)-dimensional hypercube (\(d\ge 2\)), which is subject to a harmonic potential and suspended in an ideal gas confined by two parallel walls. For elastic walls and under the infinite-mass limit, we obtain analytic expressions for the force autocorrelation function and the memory function. The transverse-direction memory function possesses a nonnegative tail decaying like \(t^{-(d-1)}\), from which anomalous diffusion is expected for \(d=2\). For \(d=3\), the position-dependent friction coefficient becomes larger than the unconfined case and the increment is inversely proportional to the square of the distance from the wall. We also perform molecular dynamics simulations with thermal walls and/or a finite-mass hypercube. We observe faster decay due to the thermal wall (\(t^{-3}\) for \(d=2\) and \(t^{-5}\) for \(d=3\) under the fully thermalizing wall) and convergence behaviors of the finite-mass memory function, which are different from the unconfined case.


Memory effects Long-time tail Finite-mass effects Anomalous diffusion Molecular dynamics 



This work was partially supported by the new DOE Center on Mathematics for Mesoscopic Modeling of Materials (CM4) and by the NSF (Grant DMS-1216437). Computations were performed at the IBM BG/Q with computer time provided by an INCITE grant.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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