Fractional kinetics in Kac-Zwanzig heat bath models

Raz Kupferman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

113 Scopus citations

Abstract

We study a variant of the Kac-Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As n → ∞ the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro-differential equation-a generalized Langevin equation (GLE) with power-law memory kernel and driven by 1/fα-noise. The limiting process exhibits fractional sub-diffusive behaviour. We further consider the approximation of non-Markovian processes by higher-dimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a so-called fractional Fokker-Planck equation in the present context. All results are supported by direct numerical experiments.

Original languageAmerican English
Pages (from-to)291-326
Number of pages36
JournalJournal of Statistical Physics
Volume114
Issue number1-2
DOIs
StatePublished - Jan 2004

Keywords

  • Fractional diffusion
  • Hamiltonian systems
  • Heat bath
  • Markovian approximation
  • Stochastic differential equations
  • Weak convergence

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