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/f α-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.
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Kupferman, R. Fractional Kinetics in Kac–Zwanzig Heat Bath Models. Journal of Statistical Physics 114, 291–326 (2004). https://doi.org/10.1023/B:JOSS.0000003113.22621.f0
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DOI: https://doi.org/10.1023/B:JOSS.0000003113.22621.f0