TY - JOUR
T1 - Fractional kinetics in Kac-Zwanzig heat bath models
AU - Kupferman, Raz
PY - 2004/1
Y1 - 2004/1
N2 - 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.
AB - 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.
KW - Fractional diffusion
KW - Hamiltonian systems
KW - Heat bath
KW - Markovian approximation
KW - Stochastic differential equations
KW - Weak convergence
UR - http://www.scopus.com/inward/record.url?scp=0347975171&partnerID=8YFLogxK
U2 - 10.1023/b:joss.0000003113.22621.f0
DO - 10.1023/b:joss.0000003113.22621.f0
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AN - SCOPUS:0347975171
SN - 0022-4715
VL - 114
SP - 291
EP - 326
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 1-2
ER -