Anomalous Diffusion in Random Dynamical Systems

Brownian motion has long been the standard paradigm for modelling random, diffusive motion, such as the haphazard movement of a dust particle floating in a fluid. This is considered to be “normal” diffusion, in which the mean square particle displacement – calculated as an average over an ensemble of particles – increases linearly in the long-time limit. Yet many examples of diffusion in nature, or in areas such as economics and finance, turn out to deviate from Brownian motion, exhibiting “anomalous” diffusion and a mean square displacement which does not grow linearly. For example, objects diffusing in crowded environments, such as organelles within biological cells or molecules passing through channels in nanoporous materials, exhibit sub-diffusion – the mean square displacement does not increase linearly in time, but more slowly, with an exponent less than one. In contrast, light propagating through disordered matter and many animals when foraging for food exhibit super-diffusion – faster than linear growth of the mean square displacement, with an exponent greater than one.
Experimental data showing anomalous diffusion can be modelled using tools from stochastic theory, including continuous time random walks, Lévy walks, generalised Langevin equations or fractional Fokker-Planck equations. Researchers using such models intentionally insert mechanisms to generate anomalous behaviour by choosing non-Gaussian probability distributions or power-law memory kernels. Less work has pursued the origin of anomalous diffusion in deterministic dynamical systems, and currently only a few mechanisms are known – the stickiness of orbits near KAM tori in Hamiltonian systems, marginally unstable fixed points in dissipative Pomeau-Manneville-like maps and non-trivial topologies exhibited by polygonal billiards.
In a recent paper, LML External Fellows Yuzuru Sato and Rainer Klages have studied a simple hybrid system at the interface between deterministic and stochastic dynamics, and shown that it offers another generic source for anomalous diffusion. Their model combines two deterministic dynamical systems, with the dynamics being driven by one system or the other based on a random choice at each time step. Under one dynamic, the motion is normal diffusion, while the other tends to drive the dynamics to fixed points. As they demonstrate, adjusting between 0 and 1 the probability p for choosing the diffusive dynamic leads to a rich transition in the observed behaviour. For p < 2/3, the system exhibits normal diffusive behaviour for long times, while for p > 2/3, all trajectories eventually get trapped. For p very close to 2/3, the dynamics shows an anomalous behaviour for a long transient time, before converging to one or the other dynamics.
As the authors note, similar models have been used to understand the convection of particles in flowing fluids including fractal clustering and path coalescence, the localisation transition in continuum percolation problems, intermittency in nonlinear electronic circuits and random attractors in stochastic climate dynamics. They expect that the study of such simplified models should stimulate useful applications to related problems.
The paper is available at

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