Abstract
The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.
Highlights
Diffusion, the stochastic motion of a tracer particle, was beautifully described by Brown in his study of pollen granules and a multitude of other molecules[1]
A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells
The motion of the lipids is Gaussian for all cases and best described as viscoelastic diffusion governed by the generalised Langevin equation (22) fuelled by power-law noise [22, 24, 25].+ As can be seen in figure 9 the mean squared displacement (MSD) of the liquid disordered lipid systems exhibits a clear crossover from subdiffusion to normal diffusion at roughly 10 nsec, the typical crossover time scale
Summary
The stochastic motion of a tracer particle, was beautifully described by Brown in his study of pollen granules and a multitude of other molecules (microscopic particles). The quantitative description of this anomalous-to-normal crossover is the topic of this paper For both the subdiffusive and superdiffusive situations we include a maximum correlation time of the driving noise and provide exact solutions for the MSD in the case of hard, exponential and power-law truncation, so-called tempering, that can be applied in the analysis of experimental or simulations data.
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