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"Diffusing diffusivity": A model of "anomalous yet Brownian" diffusion

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Wang et al.
[PNAS 106 (2009) 15160] have found that in several systems, the linear time
dependence of mean-square displacement (MSD) of diffusing colloidal particles,
typical of normal diffusion, is accompanied by a non-Gaussian displacement
distribution (DD), with roughly exponential tails at short times, a situation
termed “anomalous yet Brownian” diffusion. We point out that lack of “direction
memory” in the particle trajectory (a jump in a particular direction does not
change the probability of subsequent jumps in that direction) is sufficient for
a strictly linear MSD (assuming that the system is pre-equilibrated), but if at
the same time there is “diffusivity memory” (a particle diffusing faster than
average is likely to keep diffusing faster for some time), the DD will be
non-Gaussian at short times. A gradual change in diffusivity can be due to the
environment of the particle changing slowly on its own, the particle moving
between different environments, or both. In our model, this is represented by
the particle diffusivity itself undergoing a (perhaps biased) random walk
(“diffusing diffusivity”). Roughly exponential tails of the DD, as in
experiment, are observed in several variants of the model.