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Home page > Research topics > Slow dynamics in complex out-of-equilibrium systems: relaxation time distributions in anomalous diffusion.

Slow dynamics in complex out-of-equilibrium systems: relaxation time distributions in anomalous diffusion

MSC : Noëlle Pottier


As well-known, the generalized Langevin equation with a memory kernel decreasing as an inverse power law of time (\gamma(t)\propto t^{-\delta} with 0<\delta<2) describes the motion of an anomalously diffusing particle whose displacement variance varies at large times as \Delta x^2(t)\sim t^\delta. We now focus the interest on some new aspects of the dynamics, successively considering the memory kernel, the relaxation of the particle mean velocity, and the diffusion function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although very popular, the concept of relaxation time distribution cannot be associated with any time- decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonous function).

Technically, we use a memory kernel decreasing as a Mittag-Leffler function (the Mittag—Leffler functions interpolate between a stretched or compressed exponential behaviour at short times and an inverse power law behaviour at large times). We show that, for 0< 
\delta<1 (subdiffusive motion), a relaxation time distribution can be defined for the memory kernel and the diffusion function, but that this is not the case for the particle velocity. The situation is opposite when 1<\delta<2 (superdiffusion).


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POTTIER Noëlle

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