Levy flight processes fractional dynamics

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

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