Levy flight model

Violation of (/) can be obtained in the so-called Levy flight model [20]. The simplest instance is the discrete (in time) one-dimensional case, where the particle position x(t + 1) at the time t + 1 is obtained from x(t) as follows ... 

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]... 

B. Bergersen, Z. Racz. Dynamical generation of long-range interactions Random Levy flights in the kinetic Ising and spherical models. Phys Rev Lett 67 3047-3050, 1991. 

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. 

The Levy walk is physically more plausible than the Levy flight. How to derive the Levy walk from a Liouville approach of the kind described in Section III Here, we illustrate a path explored some years ago, to establish a connection between GME and this kind of superdiffusion [49,50]. We assume that there exists a waiting time distribution v /(x), prescribed, for instance, by the dynamic model illustrated in Section V. This function corresponds to a distribution of uncorrelated times. We can imagine the ideal experiment of creating the sequence x,, by drawing in succession the numbers of this distribution. Then we create the fluctuating velocity E,(f), according to the procedure illustrated in Section V. 

The main objective of this chapter is to establish the relation between the macroscopic equations like (3.1) and (3.5), the mesoscopic equations (3.2) and (3.3), etc., and the underlying microscopic movement of particles. We will show how to derive mesoscopic reaction-transport equations like (3.2) and (3.3) from microscopic random walk models. In particular, we will discuss the scaling procedures that lead to macroscopic reaction-transport equations. As an example, let us mention that the macroscopic reaction-diffusion equation (3.1) occurs as a result of the convergence of the random microscopic movement of particles to Brownian motion, while the macroscopic fractional equation (3.5) is closely related to the convergence of random walks with heavy-tailed jump PDFs to a-stable random processes or Levy flights. 

CTRWs with short-tailed waiting time PDFs and heavy-tailed jump length PDFs correspond to Levy flights, which model superdiffusive processes, as discussed in Chap. 3. Ldvy flights have been studied extensively [299, 74, 296] and have found... 

A complete understanding of the factors that regulate the magnitude of activity of bed sediment macrofauna is not available. However, some theory and evidence exist that allows the extrapolation of site-specific Du. data to other locales or conditions. In the case of temperature, it is safe to assume the standard correction/t = lE[(r — 20°)/33] where T is °C. This predicts a 2 x increase in Dbs for each lO C rise in temperature. In the case of carbon food source in the bed the Dbs vs burial velocity empirical correlation (Boudreau, 1997) may be used to adjust between site conditions. This relationship assumes the energy available to sustain the macrofauna community is derived from material settling from the water column onto the bed surface. In the case of corrections for population density, the Levy flight-random walk theoretical model (Reible and Mohanty, 2002) result of Dbs where n is... 

Reible, D. and S. Mohanty. 2002. A Levy flight-random walk model for bioturbation. Environ. Toxicol. Chem. 21 875-881. 

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