By M. Hamon
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Extra resources for Aïkido. Une tradition, un art, un sport French
Specifically, superdiffusion is observed for random walks with heavytailed jump length distributions and subdiffusion for heavy-tailed waiting time distributions, see Sect. 4. The latter type of distribution can be caused by “traps” that have an infinite mean waiting time . , [298, 299, 229]. 1 Continuous-Time Random Walks Anomalous diffusion is often modeled by a continuous-time random walk (CTRW) [304, 213, 298, 299, 102], though other approaches have been explored, such as stochastic switching generating superdiffusion , fractional Brownian motion [266, 258, 254], and generalized Langevin equations with a memory kernel [465– 467].
Here I0 is the modified Bessel function, ξ = Dt /τ − x , and N = 4Dτ . 14) also shows explicitly √ that perturbations governed by the telegraph equation spread with a finite speed D/τ , as expected for a damped wave equation. 3), we obtain hyperbolic reaction–diffusion equations (HRDEs): 2 τ ∂ ρ ∂t 2 2 + ∂ρ ∂ ρ = D 2 + F(ρ). 15) This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein–Gordon equation, see for example [165, 355, 366].
While it is fairly straightforward to show that the standard reaction–diffusion equation preserves positivity, the problem is much harder, not to say intractable, for other reaction–transport equations. In this context, a mesoscopic approach has definite merit. If that approach is done correctly and accounts for all reaction and transport events that particles can undergo, then by construction the resulting evolution equation preserves positivity and represents a valid reaction– transport equation.
Aïkido. Une tradition, un art, un sport French by M. Hamon