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Continuous Time Random Walk: Difference between revisions
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In the framework of transport in plasmas, it is believed that the presence of ''trapping regions'' (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to [[Non-diffusive transport|sub-diffusion]], whereas the occurrence of ''streamers'' and profile self-regulation (via [[TJ-II:Turbulence|turbulence]]) may lead to [[Non-diffusive transport|super-diffusion]]. | In the framework of transport in plasmas, it is believed that the presence of ''trapping regions'' (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to [[Non-diffusive transport|sub-diffusion]], whereas the occurrence of ''streamers'' and profile self-regulation (via [[TJ-II:Turbulence|turbulence]]) may lead to [[Non-diffusive transport|super-diffusion]]. | ||
The goal of the CTRW approach is to model the effective transport in the presence of these complex phenomena. | The goal of the CTRW approach is to model the effective transport in the presence of these complex phenomena. | ||
== Starting point == | |||
CTRW theory starts from the single-particle step distribution function (in one dimension) | |||
:<math>\xi(\Delta x, \Delta t; x, t)</math> | |||
giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''Δx'' after waiting a time ''Δt'' > 0. All particles are assumed to be identical. | |||
== The Master Equation == | |||
By making some additional assumptions regarding the nature of this single-particle step distribution, it becomes possible to compute the average behaviour of the system in the limit of infinitely many particles, and to deduce an evolution equation for the particle (probability) density. This equation is a Master Equation. | |||
== Fractional Differential Equations == | |||
For certain specific choices of the single-particle step distribution (even more restricted than the restrictions needed to derive a Master Equation), the resulting evolution equation for the particle density can be written in the form of a Fractional Differential Equation (FDE). | |||
== References == | == References == | ||
<references /> | <references /> | ||