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Continuous Time Random Walk: Difference between revisions
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→The Master Equation
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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. | 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. | |||
In standard CTRW theory, it is customary to assume that the single particle step distribution is ''separable'', i.e., that ''Δx'' is independent from ''Δt'', so that | |||
: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t) \psi(\Delta t; x,t)</math> | |||
In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''ψ'' do not depend on ''x'' and ''t''). However, recently it was shown that a Master Equation can also be derived in the case that ''p'' depends on ''x'' and ''t''. | |||
<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. Van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', Phys. Plasmas '''11''', 5 (2004) 2272]</ref> | <ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. Van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', Phys. Plasmas '''11''', 5 (2004) 2272]</ref> | ||
This | This significant extension of the standard CTRW model has led to the development of a model with very interesting properties from the point of view of plasma transport (see the cited reference). | ||
== Fractional Differential Equations == | == Fractional Differential Equations == | ||