Continuous Time Random Walk: Difference between revisions

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: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t) \psi(\Delta t; x,t)</math>
: <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 ''&psi;'' 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''.
In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''&psi;'' 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'', while ''&psi;'' depends on ''x'' (but not ''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 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).
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).
The Master Equation can be written in the form
:<math>\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int_{-\infty}^{\infty}{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int_{-\infty}^{\infty}{K(x-x',t-t',x',t')dx'}\right )dt'</math>


== Fractional Differential Equations ==
== Fractional Differential Equations ==

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