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Continuous Time Random Walk: Difference between revisions
→The Master Equation
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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 ''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''. | 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'', while ''ψ'' 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 == | ||