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Continuous Time Random Walk: Difference between revisions
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→The Master Equation
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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 and <ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</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 and <ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>). | ||
The Generalized Master Equation can be written in the form | The Generalized Master Equation (GME) can be written in the form | ||
:<math>\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt'</math> | :<math>\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt'</math> | ||
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:<math>K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x)</math> | :<math>K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x)</math> | ||
The GME is an integro-differential equation, and generalizes the usual (partial differential) equations for transport. | |||
<ref>[http://dx.doi.org/10.1088/1751-8113/41/21/215004 B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', J. Phys. A: Math. Theor. '''41''' (2008) 215004]</ref> | |||
The treatment of boundary conditions in a GME is different from standard differential equations. <ref>[http://dx.doi.org/10.1088/1751-8113/41/21/215004 B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', J. Phys. A: Math. Theor. '''41''' (2008) 215004]</ref> | |||
The final (quasi) steady state of the system is a function of the balance between sources and sinks, rather than of imposed values or gradients at the system boundaries. | |||
== Fractional Differential Equations == | == Fractional Differential Equations == | ||