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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. | This equation is a (Generalized) 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 | 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 | ||
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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 Master Equation can be written in the form | The Generalized Master Equation 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> | ||
While the Master Equation provides a very general description of transport, it does not allow a straightforward treatment of boundary conditions as with standard differential equations (except in special cases | While the Generalized Master Equation provides a very general description of transport, it does not allow a straightforward treatment of boundary conditions as with standard differential equations (except in special cases | ||
<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>). Thus, the final (quasi) steady state of the system is a function of the balance between sources and sinks. | <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>). Thus, the final (quasi) steady state of the system is a function of the balance between sources and sinks. | ||