Continuous Time Random Walk - Revision history

Admin at 10:46, 26 January 2023

2023-01-26T10:46:46Z

← Older revision		Revision as of 12:46, 26 January 2023
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	CTRW theory		CTRW theory
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, {{ISBN\|9780750310307}}</ref>
	starts from the single-particle step distribution function (in one dimension)		starts from the single-particle step distribution function (in one dimension)

Admin: Updated reference links

2018-04-03T12:22:54Z

Updated reference links

Admin: Reverted edits by Otihizuv (Talk) to last revision by Admin

2010-11-24T11:21:32Z

Reverted edits by Otihizuv (Talk) to last revision by Admin

Otihizuv at 01:05, 24 November 2010

2010-11-24T01:05:16Z

Admin at 09:21, 11 September 2009

2009-09-11T09:21:04Z

← Older revision		Revision as of 11:21, 11 September 2009
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	== Motivation ==		== Motivation ==

	Interestingly, the absence of local characteristic scales means that ''effective'' transport coefficients (the diffusivity etc.) become dependent on the system size, as is indeed suggested by experimental scaling laws for plasma confinement.		Interestingly, the absence of local characteristic scales means that ''effective'' transport coefficients (the diffusivity etc.) become dependent on the system size, as is indeed suggested by experimental [[Scaling law\|scaling laws]] for plasma confinement.

	In the framework of transport in plasmas, it is believed that the presence of ''trapping regions'' (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to [[Non-diffusive transport\|sub-diffusion]], whereas the occurrence of ''streamers'' and profile self-regulation (via [[TJ-II:Turbulence\|turbulence]]) may lead to [[Non-diffusive transport\|super-diffusion]].		In the framework of transport in plasmas, it is believed that the presence of ''trapping regions'' (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to [[Non-diffusive transport\|sub-diffusion]], whereas the occurrence of ''streamers'' and profile self-regulation (via [[TJ-II:Turbulence\|turbulence]]) may lead to [[Non-diffusive transport\|super-diffusion]].

Admin: /* The Master Equation */

2009-08-13T09:58:25Z

The Master Equation

← Older revision		Revision as of 11:58, 13 August 2009
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	== The Master Equation ==		== The Master Equation ==

	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 suitable additional assumptions regarding the nature of this single-particle step distribution, it is 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 (Generalized) Master Equation.		This equation is a (Generalized) Master Equation.

Admin: /* Fractional Differential Equations */

2009-08-12T10:19:17Z

Fractional Differential Equations

← Older revision		Revision as of 12:19, 12 August 2009
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	To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,		To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
	<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>		<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
	these distributions are taken to be Lévy distributions.		these distributions are taken to be [[:Wikipedia:Stable_distribution\|Lévy distributions]].
	While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.		While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.
	This choice allows modelling both		This choice allows modelling both

Admin: /* Fractional Differential Equations */

2009-08-12T10:15:11Z

Fractional Differential Equations

← Older revision		Revision as of 12:15, 12 August 2009
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	<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>		<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
	these distributions are taken to be Lévy distributions.		these distributions are taken to be Lévy distributions.
	(While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.)		While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.
	This choice allows modelling both		This choice allows modelling both
	[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.		[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.
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	<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>		<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
	whereas the GME must be iterated in time.		whereas the GME must be iterated in time.
	The FDE approach can be used fruitfully to model transport in fusion plasmas, in finite-size systems.		The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
	<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
	On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.		On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.

Admin: /* Fractional Differential Equations */

2009-08-12T09:17:52Z

Fractional Differential Equations

← Older revision		Revision as of 11:17, 12 August 2009
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	The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,		The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
	<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>		<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
	whereas the GME must be iterated in time. On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.		whereas the GME must be iterated in time.
			The FDE approach can be used fruitfully to model transport in fusion plasmas, in finite-size systems.
			<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
			On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
	<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>		<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>

	== References ==		== References ==
	<references />		<references />

Admin: /* Fractional Differential Equations */

2009-08-12T08:29:16Z

Fractional Differential Equations

← Older revision		Revision as of 10:29, 12 August 2009
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	If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.		If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.

	The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration, whereas the GME must be iterated in time. On the other hand, the FDE approach does not capture the (interesting) dynamical behaviour inherent in the GME approach.		The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
			<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
			whereas the GME must be iterated in time. On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
			<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>

	== References ==		== References ==
	<references />		<references />

← Older revision		Revision as of 14:22, 3 April 2018
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	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'').		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>B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', [[doi:10.1063/1.1701893\|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 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>B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, ''Uphill transport and the probabilistic transport model'', [[doi:10.1063/1.1763915\|Phys. Plasmas '''11''', 3787 (2004)]]</ref>).

	The Generalized Master Equation (GME) can be written in the form		The Generalized Master Equation (GME) can be written in the form
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	The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).		The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).

	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 treatment of boundary conditions in a GME is different from standard differential equations. <ref>B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', [[doi:10.1088/1751-8113/41/21/215004\|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.		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.

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	While the Master Equation corresponds to a CTRW in the limit of many particles, the		While the Master Equation corresponds to a CTRW in the limit of many particles, the
	Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.		Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
	<ref>~~[http://link.aps.org/doi/10.1103/PhysRevE.71.011111~~ R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>		<ref>R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', [[doi:10.1103/PhysRevE.71.011111\|Phys. Rev. E '''71''' (2005) 011111]]</ref>
	The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.		The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.

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	The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,		The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
	<ref>~~[http://dx.doi.org/10.1016/j.jcp.2003.07.008~~ V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>		<ref>V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', [[doi:10.1016/j.jcp.2003.07.008\|Journal of Computational Physics '''192''', 2 (2003) 406-421]]</ref>
	whereas the GME must be iterated in time.		whereas the GME must be iterated in time.
	The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.		The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
	<ref>~~[http://link.aip.org/link/?PHPAEN/13/082308/1~~ D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>		<ref>D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', [[doi:10.1063/1.2336114\|Phys. Plasmas '''13''' (2006) 082308]]</ref>
	On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.		On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
	<ref>~~[http://dx.doi.org/10.1088/0029-5515/47/3/004~~ B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>		<ref>B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', [[doi:10.1088/0029-5515/47/3/004\|Nucl. Fusion '''47''' (2007) 189]]</ref>

	== References ==		== References ==
	<references />		<references />

← Older revision		Revision as of 13:21, 24 November 2010
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	~~=[http://exytebuc.co.cc UNDER COSTRUCTION, PLEASE SEE THIS POST IN RESERVE COPY]=~~
	The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).		The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).

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	CTRW theory		CTRW theory
	~~<~~ref~~>~~R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307~~<~~/ref~~>~~		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	starts from the single-particle step distribution function (in one dimension)		starts from the single-particle step distribution function (in one dimension)

	:~~<~~math~~>~~\xi(\Delta x, \Delta t; x, t)~~<~~/math~~>~~		:<math>\xi(\Delta x, \Delta t; x, t)</math>

	giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''&~~amp;~~Delta;x'' after waiting a time ''&~~amp;~~Delta;t'' &~~amp;~~gt; 0. All particles are assumed to be identical.		giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''Δx'' after waiting a time ''Δt'' > 0. All particles are assumed to be identical.

	== The Master Equation ==		== The Master Equation ==
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	This equation is a (Generalized) 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 ''&~~amp;~~Delta;x'' is independent from ''&~~amp;~~Delta;t'', 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

	: ~~<~~math~~>~~\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)~~<~~/math~~>~~		: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)</math>

	In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''&~~amp;~~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 ''&~~amp;~~psi;'' depends on ''x'' (but not ''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 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 (GME) 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>

	where ''n'' is the particle (probability) density, and ''K'' a kernel of the form		where ''n'' is the particle (probability) density, and ''K'' a kernel of the form

	:~~<~~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, generalizing the usual (partial differential) equations for transport.		The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport.
	The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).		The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).

	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 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.		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.

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	While the Master Equation corresponds to a CTRW in the limit of many particles, the		While the Master Equation corresponds to a CTRW in the limit of many particles, the
	Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.		Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
	~~<~~ref~~>~~[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]~~<~~/ref~~>~~		<ref>[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>
	The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.		The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.

	To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,		To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
	~~<~~ref~~>~~B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)~~<~~/ref~~>~~		<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
	these distributions are taken to be [[:Wikipedia:Stable_distribution\|Lévy distributions]].		these distributions are taken to be [[:Wikipedia:Stable_distribution\|Lévy distributions]].
	While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since &~~amp;~~Delta;''t'' &~~amp;~~gt; 0.		While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.
	This choice allows modelling both		This choice allows modelling both
	[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard (~~"~~Fickian~~"~~) transport is recovered.		[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.
	If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.		If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.

	The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,		The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
	~~<~~ref~~>~~[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]~~<~~/ref~~>~~		<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
	whereas the GME must be iterated in time.		whereas the GME must be iterated in time.
	The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.		The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
	On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.		On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
	~~<~~ref~~>~~[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]~~<~~/ref~~>~~		<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>

	== References ==		== References ==
	~~<~~references /~~>~~		<references />

← Older revision		Revision as of 03:05, 24 November 2010
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			=[http://exytebuc.co.cc UNDER COSTRUCTION, PLEASE SEE THIS POST IN RESERVE COPY]=
	The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).		The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).

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	CTRW theory		CTRW theory
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	starts from the single-particle step distribution function (in one dimension)		starts from the single-particle step distribution function (in one dimension)

	:<math>\xi(\Delta x, \Delta t; x, t)</math>		:<math>\xi(\Delta x, \Delta t; x, t)</math>

	giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''Δx'' after waiting a time ''Δt'' > 0. All particles are assumed to be identical.		giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''&Delta;x'' after waiting a time ''&Delta;t'' &gt; 0. All particles are assumed to be identical.

	== The Master Equation ==		== The Master Equation ==
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	This equation is a (Generalized) 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 ''&Delta;x'' is independent from ''&Delta;t'', so that

	: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)</math>		: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta 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'', while ''ψ'' depends on ''x'' (but not ''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 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 (GME) 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>

	where ''n'' is the particle (probability) density, and ''K'' a kernel of the form		where ''n'' is the particle (probability) density, and ''K'' a kernel of the form

	:<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, generalizing the usual (partial differential) equations for transport.		The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport.
	The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).		The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).

	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 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.		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.

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	While the Master Equation corresponds to a CTRW in the limit of many particles, the		While the Master Equation corresponds to a CTRW in the limit of many particles, the
	Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.		Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
	<ref>[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>		<ref>[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>
	The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.		The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.

	To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,		To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
	<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>		<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
	these distributions are taken to be [[:Wikipedia:Stable_distribution\|Lévy distributions]].		these distributions are taken to be [[:Wikipedia:Stable_distribution\|Lévy distributions]].
	While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since Δ''t'' > 0.		While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since &Delta;''t'' &gt; 0.
	This choice allows modelling both		This choice allows modelling both
	[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.		[[Non-diffusive transport\|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.
	If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.		If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.

	The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,		The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
	<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>		<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
	whereas the GME must be iterated in time.		whereas the GME must be iterated in time.
	The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.		The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
	<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
	On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.		On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
	<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>		<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>

	== References ==		== References ==
	<references />		<references />