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Continuous Time Random Walk: Difference between revisions
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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]]. | ||
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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) | ||
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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> | <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> | 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> | 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> | <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> | <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> | <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> | <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 /> | ||