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Continuous Time Random Walk: Difference between revisions
→Fractional Differential Equations
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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. | |||
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, | 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. | ||