Continuous Time Random Walk: Difference between revisions

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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 &Delta;''t'' &gt; 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.
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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.

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