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Continuous Time Random Walk: Difference between revisions
→Fractional Differential Equations
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In order to proceed, 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, | In order to proceed, 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. This choice allows modelling both | 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 | |||
[[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. | ||