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Continuous Time Random Walk: Difference between revisions
→Fractional Differential Equations
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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 (quasi) 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 (quasi) 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, | |||
<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. | ||
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[[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, 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. | |||
== References == | == References == | ||
<references /> | <references /> | ||