Simple random walk in a fractal space
DOI:
https://doi.org/10.22481/exon.v10i2.20472Keywords:
Statistical Mechanics, Random Walk, Fractal Space, Fokker-PlanckAbstract
For the study of the trajectory of less massive particles, it is necessary to use Statistical Mechanics of Non-Equilibrium. The random walk in a fractal space arises from the need for the probability distribution to be dependent on a certain measure and to be non-differentiable for non-integer orders in spatial coordinates, but preserves the differentiability requirements at a certain moment t ∈ I ⊂ R+. A possible equation governing this phenomenon is the fractional Fokker-Planck equation that depends on the derivatives of the α order and the order 2α on a certain spatial extent such that α ∈ (0, 1]. The study will be based on the binomial distribution and the recurrence relation for the one-dimensional random walk, since, although the measures are not one-dimensional, the study will be based on the relation "measure versus time". Some hypotheses will be used for the demonstration of the fractional Fokker-Planck equation and the treatment with the Fractional Calculus through the local derivation operator is of paramount importance for the conclusion of the demonstration.
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