Fractional advection diffusion asymmetry equation, derivation, solution and application. (English) Zbl 1534.60161
Summary: The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian Lévy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to Lévy statistics for the number of renewals in the process.
{© 2024 IOP Publishing Ltd}
{© 2024 IOP Publishing Ltd}
MSC:
| 60K50 | Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) |
| 60K40 | Other physical applications of random processes |
| 35R11 | Fractional partial differential equations |
| 60G51 | Processes with independent increments; Lévy processes |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
continuous time random walk; anomalous diffusion; fractional advection diffusion asymmetry equation; fractional kinetic equationsSoftware:
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