Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method. (English) Zbl 1514.60113
Summary: This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete fractional Laplacian, \((-\Delta)^s\), \(s\in(0, 2)\) in combination with results from spectral and measure theory. It is a classical mathematical result that the standard Anderson model exhibits localization of energy states for all nonzero disorder in one-dimensional systems. Numerical simulations utilizing our proposed model demonstrate that this localization effect is enhanced for sub-diffusive realizations of the operator, \(s\in(1, 2)\) while the super-diffusive realizations of the operator, \(s\in(0, 1)\) can exhibit energy states with less localized features. These results suggest that the proposed method can be used to examine anomalous diffusion in physical systems where strong correlations, structural defects, and nonlocal effects are present.
MSC:
| 60K50 | Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
Keywords:
Anderson localization; anomalous diffusion; discrete fractional Laplacian; spectral approach; disordered systemsReferences:
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