| Description |
1 online resource (274 pages) |
| Contents |
Preface -- 1. Statistical grounds -- 1.1 Levy stable statistics -- 1.1.1 Generalized limit theorems -- 1.1.2 Two subclasses of stable distributions -- 1.1.3 Fractional stable distributions -- 1.1.4 Self-similar processes: Brownian motion and Levy motion -- 1.1.5 Space-fractional equations -- 1.2 Random flight models -- 1.2.1 Continuous time random flights -- 1.2.2 Counting process for number of jumps -- 1.2.3 The Poisson process -- 1.2.4 The Fractional Poisson process -- 1.2.5 Simulation of waiting times -- 1.3 Some properties of the fractional Poisson process -- 1.3.1 The nth arrival time distribution -- 1.3.2 The fractional Poisson distribution -- 1.3.3 Limit fractional Poisson distributions -- 1.3.4 Fractional Furry process -- 1.3.5 Time-fractional equation -- 1.4 Random flights on a one-dimensional Levy-Lorentz gas -- 1.4.1 One-dimensional Levy-Lorentz gas -- 1.4.2 The flight process on the fractal gas -- 1.4.3 Propagators -- 1.4.4 Fractional equation for flights on fractal -- 1.5 Subdiffusion -- 1.5.1 Integral equations of diffusion in a medium with traps -- Necessary and sufficient condition for subdiffusion -- 1.5.2 Differential equations of subdiffusion -- 1.5.3 Subdiffusion distribution density -- 1.5.4 Analysis of subdiffusion distributions -- 1.5.5 Discussion -- 2. Fractional kinetics of dispersive transport -- 2.1 Macroscopic phenomenology -- 2.1.1 A role of phenomenology in studying complex systems -- 2.1.2 Universality of transient current curves -- 2.1.3 From self-similarity to fractional derivatives -- 2.1.4 From transient current to waiting time distribution -- 2.2 Microscopic backgrounds of dispersive transport -- 2.2.1 From the Scher-Montroll model to fractional derivatives -- 2.2.2 Physical basis of the power-law waiting time distribution -- 2.2.3 Multiple trapping regime -- 2.2.4 Hopping conductivity |
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2.2.5 Bassler's model of Gaussian disorder -- 2.3 Fractional formalism of multiple trapping -- 2.3.1 Prime statements -- 2.3.2 Multiple trapping regime and Arkhipov-Rudenko approach -- 2.3.3 Fractional equations for delocalized carriers -- 2.3.4 Fractional equation for the total concentration -- 2.3.5 Two-state dynamics -- 2.3.6 Delocalized carrier concentration -- 2.3.7 Percolation and fractional kinetics -- 2.3.8 The case of Gaussian disorder -- 2.4 Some applications -- 2.4.1 Dispersive diffusion -- 2.4.2 Photoluminescence decay -- 2.4.3 Including recombination -- 2.4.4 Including generation -- 2.4.5 Bipolar dispersive transport -- 2.4.6 The family of fractional dispersive transport equations -- 3. Transient processes in disordered semiconductor structures -- 3.1 Time-of-flight method -- 3.1.1 Transient current in disordered semiconductors -- 3.1.2 Transient current for truncated waiting time distributions -- 3.1.3 Distributed dispersion parameter -- 3.1.4 Transient current curves in case of Gaussian disorder -- 3.1.5 Percolation in porous semiconductors -- 3.1.6 Non-stationary radiation-induced conductivity -- 3.2 Non-homogeneous distribution of traps -- 3.2.1 Non-uniform spatial distribution of localized states -- 3.2.2 Multilayer structures -- 3.2.3 The "disordered -- crystalline" semiconductor structure -- 3.3 Transient processes in a diode under dispersive transport conditions -- 3.3.1 Turning on by the current step -- 3.3.2 Turning off by interruption of circuit -- 3.4 Frequency properties of disordered semiconductor structures -- 3.4.1 Frequency dependence of conductivity -- 3.4.2 A diode at dispersive transport conditions -- 4. Fractional kinetics in quantum dots and wires -- 4.1 Fractional optics of quantum dots -- 4.1.1 Off- and on-intervals statistics -- 4.1.2 Physical mechanisms of power law blinking -- 4.1.3 Two-state renewal model |
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4.1.4 Fractional blinking process -- 4.1.4.1 Total fluorescence time distribution -- 4.1.5 Photon counts distribution -- 4.2 Charge kinetics in colloidal quantum dot arrays -- 4.2.1 Fractional currents in colloidal quantum dot array -- 4.2.2 Modification of the Scher-Montroll model -- 4.2.3 Current decay in the modified model -- 4.2.4 Interdot disorder -- 4.2.5 Monte Carlo simulation -- 4.3 Conductance through fractal quantum conductors -- 4.3.1 Weak localization (scattering) -- 4.3.2 Sequential incoherent tunneling -- 5. Fractional relaxation in dielectrics -- 5.1 The relaxation problem -- 5.1.1 The relaxation functions -- 5.1.2 Non-Debye empirical laws -- 5.1.3 Superposition model -- 5.1.4 Stochastic interpretations of the universal relaxation law -- 5.1.5 Random activation energy model -- 5.2 Fractional approach -- 5.2.1 Fractional derivatives for relaxation problem -- 5.2.2 Polar dielectrics: model of rotational subdiffusion -- 5.2.3 A prehistory contribution -- 5.2.4 Green's function -- 5.2.4.1 The first representation -- 5.2.4.2 The second representation -- 5.3 The Cole-Cole kinetics -- 5.3.1 Fractional generalization of the Ohm's law -- 5.3.2 Numerical demonstration of the memory effect -- 5.3.2.1 Mittag-Leffer representation -- 5.3.2.2 Monte Carlo calculations -- 5.3.3 Polarization-depolarization currents -- 5.3.4 Radiation-induced dielectric effect in polymers -- 5.3.5 Hysteresis in ferroelectric ceramics -- 5.4 The Havriliak-Negami kinetics -- 5.4.1 The Cole-Davidson response -- 5.4.2 Fractional kinetics and Havriliak-Negami response -- 5.4.3 Stochastic inversion of the Havriliak-Negami operator -- 5.4.4 Three-power term approximation of the HN-relaxation -- 5.4.5 Pass-through conductivity and Raicu's response -- 5.4.6 Fractional waves in the HN dielectrics -- 5.5 The Kohlrausch-Williams-Watts kinetics -- 5.5.1 The KWW relaxation function |
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5.5.2 Levy-stable statistics and KWW relaxation -- 5.5.2.1 Relaxation in glassy materials -- 5.5.2.2 Quantum decay theory -- 5.5.3 Fractional equation for KWW relaxation -- 6. The scale correspondence principle -- 6.1 Finity and infinity -- 6.2 Intermediate space-asymptotics -- 6.3 Intermediate time-asymptotics -- 6.4 Concluding remarks -- Appendix A One-sided stable laws -- Appendix B Fractional stable distributions -- Appendix C Fractional operators: main properties -- C.1 Axiomatics (Ross, 1975) -- C.2 Interrelations between fractional operators -- C.3 The law of exponents -- C.4 Differentiation of a product -- C.5 Integration by parts -- C.6 Generalized Taylor series -- C.7 Expression of fractional derivatives through the integers -- C.8 Indirect differentiation: chain rule -- C.9 Fractional powers of operators and Levy stable variables -- Bibliography -- Index |
| Summary |
The standard (Markovian) transport model based on the Boltzmann equation cannot describe some non-equilibrium processes called anomalous that take place in many disordered solids. Causes of anomality lie in non-uniformly scaled (fractal) spatial heterogeneities, in which particle trajectories take cluster form. Furthermore, particles can be located in some domains of small sizes (traps) for a long time. Estimations show that path length and waiting time distributions are often characterized by heavy tails of the power law type. This behavior allows the introduction of time and space derivative |
| Notes |
Print version record |
| Subject |
Chemical kinetics -- Mathematics
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Electric discharges -- Mathematical models
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Electron transport -- Mathematical models
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Fractional calculus.
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Semiconductors -- Electric properties
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Solid state physics -- Mathematics
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Electric discharges -- Mathematical models.
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Electron transport -- Mathematical models.
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Fractional calculus.
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Semiconductors -- Electric properties.
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Solid state physics -- Mathematics.
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| Form |
Electronic book
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| Author |
Sibatov, Renat
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| ISBN |
9789814355438 |
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9814355437 |
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