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.

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.

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.

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.