Reaction-subdiffusion front propagation in a comblike model of spiny dendrites.

Fractional reaction-diffusion equations are derived by exploiting the geometrical similarities between a comb structure and a spiny dendrite. In the framework of the obtained equations, two scenarios of reaction transport in spiny dendrites are explored, where both a linear reaction in spines and nonlinear Fisher-Kolmogorov-Petrovskii-Piskunov reactions along dendrites are considered. In the framework of fractional subdiffusive comb model, we develop a Hamilton-Jacobi approach to estimate the overall velocity of the reaction front propagation. One of the main effects observed is the failure of the front propagation for both scenarios due to either the reaction inside the spines or the interaction of the reaction with the spines. In the first case the spines are the source of reactions, while in the latter case, the spines are a source of a damping mechanism.

Exponential spreading and singular behavior of quantum dynamics near hyperbolic points.

Quantum dynamics of a particle in the vicinity of a hyperbolic point is considered. Expectation values of dynamical variables are calculated, and the singular behavior is analyzed. Exponentially fast extension of quantum dynamics is obtained, and conditions for this realization are analyzed.

Application of hyperbolic scaling for calculation of reaction-subdiffusion front propagation.

A technique of hyperbolic scaling is applied to calculate a reaction front velocity in an irreversible autocatalytic conversion reaction A+B → 2A under subdiffusion. The method, based on the geometric optics approach is a technically elegant observation of the propagation front failure obtained in Phys. Rev. E 78, 011128 (2008).

A toy model of fractal glioma development under RF electric field treatment.

A toy model for glioma treatment by a radio frequency electric field is suggested. This low-intensity, intermediate-frequency alternating electric field is known as the tumor-treating field (TTF). In the framework of this model the efficiency of this TTF is estimated, and the interplay between the TTF and the migration-proliferation dichotomy of cancer cells is considered. The model is based on a modification of a comb model for cancer cells, where the migration-proliferation dichotomy becomes naturally apparent. Considering glioma cancer as a fractal dielectric composite of cancer cells and normal tissue cells, a new effective mechanism of glioma treatment is suggested in the form of a giant enhancement of the TTF. This leads to the irreversible electroporation that may be an effective non-invasive method of treating brain cancer.

Generalized Lyapunov exponent and transmission statistics in one-dimensional Gaussian correlated potentials.

The distribution of the transmission coefficient T of a long system with a correlated Gaussian disorder is studied analytically and numerically in terms of the generalized Lyapunov exponent (LE) and the cumulants of ln T. The effect of the disorder correlations on these quantities is considered in weak, moderate, and strong disorder for different models of correlation. Scaling relations between the cumulants of ln T are obtained. The cumulants are treated analytically within the semiclassical approximation in strong disorder and numerically for an arbitrary strength of the disorder. A small correlation scale approximation is developed for calculation of the generalized LE in a general correlated disorder. An essential effect of the disorder correlations on the transmission statistics is found. In particular, obtained relations between the cumulants and between them and the generalized LE show that, beyond weak disorder, transmission fluctuations and deviation of their distribution from the log-normal form (in a long but finite system) are greatly enhanced due to the disorder correlations. Parametric dependence of these effects upon the correlation scale is presented.

Toy model of fractional transport of cancer cells due to self-entrapping.

A simple mathematical model is proposed to study the influence of cell fission on transport. The model describes fractional, in time, tumor development, which is a one-dimensional continuous-time random walk. The model is relevant for consideration of both solid and diffusive cancers.

Negative superdiffusion due to inhomogeneous convection.

Fractional transport of particles on a comb structure in the presence of an inhomogeneous convection flow is studied [Baskin and Iomin, Phys. Rev. Lett. 93, 120603 (2004)]. The large scale asymptotics is considered. It is shown that a contaminant spreads superdiffusively in the direction opposite to the convection flow. Conditions for the realization of this effect are discussed in detail.

Superdiffusion on a comb structure.

We study specific properties of particles transport by exploring an exact solvable model, a so-called comb structure, where diffusive transport of particles leads to subdiffusion. A performance of the Lévy-like process enriches this transport phenomenon. It is shown that an inhomogeneous convection flow is a mechanism for the realization of the Lévy-like process. It leads to superdiffusion of particles on the comb structure. This superdiffusion is an enhanced one with an arbitrary large transport exponent, but all moments are finite. A frontier case of superdiffusion, where the transport exponent approaches infinity, is studied. The log-normal distribution with the exponentially fast superdiffusion is obtained for this case.

Loschmidt echo for a chaotic oscillator.

Chaotic dynamics of a nonlinear oscillator is considered in the semiclassical approximation. The Loschmidt echo as a measure of quantum stability to a time dependent variation is calculated. It is shown that an exponential decay of the Loschmidt echo is due to a Lyapunov exponent and it has a pure classical nature. The Lyapunov regime is observed for a time scale which is of the power law in semiclassical parameter.

Semiclassical approximation for a nonlinear oscillator with dissipation.

An S-matrix approach is developed for the chaotic dynamics of a nonlinear oscillator with dissipation. The quantum-classical crossover is studied in the framework of the semiclassical expansion for the S matrix. An analytical expression for the breaking time, which is the Ehrenfest time for the dissipative system, is obtained. A correlation function of the S-matrix elements is studied as well.

Semiclassical quantization of separatrix maps.

Quantization of energy balance equations, which describe a separatrixlike motion is presented. The method is based on an exact canonical transformation of the energy-time pair to the action-angle canonical pair, (E,t)-->(I,theta). Quantum mechanical dynamics can be studied in the framework of the new Hamiltonian. This transformation also establishes a relation between a wide class of the energy balance equations and dynamical localization of classical diffusion by quantum interference, that was studied in the field of quantum chaos. An exact solution for a simple system is presented as well.

Breaking time for the quantum chaotic attractor.

A model of a quantum dissipative system is considered in the regime when the classical limit corresponds to a chaotic attractor, and the breaking time tau(Planck) of the classical-quantum correspondence is obtained. The model describes a periodically kicked harmonic oscillator (or a particle in a constant magnetic field) with a dissipation. Another analog of this problem is the dissipative kicked Harper model. It is shown that in the limit of the so-called dying attractor, the breaking time tau(Planck) can be arbitrarily large.

Quantum localization for a kicked rotor with accelerator mode islands.

Dynamical localization of classical superdiffusion for the quantum kicked rotor is studied in the semiclassical limit. Both classical and quantum dynamics of the system become more complicated under the conditions of mixed phase space with accelerator mode islands. Recently, long time quantum flights due to the accelerator mode islands have been found. By exploration of their dynamics, it is shown here that the classical-quantum duality of the flights leads to their localization. The classical mechanism of superdiffusion is due to accelerator mode dynamics, while quantum tunneling suppresses the superdiffusion and leads to localization of the wave function. Coupling of the regular type dynamics inside the accelerator mode island structures to dynamics in the chaotic sea proves increasing the localization length. A numerical procedure and an analytical method are developed to obtain an estimate of the localization length which, as it is shown, has exponentially large scaling with the dimensionless Planck's constant (tilde)h<<1 in the semiclassical limit. Conditions for the validity of the developed method are specified.

Quantum breaking time scaling in superdiffusive dynamics.

We show that the breaking time of quantum-classical correspondence depends on the type of kinetics and the dominant origin of stickiness. For sticky dynamics of quantum kicked rotor, when the hierarchical set of islands corresponds to the accelerator mode, we demonstrate by simulation that the breaking time scales as tau(Planck's over 2pi) approximately (1/Planck's over 2pi)(1/mu) with the transport exponent mu>1 that corresponds to superdiffusive dynamics [B. Sundaram and G. M. Zaslavsky, Phys. Rev. E 59, 7231 (1999)]. We discuss also other possibilities for the breaking time scaling and transition to the logarithmic one tau(Planck's over 2pi) approximately ln(1/Planck's over 2pi) with respect to Planck's over 2pi.

Quantum response for chaotic resonances

Chaotic dynamics of conducting electrons in the presence of a high-frequency electromagnetic field and a constant homogeneous magnetic field is considered. It is shown that quantum fluctuations become important in this case on a time scale shorter than mean free time. Nonperturbative approach for calculation of a kinetic coefficient (conductivity) is developed. An analytical expression for the kinetic coefficient as a function of the magnetic field and a localization length is obtained. Dependence of the conductivity on the quantum localization length is studied.

Hierarchical structures in the phase space and fractional kinetics: II. Immense delocalization in quantized systems.

Anomalous transport due to Levy-type flights in quantum kicked systems is studied. These systems are kicked rotor and kicked Harper model. It is confirmed for a kicked rotor that there exist special "magic" values of a control parameter of chaos K=K(*)=6.908 745 em leader for which an essential increasing of a localization length is obtained. Functional dependence of the localization length on both parameter of chaos and quasiclassical parameter h is studied. We also observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K(*)=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization as well as increasing of localization length can be found in other systems. (c) 2000 American Institute of Physics.

Immense delocalization from fractional kinetics.

We observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K*=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization can be found in other systems.