It is shown that a quantum Levy process in a box leads to a problem involving topological constraints in space, and its treatment in the framework of the path integral formalism with the Levy measure is suggested. The eigenvalue problem for the infin
ite potential well is properly defined and solved. An analytical expression for the evolution operator is obtained in the path integral presentation, and the path integral takes the correct limit of the local quantum mechanics with topological constraints. An example of the Levy process in oscillating walls is also considered in the adiabatic approximation.
Electrodynamics of composite materials with fractal geometry is studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integro-differentiation. The photoconductivity
in the vicinity of the electrode-electrolyte fractal interface is studied. The methods of fractional calculus are employed to obtain an analytical expression for the giant local enhancement of the optical electric field inside the fractal composite structure at the condition of the surface plasmon excitation. This approach makes it possible to explain experimental data on photoconductivity in the nano-electrochemistry.
A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed an explained in the framework of a continuous time rand
om walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension $frD<3$.
Anomalous transport and reaction dynamics are considered by providing the theoretical grounds for the possible experimental realization of actin polymerization in comb-like geometry. Two limiting regimes are recovered, depending on the concentration
of reagents (magnesium and actin). These are both the failure of the reaction front propagation and a finite speed corresponding to the Fisher-KPP long time asymptotic regime.
This chapter is a contribution in the Handbook of Applications of Chaos Theory ed. by Prof. Christos H Skiadas. The chapter is organized as follows. First we study the statistical properties of combs and explain how to reduce the effect of teeth on t
he movement along the backbone as a waiting time distribution between consecutive jumps. Second, we justify an employment of a comb-like structure as a paradigm for further exploration of a spiny dendrite. In particular, we show how a comb-like structure can sustain the phenomenon of the anomalous diffusion, reaction-diffusion and Levy walks. Finally, we illustrate how the same models can be also useful to deal with the mechanism of ta translocation wave / translocation waves of CaMKII and its propagation failure. We also present a brief introduction to the fractional integro-differentiation in appendix at the end of the chapter.
This study is concerned with destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schrodinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distingu
ished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.
