A. A. Khamzin

Detection of quasi-periodic processes in complex systems: how do we quantitatively describe their properties?

It has been shown that in reality at least two general scenarios of data structuring are possible: (a) a self-similar (SS) scenario when the measured data form an SS structure and (b) a quasi-periodic (QP) scenario when the repeated (strongly correlated) data form random sequences that are almost periodic with respect to each other. In the second case it becomes possible to describe their behavior and express a part of their randomness quantitatively in terms of the deterministic amplitude?frequency response belonging to the generalized Prony spectrum. This possibility allows us to re-examine the conventional concept of measurements and opens a new way for the description of a wide set of different data. In particular, it concerns different complex systems when the ?best-fit? model pretending to be the description of the data measured is absent but the barest necessity of description of these data in terms of the reduced number of quantitative parameters exists. The possibilities of the proposed approach and detection algorithm of the QP processes were demonstrated on actual data: spectroscopic data recorded for pure water and acoustic data for a test hole. The suggested methodology allows revising the accepted classification of different incommensurable and self-affine spatial structures and finding accurate interpretation of the generalized Prony spectroscopy that includes the Fourier spectroscopy as a partial case.

Microscopic model of dielectric α-relaxation in disordered media

The micro/mesoscopic theory of dielectric relaxation has been developed. Based on the fractional kinetics it gives a possibility to obtain the desired expression for the complex dielectric permittivity (CDP) and describe the asymmetric peaks that are created presumably by the so-called ”excess wing” located in high-frequency region. The well-known empirical Cole-Davidson expression and its generalization for the CDP were obtained from this theory. This theory is based on self-similar phenomenon and multi-channel organization of relaxation process in disordered dielectrics. The relaxation parameters are connected with the structural parameters of the medium considered.

Thermodynamics of an interacting Fermi system in the static fluctuation approximation

We suggest a new method of calculation of the equilibrium correlation functions of an arbitrary order for the interacting Fermi-gas model in the framework of the static fluctuation approximation method. This method based only on a single and controllable approximation allows obtaining the so-called far-distance equations. These equations connecting the quantum states of a Fermi particle with variables of the local field operator contain all necessary information related to the calculation of the desired correlation functions and basic thermodynamic parameters of the many-body system. The basic expressions for the mean energy and heat capacity for the electron gas at low temperatures in the high-density limit were obtained. All expressions are given in the units of rs, where rs determines the ratio of a mean distance between electrons to the Bohr radius a0. In these expressions, we calculate terms of the respective order rs and rs2. It is also shown that the static fluctuation approximation allows finding the terms related to higher orders of the decomposition with respect to the parameter rs.

Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism

Using the memory conception developed in the framework of the Mori-Zwanzig formalism, the kinetic equations for relaxation functions that correspond to the previously suggested empirical functions (Cole-Davidson and Havriliak-Negami) are derived. The obtained kinetic equations contain differential operators of non-integer order and have clear physical meaning and interpretation. The derivation of the memory function corresponding to the Havriliak-Negami relaxation law in the frame of Mori-Zwanzig formalism is given. A physical interpretation of the power-law exponents involved in the Havriliak-Negami empirical expression is provided too.

Superfluidity of heated Fermi systems in the static fluctuation approximation

Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.

Temporal multi-sensor system for voltammetric recognition of L- and D-tryptophan enantiomers based on generalized principal component analysis

The results of a quantitative reading of the cyclic voltammetry behavior of the tryptophan (Trp) enantiomers deposited on an electrochemically activated glassy carbon electrode (GCE) are presented. A combined method for “reading” the analytical signals based on the modified Fourier transform and GPCA was applied. It allows a quantitative description of a derivative (dJ/dU) of the measured VAG. This combined method essentially increases the limits of conventional PCA and can be applied to a wide set of electrochemical data. This combined method allows the differentiation and quantitative description of signals governed by the different structures of L- and D-Trp(s) in aqueous solutions.

Log-periodic oscillations in the specific heat behaviour for self-similar Ising type spin systems

The self-similar model of spin-system of the Ising type is formulated. The thermodynamic properties of this model are considered. Analytically and numerically the specific heat of this system is calculated in the nearest neighbor approximation (only the influence of two neighboring spins was taken into account). It is shown that in temperature dependence of the specific heat the log-periodic oscillations are appeared. These oscillations are imposed on the expected power-law dependence.

Dynamics of nuclear spins with equal constants of spin-spin interaction

The dynamics of a system of interacting nuclear spins placed in a nanocontainer are considered. An exact solvable model for describing the dynamics of the investigated system is formulated. Relations that express free induction decay and the shape of the NMR line that are valid for low temperatures are found. The possibility of comparing theoretical calculations to the experimental data in order to control nanocavities in materials and to investigate ordered spin structures is discussed.

On the moment of inertia of a quantum harmonic oscillator

An original method for calculating the moment of inertia of the collective rotation of a nucleus on the basis of the cranking model with the harmonic-oscillator Hamiltonian at arbitrary frequencies of rotation and finite temperature is proposed. In the adiabatic limit, an oscillating chemical-potential dependence of the moment of inertia is obtained by means of analytic calculations. The oscillations of the moment of inertia become more pronounced as deformations approach the spherical limit and decrease exponentially with increasing temperature.

The NMR line shape of magneto-active nanoclusters in moveable nano-containers with self-similar stochastic dynamics

NMR line shape of isolated spin clusters in moving nanocontainers with self-similar correlation law was calculated. It was shown that taking into account self-similarity in stochastic dynamics of pores leads to new shapes of NMR lines which differ from traditional shapes of Gauss and Lorentz types. Fractal dimension of spatial-temporal ensemble can serve as convenient fitting parameter for experimental data interpretation.