Fractional integral and its physical interpretation

A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It follows from analysis of the results that equations in fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the system that are preserved during evolution timet. Such systems can be classified as systems with residual memory, and they occupy an intermediate position between systems with complete memory, on the one hand, and Markov systems, on the other. The use of such equations to describe transport and relaxation processes is discussed. Some generalizations that extent the domain of applicability of the fractional derivative concept are obtained.Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 354–368, March, 1992.     

The generalized Jonscher's relationship for conductivity and its confirmation for porous structures

The theoretical generalization of the Jonscher's relationship for the complex conductivity of carriers moving in self-similar medium is derived. It is shown that the correction derived enters to more general expression, which, in turn, we define as the generalized Jonscher's relationship. The basic idea which was used for the derivation of the relationship is based on the supposition that disordered medium has self-similar property. The derived relationship is confirmed on dielectric spectroscopy data related to sodium nitrite embedded to porous glasses. Based on new relationship there is a possibility to extract additional information about relaxation processes of a system of dipoles from the processes related to conductivity. It is important in the cases when the contribution to relaxation peaks is small and unnoticeable on the background of essential domination of processes related to conductivity.     

Liquid Helium-4 in the Static Fluctuation Approximation

In this work liquid helium-4 is studied for the first time within the framework of the so-called static fluctuation approximation. This is based on the replacement of the square of the local-field operator with its mean value. A closed set of nonlinear integral equations is derived for weakly as well as for strongly interacting systems. This set is solved numerically by an iteration method for a realistic interhelium potential. The thermodynamic properties are then obtained for both the weakly interacting system, liquid ⁴He in Vycor glass, and the strongly interacting system, liquid ⁴He. It turns out, however, that the present quadratic-fluctuation approximation is valid in the latter, strongly interacting case only in the low-temperature limit (0.15 K). Our results are presented in a set of figures. The role of the interaction is emphasized and the functional dependence of key thermodynamic quantities on the temperature is derived for both weakly and strongly interacting ⁴He systems.     

One-, Two-, and Three-Dimensional Ising Model in the Static Fluctuation Approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator _f ^z withquadratic fluctuations is replaced with its mean value [( _f ^z )² ( _f ^z )²]. This means that the true quantum mechanical spectrum of the operator _f ^z is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.     

The effect of clay morphology on water relaxation

The frequency dependence of the permittivity of water in calcium kaolinite (clay) is measured. It is shown that two mechanisms contribute to dipole relaxation of water. One refers to water in the free volume of pores in the clay. The other is associated with bound water covering the porous surface. Experimental data are treated in terms of a fractal model of the medium. The frequency dependence of the permittivity in a wide range of water content in the clay is accounted for theoretically.