Particle fluctuations in nonuniform and trapped Bose gases

The problem of particle fluctuations in arbitrary nonuniform systems with Bose-Einstein condensate is considered. This includes the case of trapped Bose atoms. It is shown that the correct description of particle fluctuations for any nonuniform system of interacting atoms always results in thermodynamically normal fluctuations.Comment: Latex file, 16 page

Turbulent superfluid as continuous vortex mixture

A statistical model is advanced for describing quantum turbulence in a superfluid system with Bose-Einstein condensate. Such a turbulent superfluid can be realized for trapped Bose atoms subject to either an alternating trapping potential or to an alternating magnetic field modulating the atomic scattering length by means of Feshbach resonance. The turbulent system is represented as a continuous mixture of states each of which is characterized by its own vorticity corresponding to a particular vortex.Comment: Latex file, 22 pages, one figur

Method of self-similar factor approximants

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.Comment: Latex file, 21 pages, 4 tables, 4 figure

Temporal Dynamics in Perturbation Theory

Perturbation theory can be reformulated as dynamical theory. Then a sequence of perturbative approximations is bijective to a trajectory of dynamical system with discrete time, called the approximation cascade. Here we concentrate our attention on the stability conditions permitting to control the convergence of approximation sequences. We show that several types of mapping multipliers and Lyapunov exponents can be introduced and, respectively, several types of conditions controlling local stability can be formulated. The ideas are illustrated by calculating the energy levels of an anharmonic oscillator.Comment: 1 file, 21 pages, RevTex, 2 table

Self-Similar Law of Energy Release before Materials Fracture

A general law of energy release is derived for stressed heterogeneous materials, being valid from the starting moment of loading till the moment of materials fracture. This law is obtained by employing the extrapolation technique of the self-similar approximation theory. Experiments are accomplished measuring the energy release for industrial composite samples. The derived analytical law is confronted with these experimental data as well as with the known experimental data for other materials.Comment: Latex, 15 pages, no figure

Statistical Mechanics of Structural Fluctuations

The theory of mesoscopic fluctuations is applied to inhomogeneous solids consisting of chaotically distributed regions with different crystalline structure. This approach makes it possible to describe statistical properties of such mixture by constructing a renormalized Hamiltonian. The relative volumes occupied by each of the coexisting structures define the corresponding geometric probabilities. In the case of a frozen heterophase system these probabilities should be given a priori. And in the case of a thermal heterophase mixture the structural probabilities are to be defined self-consistently by minimizing a thermodynamical potential. This permits to find the temperature behavior of the probabilities which is especially important near the points of structural phase transitions. The presense of these structural fluctuations yields a softening of a crystal and a decrease of the effective Debye temperature. These effects can be directly seen by nuclear gamma resonance since the occurrence of structural fluctuations is accompanied by a noticeable sagging of the M\"ossbauer factor at the point of structural phase transition. The structural fluctuations also lead to the attenuation of sound and increase of isothermic compressibility.Comment: 1 file, 18 pages, RevTex, no figure

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.Comment: Latex file, 27 pages, 2 figures, 5 table

Self-Consistent Theory of Bose-Condensed Systems

In the theory of Bose-condensed systems, there exists the well known problem, the Hohenberg-Martin dilemma of conserving versus gapless approximations. This dilemma is analysed and it is shown that it arises because of the internal inconsistency of the standard grand ensemble, as applied to Bose-systems with broken global gauge symmetry. A solution of the problem is proposed, based on the notion of representative statistical ensembles, taking into account all constraints imposed on the system. A general approach for constructing representative ensembles is formulated. Applying a representative ensemble to Bose-condensed systems results in a completely self-consistent theory, both conserving and gapless in any approximation.Comment: Latex file, 12 page

Summation of Power Series by Self-Similar Factor Approximants

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the self-similar renormalization to the latter rather to the former. This results in self-similar factor approximants extrapolating the sought functions from the region of asymptotically small variables to their whole domains. The method of constructing crossover formulas, interpolating between small and large values of variables is also analysed. The techniques are illustrated on different series which are typical of problems in statistical mechanics, condensed-matter physics, and, generally, in many-body theory.Comment: 30 pages + 5 ps figures, some misprints have been correcte

Self-Similar Crossover in Statistical Physics

An analytical method is advanced for constructing interpolation formulae for complicated problems of statistical mechanics, in which just a few terms of asymptotic expansions are available. The method is based on the self-similar approximation theory, being its variant where control functions are defined from asymptotic crossover conditions. Several examples from statistical physics demonstrate that the suggested method results in rather simple and surprisingly accurate formulae.Comment: 1 file, 23 pages, LaTe