A. S. Gevorkyan

On the motion of a three-body system on hypersurface of proper energy

Based on the fact that for hamiltonian system there exists equivalence between phase trajectories and geodesic trajectories on the Riemannian manifold, the classical three-body problem is formulated in the framework of six ordinary differential equations (ODEs) of the second order on the energy hypersurface of body system. It is shown that in the case when the total interaction potential of the body system depends on the relative distances between particles, the three of six geodesic equations describing rotations of formed by three bodies triangle are solved exactly. Using this fact, it is shown that the three-body problem can be described in the limits of three nonlinear ODEs of canonical form, which in phase space is equivalent to the autonomous sixth-order system. The equations of geodesic deviations on the manifold Open image in new window (the space of relative distances between particles) are derived in an explicit form. A system of algebraic equations for finding the homographic solutions of restricted three-body problem is obtained. The initial and asymptotic conditions for solution of the classical scattering problem are found.

A new parallel algorithm for simulation of spin glasses on scales of space-time periods of external fields with consideration of relaxation effects

We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.

Quantum‐Mechanical Channel of Interactions between Macroscopic Systems

The macroscopic experimental phenomena which cannot be explained in the frames of classical concepts are described. The attempt of qualitatively understanding of observed effects based on Bohm’s representation of Schrodinger equation for arbitrary system of interacting particles is presented.

On reduction of the general three-body Newtonian problem and the curved geometry

In the framework of an idea of separation of rotational and vibrational motions, we have examined the problem of reducing the general three-body problem. The class of differentiable functions allowing transformation of the 6D Euclidean space to the 6D conformal-Euclidean space is defined. Using this fact the general classical three-body problem is formulated as a problem of geodesic flows on the energy hypersurface of the bodies system. It is shown that when the total potential depends on relative distances between the bodies, three from six ordinary differential equations of second order describing the non-integrable hamiltonian system are integrated exactly, thus allowing reducing the initial system in the phase space to the autonomous system of the 6th order. In the result of reducing of the initial Newtonian problem the geometry of reduced problem becomes curved. The latter gives us new ideas related to the problem of geometrization of physics as well as new possibilities for study of different physical problems.

Modeling of the potential energy surface of regrouping reaction in collinear three-atom collision system using nonlinear optimization

A two-dimensional analytical model with a set of adjusting parameters [1] is proposed for an interaction potential of three-atom collinear system. Different numerical methods for obtaining the optimal set of parameters are considered in the present work. These parameters are used to approximate a two-dimensional numerical array (as obtained from another quantum-chemical ab initio calculations for reaction surfaces) with model potential to given accuracy. Appropriate methods of numerical simulation have been analyzed and the optimal one was selected. Based on this model an algorithm for numerical solution of the problem of finding the adjusting parameters has been realized using successive iterations. It was implemented for two specific cases discussed below.

Deterministic Computation towards Indeterminism

In the present work we propose some interpretation of the results of the direct simulation of quantum chaos.

Use of the Internet for Distributed Computing of Quantum Evolution

The experiment of complex physical problem solution on two remotehigh-performance systems is described. The new algorithm of the quantum evolution computation made it possible to organize computation on a cluster with three subsystems and low exchange of information between them.

NEW ALGORITHM FOR SIMULATION OF 3D CLASSICAL SPIN GLASSES UNDER THE INFLUENCE OF EXTERNAL ELECTROMAGNETIC FIELDS

We study statistical properties of 3D classical spin-glass under the influence of external fields. It is proved that in the framework of the nearest-neighboring model 3D spin-glass problem at performing of Birkhoff’s ergodic hypothesis regarding to orientations of spins in the 3D space can be reduced to the problem of disordered 1D spatial spin-chains (SSC) ensemble where each spin-chain interacts with a random environment. The 1D SSC is defined as a periodic 1D lattice, where spins in nodes are randomly oriented in 3D space, in addition they all interact with each other randomly. For minimization of the Hamiltonian in an arbitrary node of the 1D lattice lattice, we obtained recurrent equations and corresponding Sylvester criterion, which allow to find energy local minimum. On the bases of these equations the high-performance parallel algorithm is developed which allows to calculate all statistical parameters of 3D spin glass, including distribution of a constant of spin-spin interaction, from the first principles of the classical mechanics.

Classical spin glass system in external field with taking into account relaxation effects

We study statistical properties of disordered spin systems under the influence of an external field with taking into account relaxation effects. For description of system the spatial 1D Heisenberg spin-glass Hamiltonian is used. In addition, we suppose that interactions occur between nearest-neighboring spins and they are random. Exact solutions which define angular configuration of the spin in nodes were obtained from the equations of stationary points of Hamiltonian and the corresponding conditions for the energy local minimum. On the basis of these recurrent solutions an effective parallel algorithm is developed for simulation of stabile spin-chains of an arbitrary length. It is shown that by way of an independent order of N2 numerical simulations (where N is number of spin in each chain) it is possible to generate ensemble of spin-chains, which is completely ergodic which is equivalent to full self-averaging of spin-chains’ vector polarization. Distributions of different parameters (energy, average polarization by coordinates, and spin-spin interaction constant) of unperturbed system are calculated. In particular, analytically is proved and numerically is shown, that for the Heisenberg nearest-neighboring Hamiltonian model, the distribution of spin-spin interaction constants as opposed to widely used Gauss-Edwards-Anderson distribution satisfies Levy alpha-stable distribution law. This distribution is nonanalytic function and does not have variance. In the work we have in detail studied critical properties of an ensemble depending on value of external field parameters (from amplitude and frequency) and have shown that even at weak external fields the spin-glass systemis strongly frustrated. It is shown that frustrations have fractal behavior, they are selfsimilar and do not disappear at scale decreasing of area. By the numerical computation is shown that the average polarization of spin-glass on a different coordinates can have values which can lead to catastrophes in the equation ofClausius-Mossotti for dielectric constant. In other words, for some values of external field parameter, a critical phenomenon occurs in the system which is impossible to describe by the real-valued Heisenberg spin-glass Hamiltonian. For the solution of this problem at first the complex-valued disordered Hamiltonian is used. Physically this type of extension of Hamiltonian allows to consider relaxation effects which occur in the system under the influence of an external field. On the basis of developed approach an effective parallel algorithm is developed for simulation of statistic parameters of spin-glass system under the influence of an external field.

Quantum 3D spin-glass system on the scales of space-time periods of external electromagnetic fields

A dielectric medium consisting of rigidly polarized molecules has been treated as a quantum 3D disordered spin system. It is shown that using Birkhoff’s ergodic hypothesis the initial 3D disordered spin problem on scales of space-time periods of external field is reduced to two conditionally separable 1D problems. The first problem describes a 1D disordered N-particle quantum system with relaxation in random environment while the second one describes statistical properties of ensemble of disordered 1D steric spin chains of certain length. Basing on constructions which are developed in both problems, the coefficient of polarizability related to collective orientational effects under the influence of external field was calculated. On the basis of these investigations the equation of Clausius-Mossotti (CM) has been generalized as well as the equation for permittivity. It is shown that under the influence of weak standing electromagnetic fields in the equation of CM arising of catastrophe is possible, that can substantially change behavior of permittivity in the X-ray region on the macroscopic scale of space.