Alexander I. Olemskoi

Synergetic theory for a jamming transition in traffic flow.

The theory of a jamming transition is proposed for the homogeneous car-following model within the framework of the Lorenz scheme. We represent a jamming transition as a result of the spontaneous deviations of headway and velocity that is caused by the acceleration/braking rate to be higher than the critical value. The stationary values of headway and velocity deviations and time of acceleration/braking are derived as functions of control parameter (time needed for car to take the characteristic velocity).

Self-organization of quasi-equilibrium steady-state condensation in accumulative ion-plasma devices

We consider both theoretically and experimentally self-organization process of quasi-equilibrium steady-state condensation of sputtered substance in accumulative ion-plasma devices. It has been shown that the self-organization effect is caused by self-consistent variations of the condensate temperature and the supersaturation of depositing atoms. Two possible types of self-organization process have been found out on the basis of the phase-plane method. The aluminium condensation experimental data confirming the self-organization nature of quasi-equilibrium steady-state condensation are discussed.

Theory of hierarchical coupling

Abstract A recursion relation between the intensity of hierarchical objects at neighbouring levels of a hierarchical tree, the strength of the coupling between them and the level distribution of nodes of the hierarchical tree are proposed. Regular (including Fibonacci), degenerate and irregular trees are considered. It is shown that the strength of hierarchical coupling is an exponentially, logarithmically or power law decreasing function of distance from a common ancestor, respectively.

Characteristics of the evolution of defect structure during phase transformations in the Pd-H system

The pattern of evolution of the defect structure in the Pd-H system during phase transformations is formulated on the basis of x-ray data. It is shown that once the random dislocations formed during phase transformations in the a phase reach their critical density, they assemble into dislocation walls. This process results in the formation of a cellular dislocation substructure in the α phase. After the formation of the cellular substructure in the α phase the random dislocations created during phase transformations climb into the hydride phase, thereby curtailing the evolution of defect structure in the α phase. The subsequent influx of dislocations into the β phase maintains continued evolution of the defect structure (from cellular to block dislocation substructure). Not until that time is it possible for the evolution of the defect structure in the α phase to terminate, culminating in the formation of a block substructure. The nature of the observed phenomenon is discussed.

The theory of spatiotemporal pattern in nonequilibrium systems

Abstract Stationary states and evolution of the spatially periodic and self-similar structures arising in the course of phase transformations are considered. Stationary inhomogeneous structures are studied in the framework of the thermodynamic approach, investigation of the dynamics of their formation is based on the field representation which makes it possible to describe behavior of the most probable values of stochastic variables. In the framework of unified conceptions, peculiarities of the behavior of thermodynamic, self-organizing and stochastic systems with multiplicative noise are studied. The concepts of the fractional integral and derivative are introduced which allow one to write the equation of evolution of an arbitrary system in a general form. Investigation of stationary inhomogeneous distributions is carried out by using the two- and one-component representations of the order parameter. A qualitative analysis based on the method of phase plane shows that under the transition from the disordered phase into the ordered one the harmonic distribution with a short period transforms into the step-like distribution with a long period. In the framework of the η4-model the stationary distribution of the order parameter reduces to the elliptic cosine and passing to the more complicated η6-model results in the fact that the (approximate) solutions will be the elliptic sine, cosine and delta of amplitude. The first of them describes the order parameter distribution, the second describes the antiphase boundaries and the form of the last one is defined by the two first ones. Creation of incommensurate long-period structures in ordered alloys is connected with the attraction of antiphase boundaries via the optical waves of atomic displacements. On the basis of the synergetic approach it is shown that, with increasing the concentration of the antiphase boundaries the optical-phonon exchange leads to stabilization of the long-period structure. The value of the force of the coherent bond between the boundaries is found. It is shown that creation of the long period is realized by the mechanism of phase transition of the first order. When describing the kinetics of transition into the stationary state, the simplest picture is first studied, in the framework of which the phase transition is represented by a single-order parameter. It is shown that under the parallel regime of phase transition the Debye character of relaxation is transformed into slowly decreasing dependencies of the type of the stretched Kohlrausch exponent, power, logarithmic and double logarithmic ones. The synergetic picture of phase transformations of the first and second orders is investigated, which is represented by the order parameter, conjugate field and control parameter. Analytical and numerical investigations of phase portraits have been carried out in different kinetic regimes. It is shown that owing to the critical increase of the times of relaxation of the order parameter and conjugate field the oscillatory behavior is realized if the initial relaxation time of the control parameter is much larger than its value for other degrees of freedom. In the opposite case, all the phase trajectories quickly converge to the universal zone. For a stochastic system with additive noise a self-consistent evolution of the order parameter and the amplitude of conjugate field fluctuations is considered. Investigation of the corresponding phase portraits shows that, depending on the ratio of the inhomogeneity scales, stable and unstable stationary states are possible. When passing to the multiplicative noise with an amplitude depending on the order parameter, the phase plane is divided into isolated domains of large, intermediate and small values of the order parameter. In the first one, in the course of time the trajectories converge to infinite values and the probability of their realization is vanishing. In the intermediate domain, the configurational point tends to the attractive node which corresponds to the stationary ordered state. Finally, in the region of small values of the order parameter, an absorbing state can arise where the system behaves in a deterministic way. The investigation of quasiperiodic distribution of the type which is observed in quasicrystals is based on the fact that it is generated by the same class of mapping, as incommensurate structures: the long-period structures correspond to points of the monofractal set which is contained in the given multifractal to the maximum extent; the quasicrystal sequence is the most rarely realized. A regular method for constructing this sequence is described and the distribution of wave vectors for which the radiation penetrating the quasicrystal yields the diffraction maxima is found. The mode-locking phenomenon is considered, whose spectra of frequencies and wave vectors represent “the devils staircase”. In the framework of fractal ideology, the processes are considered which evolve in space–time in a non-local way. It is shown that in the presence of the non-perfect memory the generalized force leads to a flux which is expressed in the form of a fractional integral. Accordingly, increase of the share of dissipative channels leads to the transformation of the wave-type equation into the heat conduction equation. With decrease of the number of channels with the conserved order parameter, a smooth decrease of the order of the spatial derivative occurs. The method developed allows one to obtain not only a linear fractional-order equation of motion, but also to generalize it for the non-linear case. The latter case contains, in particular, such expressions, as the non-linear Schrodinger equation, the Korteweg–de Vries and sine Gordon equations.

Phase transitions induced by noise cross-correlations

A general approach for treating the spatially extended stochastic systems with the nonlinear damping and correlations between additive and multiplicative noises is developed. Within the modified cumulant expansion method, we derive an effective Fokker-Planck equation with stationary solutions that describe the character of the ordered state. We find that the fluctuation cross-correlations lead to a symmetry breaking of the distribution function even in the case of zero-dimensional system. In a general case, continuous, discontinuous and reentrant noise induced phase transitions take place. It appears that the cross-correlations play the role of bias field which can induce a chain of phase transitions of different nature. Within the mean field approach, we give an intuitive explanation of the system behavior by an effective potential of the thermodynamic type. This potential is written in the form of an expansion with coefficients defined by the temperature, intensity of spatial coupling, autocorrelation and cross-correlation times and intensities of both additive and multiplicative noises.

Axiomatic theory of self-organizing system

Mutually conjugated synergetic schemes are assumed to address the evolution of nonequilibrium self-organizing system. Within the framework of the former, the system is parameterized by a conserving order parameter using density, a conjugate field reducing to a gradient of related flux, and control parameter, whose driven magnitude fixes stationary state. We show that the introduced conjugate field and the control parameter are relevant to entropy and internal energy, so that self-organization effect appears as a negative temperature. Along the line of the conjugated scheme, roles of order parameter, conjugate field and control parameter are played with a flux of conserving value, and gradients of both chemical potential and temperature. With the growth of the latter, a relevant value of the entropy shows a decrease in the supercritical regime related to spontaneous flux-state. We prove that both the approaches stated on using density and conjugated flux as order parameters follow from unified field theory related to the simplest choice of both the Lagrangian and dissipative function.

Evolution of the system with multiplicative noise

The governed equations for the order parameter, one- and two-time correlators are obtained for systems with white multiplicative noise. We consider the noise whose amplitude depends on stochastic variable as xa where 0 12, when the system is disordered, the correlator behaves in the course of time non-monotonically, whereas the autocorrelator increases monotonically. At a<12 the phase portrait of the system divides into two domains: at small initial values of the order parameter, the system evolves to a disordered state, as above; within the ordered domain it is attracted to the point with finite values of the autocorrelator and order parameter. The long-time asymptotes are defined to show that, within the disordered domain, the autocorrelator decays hyperbolically and the order parameter behaves as a power-law function with fractional exponent −2(1−a). Correspondingly, within the ordered domain, the behaviour of both dependencies is exponential with an index proportional to −tlnt.

Generalization of multifractal theory within quantum calculus

On the basis of the deformed series in quantum calculus, we generalize the partition function and the mass exponent of a multifractal, as well as the average of a random variable distributed over a self-similar set. For the partition function, such expansion is shown to be determined by binomial-type combinations of the Tsallis entropies related to manifold deformations, while the mass exponent expansion generalizes the known relation τq=Dq(q-1). We find the equation for the set of averages related to ordinary, escort, and generalized probabilities in terms of the deformed expansion as well. Multifractals related to the Cantor binomial set, exchange currency series, and porous-surface condensates are considered as examples.

Supersymmetric field theory of non-equilibrium thermodynamic system

On the basis of Langevin equation the optimal SUSY field scheme is formulated to describe a non-equilibrium thermodynamic system with quenched disorder and non-ergodicity effects. Thermodynamic and isothermal susceptibilities, memory parameter and irreversible response are determined at different temperatures and quenched disorder intensities.