Nonequilibrium Dynamics Of Systems Research Articles

Models of generation of power laws of distribution in the processes of seismicity and in formation of oil fields and ore deposits

[1] A feature indicative of non-equilibrium dynamic system is a fulfillment of a power law of distribution. We consider a number of simple models of generation of power-law distributions in seismic processes and in the processes of mineral deposits formation. As for seismicity, it is shown that deterministic earthquake prediction does not appear to be possible within the used model, whereas a probabilistic prognosis is feasible. Thus it seems possible to reconcile the well-known theoretical concept of unpredictability of earthquakes with the actual existence of algorithms realizing a non-trivial prognosis of strong earthquakes. As for processes of mineral deposits formation, models are discussed that permit to explain a generation of a power-law distribution of a number of large hydrocarbon and ore deposits from the stock values. A new interpretation is proposed of relatively less (than could be expected from the extrapolation of a power distribution law) number of deposits with relatively smaller stock values.

SELF-ORGANIZATION AND NONEQUILIBRIUM STRUCTURES IN THE PHASE SPACE

We consider the possibilities of the formation of quasistationary distributions of particles over energy with power asymptotics in nonequilibrium systems and dynamical systems with couplings. It is shown that the Tsallis distribution is related to the exact solutions of a kinetic equation of the Boltzmann type and those of covariant kinetic equations of the Vlasov nonlocal statistical mechanics. We have studied the connection of the power-like solutions of kinetic equations with the eigenfunctions of fractional integro-differential operators and Jackson operators within quantum analysis, and that of nonextensiveness parameters in the framework of the Tsallis thermostatics, with flows in the phase space. It is shown that the processes running in a nonequilibrium nonconservative medium can be described by the solutions of the equation with fractional derivatives or Jackson derivatives for oscillations.

Dissipative structures in fault zones and their diagnostic criteria (from physical modeling data)

A fault zone is considered an open nonequilibrium dynamic system in which its constituent subsystems evolve by self-organization in a nonlinear synergetic process of successive changes of structural levels (stages). A new term fault dissipative structure is suggested for an ordered 3D space-time structure produced in a fault zone by self-organization of fracturess of different scales. The approach is used in physical modeling to simulate the evolution of a strike-slip fault zone in an elastic-viscoplastic lithosphere and to recognize space and time criteria diagnostic of fault dissipative structures through fractal and entropy analyses.

Precipitate stability and morphology in irradiation environments

A solid under irradiation constitutes a nonequilibrium dynamical system, and therefore the modeling of phase stability in such systems requires a kinetic approach. This paper presents some recent advances in the atomistic modeling of precipitation under irradiation, using kinetic Monte Carlo simulations. It illustrates that vacancy-assisted and thermodynamically driven phase separation, in competition with forced atomic mixing promoted by nuclear collisions, can lead to three very different outcomes: dissolution of pre-existing precipitates, the formation of compositional patterns at the nanoscale, or macroscopic decomposition. Interstitial transport, on the other hand, when coupled with solute fluxes, can induce the precipitation of mushy precipitates, even in ideal alloy systems. These diverse phase evolutions are shown to be rationalized by continuum modeling.

Numerical simulations of periodic travelling waves to a generalized Ginzburg–Landau equation

In this paper, we consider a generalized Ginzburg–Landau equation, which describes various pattern formation and the onset of instabilities in nonequilibrium fluid dynamical systems, as well as in the theory of phase transitions and superconductivity. By the means of the Hopf bifurcation theorem, we obtain the constant-amplitude, small-amplitude and variable-amplitude periodic travelling wave solutions to the generalized Ginzburg–Landau equation. Finally we give some numerical simulations.

Interface motion and pinning in small-world networks.

We show that the nonequilibrium dynamics of systems with many interacting elements located on a small-world network can be much slower than on regular networks. As an example, we study the phase ordering dynamics of the Ising model on a Watts-Strogatz network, after a quench in the ferromagnetic phase at zero temperature. In one and two dimensions, small-world features produce dynamically frozen configurations, disordered at large length scales, analogous to random field models. This picture differs from the common knowledge (supported by equilibrium results) that ferromagnetic shortcut connections favor order and uniformity. We briefly discuss some implications of these results regarding the dynamics of social changes.

Expansion exponents for nonequilibrium systems

Local expansion exponents for nonequilibrium dynamical systems, described by partial differential equations, are introduced. These exponents show whether the system phase volume expands, contracts, or is conserved in time. The ways of calculating the exponents are discussed. The principle of minimal expansion provides the basis for treating the problem of pattern selection. The exponents are also defined for stochastic dynamical systems. The analysis of the expansion-exponent behavior for quasi-isolated systems results in the formulation of two other principles: The principle of asymptotic expansion tells that the phase volumes of quasi-isolated systems expand at asymptotically large times. The principle of time irreversibility follows from the asymptotic phase expansion, since the direction of time arrow can be defined by the asymptotic expansion of phase volume.

THE KINEMATICS OF DISTRIBUTED COMPUTER TRANSACTIONS

A causal, stochastic model of networked computers, based on information theory and nonequilibrium dynamical systems is presented. This provides a simple explanation for recent experimental results revealing the structure of information in network transactions. The model is based on non-Poissonian stochastic variables and pseudo-periodic functions. It explains the measured patterns seen in resource variables on computers in network communities. Weakly non-Poissonian behavior can be eliminated by a conformal scaling transformation and leads to a mapping onto statistical field theory. From this, it is possible to calculate the exact profile of the spectrum of fluctuations. This work has applications to anomaly detection and time-series analysis of computer transactions.

Principle of pattern selection for nonequilibrium phenomena

A general principle is advanced allowing the classification of nonunique solutions to nonlinear evolution equations, corresponding to different spatio-temporal patterns. This is done by defining the probability distribution of patterns, which characterizes multiple solutions as more or less probable with respect to each other. The most probable pattern is naturally defined by the maximum of the pattern distribution. This maximum is shown to be equivalent to the minimum of local contraction. The formulated principle plays for nonequilibrium dynamical systems the same ordering role as the condition of minimal free energy for equilibrium statistical systems. The generality of the principle is illustrated by several examples of different nature.

Nonequilibrium Thermodynamics of Autowaves of Laminar Combustion

Within the framework of the Zel'dovich–Frank‐Kamenetskii theory of thermal propagation of flame, thermodynamic properties of an open nonlinear system are considered, and nonequilibrium entropy of a steady combustion wave is constructed. The nonequilibrium dynamic system is analyzed qualitatively and quantitatively, and the distribution functions of local entropy production in terms of the spatial variable are constructed. It is shown that total entropy production in the system is a functional on integral curves that possess extreme properties, and its minimum corresponds to a unique physically sustainable solution of the problem. The procedure of “cutting” (vanishing) of the reaction rate is justified by methods of nonequilibrium thermodynamics. A variational formulation of the problem is presented for calculation of a steady combustion wave.

Statistics of persistent events: An exactly soluble model

It was recently realized that the persistence exponent appearing in the dynamics of nonequilibrium systems is a special member of a continuously varying family of exponents, describing generalized persistence properties. We propose and solve a simple stochastic spin model, where time intervals between spin flips are independent, and distributed according to a L\'evy law. Both the limit distribution of the mean magnetization and the generalized persistence exponents are obtained exactly. We discuss the relevance of this model for phase ordering, spin glasses, and random walks.

Critical phenomena of nonequilibrium dynamical systems with two absorbing states

We study nonequilibrium dynamical models with two absorbing states: interacting monomer-dimer models, probabilistic cellular automata models, nonequilibrium kinetic Ising models. These models exhibit a continuous phase transition from an active phase into an absorbing phase which belongs to the universality class of the models with the parity conservation. However, when we break the symmetry between the absorbing states by introducing a symmetry-breaking field, Monte Carlo simulations show that the system goes back to the conventional directed percolation universality class. In terms of domain wall language, the parity conservation is not affected by the presence of the symmetry-breaking field. So the symmetry between the absorbing states rather than the conservation laws plays an essential role in determining the universality class. We also perform Monte Carlo simulations for the various interface dynamics between different absorbing states, which yield new universal dynamic exponents. With the symmetry-breaking field, the interface moves, in average, with a constant velocity in the direction of the unpreferred absorbing state and the dynamic scaling exponents apparently assume trivial values. However, we find that the hyperscaling relation for the directed percolation universality class is restored if one focuses on the dynamics of the interface on the side of the preferred absorbing state only.

Coevolutionary Chase in Two-species Systems with Applications to Mimicry

We study a general dynamical model describing coevolution of two haploid populations with two alleles at a single locus under weak linear symmetric frequency-dependent selection. A novel and more realistic element of our modeling approach is that both species are allowed to evolve. We analyse conditions for “evolutionary chase” between two phenotypically similar species in which one species evolves to decrease its resemblance with the other species while this other species evolves to increase its resemblance with the first species. We apply our results to a series of simple population genetics models describing classical Müllerian and Batesian mimicries as well as intermediate cases. We show that one of the most important factors influencing the plausibility of non-equilibrium dynamics in systems of mimicry is the relationship between the strength of between-species and within-species interactions. This indicates that this relationship should be the focus of both experimental and theoretical work. Our results suggest that systematic studies of frequencies of different mimicry morphs through time may be very useful.

Soliton interaction in nonequilibrium dynamical systems.

We propose an approach to numerical analysis of soliton interactions in dynamical systems described by the Ginzburg-Landau equation. This approach is based on the analysis of soliton trajectories in a phase plane. The main features of the interaction are represented in sufficient detail to permit understanding of the formation of bound states and their stability. Among the bound states we find some with \ensuremath{\pi}/2 out-of-phase solitons. \textcopyright{} 1996 The American Physical Society.

Mean-field theory for Lyapunov exponents and Kolmogorov-Sinai entropy in Lorentz lattice gases.

Cellular automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statisitcal mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one-dimensional example using a mean-field theory, and the results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory, and consistent with the escape-rate formalism for the coefficient of diffusion.

Die Auflösung des Sozialen

Abstract Coleman's Foundations of Social Theory is based on a theory of rational action in the specific version of utility maximization embedded in a macro-micro-macro-framework. This approach is logically and empirically deficient in the following aspects: The social constitution of the situation and the actor himself is excluded from the theoretical consideration (macro to micro transition). Limitations to rationality remain implicit and are not integrated into an empirically founded theory of bounded rationality (microtheory). The dynamics of non-equilibrium systems are outside the realm of the theory (micro to macro transition). Central concepts of social theory, as for instance rights and norms, are reduced to a power based balance of interests. This results in a dissolution of social structure and in an infinite regress of contingent conditions.

Ion pores in biological membranes as self-organized bistable systems.

It is shown that an ionic channel in biological membranes is a self-organized nonequilibrium dynamic system functioning in a multistable regime. The different components of such a regime can be compared with discrete conductivity levels of the ionic channel. For the case of a channel with two binding sites we have shown that such an effect is a consequence of organization of the conformational channel structure at the expense of the energy of an ion flux itself. The possibility of the control of the channel functioning due to the external electric field is also considered. Calculations show that one may effectively modulate channel lifetimes in their open and closed states using the applied field. It is also shown that the ensemble of channels which mutually interact in a cooperative manner via the bulk solution under the sufficiently strong dependence of the rate constants upon the conformational variable may possess various types of dynamic behavior such as monostability, bistability, and relaxation oscillations.

Dynamics of Nonequilibrium Systems in the Weakly Nonlinear Regime

Dynamics of Nonequilibrium Systems in the Weakly Nonlinear Regime

Stochastic dynamics of nonequilibrium systems near the steady state

An approximation method is proposed to obtain time dependent solutions of discrete master equations and Fokker-Planck equations. We restrict ourselves to one-dimensional one-step master equations and to one-dimensional F-P equations. Approximate evolution equations for certain statistical averages, e.g. for statistical moments, are formulated and applied for two examples of nonequilibrium phase transitions.