Equation For Order Parameter Research Articles

Distribution of barriers in spin glasses

We discuss a general formalism that allows study of transitions over barriers in spin glasses with long-range interactions that contain large but finite number, $N$, of spins. We apply this formalism to the Sherrington-Kirkpatrick model with finite $N$ and derive equations for the dynamical order parameters which allow ''instanton'' solutions describing transitions over the barriers separating metastable states. Specifically, we study these equations for a glass state that was obtained in a slow cooling process ending a little below $T_{c}$ and show that these equations allow ''instanton'' solutions which erase the response of the glass to the perturbations applied during the slow cooling process. The corresponding action of these solutions gives the energy of the barriers, we find that it scales as $\tau ^{6}$ where $\tau $ is the reduced temperature.

Interface pinning and slow ordering kinetics on infinitely ramified fractal structures

We investigate the time-dependent Ginzburg-Landau (TDGL) equation for a nonconserved order parameter on an infinitely ramified (deterministic) fractal lattice employing two alternative methods: the auxiliary field approach and a numerical method of integration of the equations of evolution. In the first case the domain size evolves with time as $L(t)\ensuremath{\sim}{t}^{{1/d}_{w}}$, where ${d}_{w}$ is the anomalous random-walk exponent associated with the fractal and differs from the normal value $2$, which characterizes all Euclidean lattices. Such a power-law growth is identical to the one observed in the study of the spherical model on the same lattice, but fails to describe the asymptotic behavior of the numerical solutions of the TDGL equation for a scalar order parameter. In fact, the simulations performed on a two dimensional Sierpinski carpet indicate that, after an initial stage dominated by a curvature reduction mechanism in the manner of Allen and Cahn [Acta. Metall. 27, 1085 (1979)], the system enters in a regime where the domain walls between competing phases are pinned by lattice defects. The lack of translational invariance determines a rough free-energy landscape, the existence of many metastable minima, and the suppression of the marginally stable modes, which in translationally invariant systems lead to power-law growth and self-similar patterns. On fractal structures, as the temperature vanishes the evolution is frozen since only thermally activated processes can sustain the growth of pinned domains.

Adsorption of polyelectrolytes and inter-colloidal forces

The effect of charged polymers on the interactions between charged surfaces in solution is investigated theoretically. We use mean-field theory to derive a self-consistent field equation for the polymer order parameter, and a modified Poisson–Boltzmann equation for the electrostatic potential. The two differential equations together with the appropriate boundary conditions are solved numerically. We use simple scaling arguments to investigate the adsorption characteristics to a single surface in the strong electrostatic regime. We obtain simple expressions for the dependence of the adsorbed amount Γ on the charge fraction p and the salt concentration c b , in good agreement with experiments. The forces between two charged surfaces in the presence of polyelectrolytes are calculated using a generalized contact theorem. We obtain repulsion as well as attraction between equally charged surfaces.

Moving spatial solitons in active nonlinear-optical resonators

We investigate spatial solitons in a resonator with a narrow-band gain element and a saturable absorber placed in Fourier-conjugated resonator planes. Solitons are stationary or move at discrete velocities depending on the resonator tuning. The modulus of the velocity of moving solitons is fixed, but the direction of their motion is arbitrary. Solitons compete in velocity space. The experiments are conducted on a photorefractive oscillator with bacteriorhodopsin saturable absorber. Observations agree well with solutions of a general order parameter equation for such resonators.

Formation of periodic and localized patterns in an oscillating granular layer

A simple phenomenological model for pattern formation in a vertically vibrated layer of granular particles is proposed. This model exhibits a variety of stable cellular patterns including standing rolls and squares as well as localized excitations ( oscillons and worms), similar to recent experimental observations (Umbanhowar et al., 1996). The model is an order parameter equation for the parametrically excited waves coupled to the mass conservation law. The structure and dynamics of the solutions resemble closely the properties of patterns observed in the experiments.

Visions of Synergetics

Synergetics is an interdisciplinary field of research. It deals with open systems that are composed of many individual parts that interact with each other and that can form spatial, temporal, or functional structures by self-organization. The research goal of synergetics is three-fold: (1) Are there general principles of self-organization? (2) Are there analogies in the behavior of self-organizing systems? (3) Can new devices be constructed because of the results (1) and (2)? From a mathematical point of view, synergetics deals with nonlinear partial stochastic differential equations and studies their solutions close to those points where the solutions change their behavior qualitatively. As I will show in my article, synergetics in its present form is based on the concepts of stability and instability, control parameters, order parameters, and the slaving principle. The slaving principle allows one to compress the information that is necessary to describe complex systems into a few order parameters. This is possible if systems are close to their instability points. But it appears that the order parameter concept is also applicable to situations away from such instabilities. At the level of the order parameter equations, profound analogies between otherwise quite different systems become visible. This allows one to realize the same process (for instance dealing with information) on quite different material substrates. The order parameter concept and the slaving principle are explained and their extension to discrete noisy maps and to delay equations are mentioned. These results can be applied to pattern formation in fluids, lasers, semiconductors, plasmas, and other fields. A section is devoted to the analysis of spatio-temporal patterns in terms of order parameters and the slaving principle. It is shown how the concepts of synergetics can be utilized to devise a new type of computer for pattern recognition. In connection with preprocessing, it can recognize patterns that are shifted, scaled or rotated in space and that are deformed. It can recognize scenes and also facial expressions as well as movement patterns. The learning procedure is briefly outlined. Because of the analogy principle of synergetics, this computer allows for hardware realizations by means of semiconductors and lasers. Decision-making by humans or machines is interpreted by means of an analogy with pattern recognition. Further sections are devoted to recognition of dynamic processes and learning by machines. In this author's opinion, synergetics will find important applications in medicine, for instance in the analysis of MEG and EEG patterns, and in the development of devices with brain-like functions. Future tasks of synergetics will be the application of the order parameter concept and the slaving principle to the integration of specialized computers or computer algorithms, for instance for the recognition of faces, movement patterns, and so on, into a computer network for scene analysis and decision making. This will also hold for complicated production processes and so on. Generally speaking, the potentialities of synergetics are based on its self-organization principles.

Visions of synergetics

Synergetics is an interdisciplinary field of research. It deals with open systems that are composed of many individual parts which interacts with each other and can form spatial, temporal, or functional structures by self-organization. The research goal of synergetics is three-fold: (1) are there general principles of self-organization? (2) are there analogies in the behavior of self-organizing systems? (3) can new devices be constructed because of the results (1) and (2)? From a mathematical point of view, synergetics deals with nonlinear partial stochastic differential equations and studies their solutions close to those points where the solutions change their behavior qualitatively. As will be shown in this article, synergetics in its present form is based on the concepts of stability and instability, control parameters, order parameters and the slaving principle. The slaving principle allows one to compress the information that is necessary to describe complex systems into a few order parameters. This is possible if systems are close to their instability points, but it appears that the order parameter concept is also applicable to situations away from such instabilities. At the level of the order parameter equations, profound analogies between otherwise quite different systems become visible. This allows one to realize the same process (for instance, dealing with information) on quite different material substrates. The order parameter concept and the slaving principle are explained and their extension to discrete noisy maps and to delay equations are mentioned. These results can be applied to pattern formation in fluids, lasers, semiconductors, plasmas and other fields. A section is devoted to the analysis of spatio-temporal patterns in terms of order parameters and the slaving principle. It is shown how the concepts of synergetics can be utilized to devise a new type of computer for pattern recognition. In connection with preprocessing, it can recognize patterns that are shifted, scaled or rotated in space and that are deformed. It can recognize scenes and also facial expressions as well as movement patterns. The learning procedure is briefly outlined. Because of the analogy principle of synergetics, this computer allows for hardware realizations by means of semiconductors and lasers. Decision making by humans or machines is interpreted by means of an analogy with pattern recognition. Further sections are devoted to recognition of dynamic processes and learning by machines. In this author's opinion, synergetics will find important applications in medicine, for instance in the analysis of MEG and EEG patterns, and in the development of devices with brain-like functions. Future tasks for synergetics will be the application of the order parameter concept and the slaving principle to the integration of specialized computers or computer algorithms, for instance for the recognition of faces, movement patterns and so on, into a computer network for scene analysis and decision making. This will hold also for complicated production processes, and so on. Generally speaking, the potentialities of synergetics are based on its self-organization principles.

Order-parameter equations for transverse pattern formation in nonlinear optical systems with nonplanar resonators

Order-parameter equations for transverse pattern formation in nonlinear optical systems with nonplanar resonators

Pattern formation near an oscillatory instability for systems without “up-down” symmetry

We present the results of our numerical investigations of the order parameter equation associated with an oscillatory instability with a small onset frequency of systems lacking “up-down” symmetry. Qualitatively different patterns arise depending on whether periodic or more realistic boundary conditions are used. Among the patterns found are blinking hexagons, traveling rectangular patterns, and states that are disordered in space and time, which dominate for realistic boundary conditions. Experimentally accessible systems for which our predictions could be checked might include non-Boussinesq convection in binary fluid mixtures.

Subhexagons and ultrahexagons as a result of a secondary instability

Simulating two-dimensional optical pattern formation in sodium vapor, we show that hexagonal patterns can be unstable against their own spatial harmonics or subharmonics. The instability results in ultrahexagons or subhexagons, respectively. The formation of ultrahexagons is due to interaction with a second solution branch with a wave number nearly resonant to the harmonics of the primary hexagon. The mechanism of the instabilities is discussed on the basis of order parameter equations obtained from symmetry considerations.

Quantum lattice fluctuations in nearly degenerate quasi one-dimensional polymers

It has been shown that the linear and nonlinear optical behaviors are quite different for degenerate and nondegenerate polymers. Quantum lattice fluctuations are believed to play an important role in the understanding of these optical properties. In this work, we study the quantum lattice fluctuations in a nondegenerate conjugated polymer by means of a modified Su-Schrieffer-Heeger-type model Hamiltonian. We first integrate out the electronic variables of the partition function and obtain an effective phonon action. Then an equation for the phonon order parameter is derived within a one-loop approximation. We find that the quantum lattice fluctuations is reduced by the nondegenerate ground state, especially in the weak electron-phonon coupling. This result is compatible with the optical properties of degenerate and nondegenerate polymers.

The Dynamics of Curved Fronts: Beyond Geometry

We derive a new set of kinematic equations for front motion in two-dimensional bistable media. The equations generalize the geometric approach by complementing the equation for the front curvature with an order parameter equation associated with a nonequilibrium Ising-Bloch bifurcation. The resulting equations capture the core structure of spiral waves and spontaneous spiral-wave nucleation.

Quantum lattice fluctuations in the ground state of one-band MX complexes

The effects of quantum lattice fluctuations on the charge density wave (CDW) of a one-dimensional one-band model for halogen-bridged mixed-valence linear chain complexes are studied by a functional integral approach. An equation for the phonon order parameter is derived within a one-loop approximation for all values of the ionic mass. We find that the dimerized CDW survives the quantum lattice fluctuations for arbitrarily small electron-lattice coupling. The result is the same as that of the Su-Schrieffer-Heeger model, although these two models are different for the interactions. The electron-phonon interactions are site-diagonal for the MX model and site-off-diagonal for the SSH model. Furthermore, in the case of finite ionic mass, the low-energy properties of the system are governed by the limit of M = 0 for a weak electron-phonon interaction, whereas they are governed by the adiabatic limit ( M = ∞) for a stronger electron-phonon interaction.

Sscillatory Cortical Activity during Visual Hallucinations.

From a theoretical point of view we investigate cortical activity patterns causing dynamic visual hallucinations. For this reason we analyze an oscillatory instability of the dynamics ofthe activator-inhibitor model of Ermentrout and Cowan.Such an oscillatory instability occurs as a result of several disease mechanisms.We explicitly derive the order parameter equation. By means of the averaging theorem, we obtain the averaged order parameter equation.The latter enables us to determine stable and unstable bifurcating cortical activity patterns analytically in lowest order.Depending on model parameters as well as on initial conditions two types of cortical activity patterns occur: travelling waves and 'blinking rolls', i.e.standing waves oscillating with the same frequency and with a phase shift of π/2. In contrast to cortical activitypatterns caused by non-oscillatory instabilitiesanalyzed by Ermentrout and Cowan and by the author the travelling waves and the blinking rolls lead to a variety of dynamic visual hallucinations which are discussed indetail.

A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations

Motivated by generic scale invariance, we examine the behavior of topological defect-defect correlation functions in two-dimensional systems driven out of equilibrium to regimes where they exhibit “defect chaos”. Using the topological nature of the defects, we show that these defect-defect correlations cannot decay as slowly as predicted by generic scale invariance. We also provide numerical calculations that yield defect-defect correlation functions in the defect turbulence regime of the two-dimensional, anisotropic complex Ginzburg-Landau equation. These numerical results, which test the specific regime of broken square symmetry, do not appear to decay as slowly as predicted by the ideas of generic scale invariance. These results are in agreement with the analytical predictions.

Quantum fluctuations and lattice dimerization in coupled polymer chains

Effects of quantum lattice fluctuations on the Peierls dimerization are considered for two neighboring polymer chains, which are modeled individually by the Su, Schrieffer, and Heeger Hamiltonian and coupled by an interchain electron-transfer term. By a functional-integral approach, an equation for the dimerization order parameter is obtained within a one-loop approximation. It is found that the quantum fluctuations are partly suppressed by the interchain coupling and then the system is dimerized for an arbitrary coupling constant and phonon frequency even for the spinless fermion system.

Slaving principle revisited

After a brief reminder of the Lyapunov-Schmidt procedure of bifurcation theory, the adiabatic elimination technique and scaling procedures, the basic content of the slaving principle is mentioned. Then it is shown how close to instability points the state vector that obeys a nonlinear evolution equation can be parametrized by order parameters (slaving principle). It is shown how this parametrization as well as the order parameter equations can be derived in a straightforward and simultaneous way.

Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns

We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) excetp at points, we are able to derive explicit expressions which capture the anatomies of point and line (and surface) defects in two and three dimensional patterns, together with their topological characters and energetic constraints.

Complex spatiotemporal convection patterns.

This paper reviews recent efforts to describe complex patterns in isotropic fluids (Rayleigh-Benard convection) as well as in anisotropic liquid crystals (electro-hydrodynamic convection) when driven away from equilibrium. A numerical scheme for solving the full hydrodynamic equations is presented that allows surprisingly well for a detailed comparison with experiments. The approach can also be useful for a systematic construction of models (order parameter equations). (c) 1996 American Institute of Physics.

Predicting the nonlinear shift of stable equilibria in interlimb rhythmic coordination

A major prediction of the elementary coordination dynamics of two contralateral limb segments in 1:1 frequency locking was tested. A shift in stable steady-state relative phase φ from 0 and π radians is induced by a difference Δω in the uncoupled frequencies of the segments. The elementary coordination dynamics, an order parameter equation in φ, predicts that equilibrium shift will be a third-order polynomial function of Δω with a cubic coefficient that is 1. (a) positive when the control parameter is constant 2. (b) negative when the control parameter decreases with Δω. The prediction was confirmed in an experiment that manipulated Δω through differential loadings and the control parameter through coupled frequency. Implications for the dynamical modelling of coordination were discussed.