Dynamic Order Parameter Research Articles

Dynamic Dipole and Quadrupole Phase Transitions in the Kinetic Metamagnetic Spin –3 / 2 Blume-Emery-Griffiths Model

Abstract We study, within a mean-field approach, the stationary states of the kinetic metamagnetic spin −3/2 Blume-Emery-Griffiths model under the presence of a time-varying (sinusoidal) magnetic field.We use the Glauber-type stochastic dynamics to describe the time evolution of the system. The behaviour of the time dependence of the average order parameters in a period, which are also called the dynamic order parameters, as functions of the reduced temperature are investigated. The natures (continuous or discontinuous) of the transitions are characterized by investigating the behaviours of the thermal variations of the dynamic order parameters. The dynamic phase transition points are obtained and the phase diagrams are constructed in the plane of the reduced temperature (T) and the amplitude of the magnetic field (h), and sixteen fundamental types of phase diagrams are found. The phase diagrams exhibit one, two, three or four dynamic tricritical points and a dynamic double-critical end point, and, besides a disordered and three ordered phases, seven coexistence regions or mixed phases depending on the interaction parameters. We also investigate the influence of the reduced biquadratic exchange parameter (k) and obtain nine different phase diagram topologies in the (k,T) plane

Columnar Packing Motifs of Functionalized Perylene Derivatives: Local Molecular Order Despite Long-Range Disorder

We elucidate local packing motifs and dynamical order parameters in a perylene tetracarboxydiimide derivative (C(8,7)-PDI), one of the most promising candidates for rationally designed, self-assembling, and self-healing molecular wires. Spectroscopic fingerprints obtained from solid-state NMR spectroscopy are interpreted by means of first-principles calculations and molecular dynamics simulations. The interplay of steric repulsion, H bonding, and pi-pi packing effects leads to a specific relative molecular pitch angle of approximately 35 +/- 10 degrees between successive molecules in the stack. Dynamical order parameters, determined from NMR sideband patterns as a measure of molecular motion, yield values of S approximately = 1.0 in the core of the columnar stack, corresponding to an almost frozen molecular dynamics at ambient temperature. This rigidity is compatible with characteristic intermolecular distances obtained from dipolar couplings between specific hydrogens via double-quantum NMR experiments and further supported by ab initio calculations.

Order Parameter Dynamics of Body-scaled Hysteresis and Mode Transitions in Grasping Behavior

Several experimental studies have shown that human grasping behavior exhibits a transition from one-handed to two-handed grasping when to-be-grasped objects become larger and larger. The transition point depends on the relative size of objects measured in terms of human body-scales. Most strikingly, the transitions between the two different behavioral 'modes' of grasping exhibit hysteresis. That is, one-to-two hand transitions and two-to-one hand transitions occur at different relative object sizes when objects are scaled up or down in size. In our study we approach body-scaled hysteresis and mode transitions in grasping by exploiting the notion that human behavior in general results from self-organization and satisfies appropriately-defined order parameter equations. To this end, grasping transitions and grasping hysteresis are discussed from a theoretical perspective in analogy to cognitive processes defined by Haken's neural network model for pattern recognition. In doing so, issues such as the exclusivity of grasping modes, biomechanical constraints, mode-mode interactions, single subject behavior and population behavior are explored.

Dynamic phase transitions and dynamic phase diagrams in the kinetic mixed spin-1 and spin-2 Ising system in an oscillating magnetic field

We study the nonequilibrium aspects of the kinetics of a mixed spin-1 and spin-2 Ising system including the biquadratic nearest-neighbor pair interaction (K) and crystal-field interaction (D) under the presence of a time-dependent oscillating external magnetic field within a mean-field approach. In particular, we calculate the dynamic phase transition (DPT) temperatures and present dynamic phase diagrams. The set of mean-field dynamic equations is obtained by employing Glauber transition rates. We investigate the time variations of average order parameters to find the phases in the system. We also study the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the phase transitions and obtain the DPT points. The dynamic phase diagrams are presented in three different planes. Phase diagrams contain disordered (d), ferrimagnetic (i), antiquadrupolar or staggered (a) phases, and four coexistence or mixed phase regions, namely i+d, i+a, i+a+d and a+d mixed phases, that strongly depend on interaction parameters.

Slow Motions in Chicken Villin Headpiece Subdomain Probed by Cross-Correlated NMR Relaxation of Amide NH Bonds in Successive Residues

The villin headpiece subdomain (HP36) is a widely used system for protein-folding studies. Nuclear magnetic resonance cross-correlated relaxation rates arising from correlated fluctuations of two N-HN dipole-dipole interactions involving successive residues were measured at two temperatures at which HP36 is at least 99% folded. The experiment revealed the presence of motions slower than overall tumbling of the molecule. Based on the theoretical analysis of the spectral densities we show that the structural and dynamic contributions to the experimental cross-correlated relaxation rate can be separated under certain conditions. As a result, dynamic cross-correlated order parameters describing slow microsecond-to-millisecond motions of N-H bonds in neighboring residues can be introduced for any extent of correlations in the fluctuations of the two bond vectors. These dynamic cross-correlated order parameters have been extracted for HP36. The comparison of their values at two different temperatures indicates that when the temperature is raised, slow motions increase in amplitude. The increased amplitude of these fluctuations may reflect the presence of processes directly preceding the unfolding of the protein.

Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker–Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε , the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O ( ε 2 ) . Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O ( ε 2 ) solution of the Liouville equation yields an equation for W to O ( ε 3 ) . An augmented Smoluchowski equation for W is obtained from the O ( ε 2 ) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.

Effect of meta-Carborane on Segmental Dynamics in a Bimodal Poly(dimethylsiloxane) Network

Bimodal networks of polydimethylsiloxane (PDMS) filled with varying amounts of icosahedral meta-carborane (m-CB) have been developed and characterized by broadband dielectric spectroscopy (BDS) and static 1H multiple quantum nuclear magnetic resonance (MQ NMR). Both BDS and MQ NMR showed evidence for a decrease in the polymer chain dynamics. BDS spectra quantified a normal-mode relaxation near 40 Hz at 40 °C. The frequency maximum observed for filled samples decreased with increasing m-CB content until contents greater than 5 wt %. The width of the relaxation spectrum increased with the addition of small quantities of filler and decreased with filler contents greater that 5 wt %. Agglomeration effects were observed at loadings greater than 5 wt % as manifest by the onset of low frequency Maxwell−Wagner−Sillars (MWS) processes. The MQ NMR data allowed the characterization of distributions of the residual dipolar couplings, <Ωd> and, thus, in the dynamic order parameter, Sb, consistent with the bimodal network architecture expected from the synthesis protocol used. Upon addition of less than 10 wt % m-CB filler, the mean <Ωd> for the longer chains increased by 46% and the width of the distribution increased by 33%. The mean <Ωd> for the shorter chains increased by much less, indicative of preferential dispersion of the filler particles in the long chain domains of the network structure. We conclude that the mechanism of reinforcement is likely free volume space filling at low loadings transitioning to complex molecular filler and polymer chain interaction phenomena at higher loadings.

Experimental Test of Curvature-Driven Dynamics in the Phase Ordering of a Two Dimensional Liquid Crystal

We study electric field driven deracemization in an achiral liquid crystal through the formation and coarsening of chiral domains. It is proposed that deracemization in this system is a curvature-driven process. We test this prediction using the recently obtained exact result for the distribution of hull-enclosed areas in two-dimensional coarsening with nonconserved scalar order parameter dynamics [J. J. Arenzon et al., Phys. Rev. Lett. 98, 145701 (2007)]. The experimental data are in very good agreement with the theory. We thus demonstrate that deracemization in such bent-core liquid crystals belongs to the Allen-Cahn universality class, and that the exact formula, which gives us the statistics of domain sizes during coarsening, can also be used as a strict test for this dynamic universality class.

Polar and nonpolar orderings in the electrically induced isotropic-nematic phase transition

We study the dynamics of the isotropic-nematic phase transition caused by an applied electric field at the time scales of dielectric relaxation. In the classic Landau-Khalatnikov theory of the phase transition dynamics, the nematic (nonpolar) order parameter is an instantaneous function of the applied field. We demonstrate that, when the field is changing faster than the time of dielectric relaxation, the induced polar order dynamics influences the dynamics of the nonpolar order parameter. We develop a model based on the Langevin equation to describe the simultaneous dynamics of both polar and nonpolar order parameters; the model is supported by experiment.

Nonequilibrium phase transition in the kinetic Ising model on a two-layer square lattice under the presence of an oscillating field

The nonequilibrium or dynamic phase transitions are studied, within a mean-field approach, in the kinetic Ising model on a two-layer square lattice consisting of spin- 1/2 ions in the presence of a time varying (sinusoidal) magnetic field has been studied by using Glauber-type stochastic dynamics. The dynamic equations of motion are obtained in terms of the intralayer coupling constants J 1 and J 2 for the first and second layer, respectively, and interlayer coupling constant J 3 between these two layers. The nature (first- or second-order) of the transitions is characterized by investigating the behavior of the thermal variations of the dynamic order parameters. The dynamic phase transitions are obtained and the dynamic phase diagrams are constructed in the plane of the reduced temperature versus the amplitude of the magnetic field and found fourteen fundamental types of phase diagrams. Phase diagrams exhibit one, two or three dynamic tricritical points for various values of J 2 / | J 1 | and J 3 / | J 1 | . Besides the paramagnetic (p), ferromagnetic (f) and compensated (c) phases, there were the f+c,f+sf,c+sf,af+p,m+p,f+m and c+af, where the af, sf and m are the antiferromagnetic, surface ferromagnetic and mixed phases respectively. Coexistence phase regions also exist in the system.

Theory of physical aging in polymer glasses

A statistical segment scale theory for the physical aging of polymer glasses is proposed and applied. The approach is based on a nonlinear stochastic Langevin equation of motion and the concept of an effective free energy which quantifies temporary localization, collective barriers, and the activated segment hopping process. The key collective structural variable that plays the role of the dynamic order parameter for aging is the experimentally measurable nanometer and longer wavelength amplitude of density fluctuations, S0 . The degree of local cooperativity, and the bare activation energy of the high-temperature Arrhenius process, are determined in the molten state by utilizing experimental measurements of the glass temperature and dynamic crossover time, respectively. A first-order kinetic equation with a time varying rate is proposed for the temporal evolution of S0 which is self-consistently and nonlinearly coupled with the mean segmental relaxation time. The theory has been applied to study physical aging of the alpha relaxation time, shear relaxation modulus, amplitude of density fluctuations, cohesive energy, absolute yield stress, and fictive temperature of polymethylmethacrylate and other glasses over a range of temperatures. Temperature-dependent logarithmic and effective power-law aging is predicted at intermediate times. Time-aging time superposition is found for the mechanical relaxation function. A strongly asymmetric aging response is predicted for up and down temperature jump experiments. Comparison of the approach with the classic phenomenological model is presented.

Observing bifurcation and resonance in a mean-field coupled periodically driven noisy overdamped oscillators by the method of moments

A method of moments for calculating long-term order parameter dynamics of mean-field coupled periodically driven overdamped oscillators is described based on nonlinear Fokker–Planck equation. Using this technique the bifurcation behavior and stochastic resonance of the order parameter is observed, and the accuracy is verified by comparing with Gaussian moment method and stochastic simulation. The technique can be used in nonharmonic external excitation case.

A semi-analytic method for computing the long-time order parameter dynamics in mean-field coupled overdamped oscillators with colored noises

The order parameter dynamics of a mean-field model is frequently investigated in macroscopic cumulant dynamics, from which a bifurcation can be predicted qualitatively. In this Letter, for quantitatively investigating the long-time order parameter dynamics, a semi-analytic method is proposed based on approximate nonlinear Fokker–Planck equations. Applying the new method to the mean-field model of periodically driven overdamped bistable oscillators with colored noise, we exhibit the bifurcation behavior and the nonlinear stochastic resonance of the order parameter by tuning noise intensity or coupling coefficient, and the accuracy of the new method are verified by direct simulation. Our observations disclose some new properties about the order parameter dynamics of the mean-field model. For example, the periodic signal shifts the critical coupling coefficient to a larger value, while the nonzero correlation time of the colored noise shifts it to a lower value. Our observation also discloses that there is no quantitatively corresponding relation between the resonant peak and the critical bifurcation parameter of the Gaussian moment system.

Market response to external events and interventions in spherical minority games

We solve the dynamics of large spherical minority games (MG) in the presence of non-negligible time-dependent external contributions to the overall market bid. The latter represent the actions of market regulators or other major natural or political events that impact on the market. In contrast to non-spherical MGs, the spherical formulation allows one to derive closed dynamical order parameter equations in an explicit form and work out the market's response to such events fully analytically. We focus on a comparison between the response to stationary versus oscillating market interventions, and reveal profound and partially unexpected differences in terms of transition lines and the volatility.

Effective-field theory on the kinetic Ising model

As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the square lattice ( Z = 4 ) and the simple cubic lattice ( Z = 6 ) , respectively. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. In the field amplitude h 0 / Z J –temperature T / Z J plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn, and the dynamical tricritical point has been observed. We also make the compare results of EFT with that given by using the mean field theory (MFT).

Kinetics of a mixed spin-1 and spin-3/2 Ising system under a time-dependent oscillating magnetic field

We present a study, within a mean-field approach, of the kinetics of the mixed spin-1 and spin-3/2 Ising model Hamiltonian with bilinear and biquadratic nearest-neighbor exchange interactions and a single-ion potential or crystal-field interaction in the presence of a time-dependent oscillating external magnetic field. We employ the Glauber transition rates to construct the mean-field dynamical equations. We investigate the time dependence of average magnetizations and the quadrupole moments, and the thermal behavior of the dynamic order parameters. From these studies, we obtain the dynamic phase transition (DPT) points and construct the phase diagrams in three different planes. Phase diagrams contain disordered (d) , ferrimagnetic (i) , the antiquadrupolar or staggered (a) phases, and four coexistence or mixed phase regions, namely, the i+d , i+a , i+a+d , and a+d , that strongly depend on interaction parameters. The system also exhibits the dynamic tricritical behavior in most cases, the reentrant behavior in few cases.

How do ensembles occupy space?

To find an answer to the title question, an attractiveness function between agents and locations is introduced yielding a phenomenological but generic model for the search for optimal distributions of agents over space. Agents can be seen as, e.g., members of biological populations like colonies of bacteria, swarms, and so on. The global attractiveness between agents and locations is maximized causing (self-propelled) `motion' of agents and, eventually, distinct distributions of agents over space. At the same token spontaneous changes or `decisions' are realized via competitions between agents as well as between locations. Hence, the model's solutions can be considered a sequence of decisions of agents during their search for a proper location. Depending on initial conditions both optimal as well as suboptimal configurations can be reached. For the latter early decision-making are important for avoiding possible conflicts: if the proper moment is missed, then only a few agents can find an optimal solution. Indeed, there is a delicate interplay between the values of the attractiveness function and the constraints as can be expressed by distinct terms of a potential function containing different Lagrange parameters. The model should be viewed as a top-down approach as it describes the dynamics of order parameters, i.e. macroscopic variables that reflect affiliations between agents and locations. The dynamics, however, is modified via so-called cost functions that are interpreted in terms of affinity levels. This interpretation can be seen as an original step towards an understanding of the dynamics at the underlying microscopic level. When focusing on the agent, one may say that the dynamics of an order parameter shows the evolution of an agent's intrinsic `map' for solving the problem of space occupation. Importantly, the dynamics does not necessarily distinguish between evolving (or moving) agents and evolving (or moving) locations though agents are more likely to be actors than the locations. Put differently, an order parameter describes an internal map which is linked to the expectation of an agent to find a certain location. Owing to the dynamical representation, we can therefore follow up the change of these maps over time leading from uncertainty to certainty.

Enveloped viruses understood via multiscale simulation: computer-aided vaccine design

Enveloped viruses are viewed as an opportunity to understand how highly organized and functional biosystems can emerge from a collection of millions of chaotically moving atoms. They are an intermediate level of complexity between macromolecules and bacteria. They are a natural system for testing theories of self-assembly and structural transitions, and for demonstrating the derivation of principles of microbiology from laws of molecular physics. As some constitute threats to human health, a computer-aided vaccine and drug design strategy that would follow from a quantitative model would be an important contribution. However, current molecular dynamics simulation approaches are not practical for modeling such systems. Our multiscale approach simultaneously accounts for the outer protein net and inner protein/genomic core, and their less structured membranous material and host fluid. It follows from a rigorous multiscale deductive analysis of laws of molecular physics. Two types of order parameters are introduced: (1) those for structures wherein constituent molecules retain long-lived connectivity (they specify the nanoscale structure as a deformation from a reference configuration) and (2) those for which there is no connectivity but organization is maintained on the average (they are field variables such as mass density or measures of preferred orientation). Rigorous multiscale techniques are used to derive equations for the order parameters dynamics. The equations account for thermal-average forces, diffusion coefficients, and effects of random forces. Statistical properties of the atomic-scale fluctuations and the order parameters are co-evolved. By combining rigorous multiscale techniques and modern supercomputing, systems of extreme complexity can be modeled.

Multicritical dynamical phase diagrams of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling in an oscillating field

We study, within a mean-field approach, the stationary states of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling under the presence of a time-varying (sinusoidal) magnetic field. We employ the Glauber-type stochastic dynamics to construct set of dynamic equations of motion. The behavior of the time dependence of the order parameters and the behavior of the average order parameters in a period, which is also called the dynamic order parameters, as functions of the reduced temperature are investigated. The dynamic phase transition points are calculated and phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane. The dynamical transition from one regime to the other can be of first- or second order depending on the region in the phase diagram. According to the values of the crystal field interaction or single-ion anisotropy constant and biquadratic exchange constant, we find 20 fundamental types of phase diagrams which exhibit many dynamic critical points, such as tricritical points, zero-temperature critical points, double critical end points, critical end point, triple point and multicritical point. Moreover, besides a disordered and ordered phases, seven coexistence phase regions exist in the system.

Kinetics of the spin-2 Blume-Capel model under a time-dependent oscillating external field

Within a mean-field approach and using the Glauber-type stochastic dynamics, we study the kinetics of the spin-2 Blume-Capel model in the presence of a time-varying (sinusoidal) magnetic field. We investigate the time dependence of the average order parameter and the behavior of the average order parameter in a period, which is also called the dynamic order parameter, as a function of the reduced temperature. The nature (continuous and discontinuous) of the transition is characterized by the dynamic order parameter. The dynamic phase transition points are obtained and the phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane. The phase diagrams exhibit one dynamic tricritical point; besides a disordered and an ordered phases, there are three phase coexistence regions that are strongly dependent on the interaction parameter.