Absorption of intense regular and noise waves in relaxing media

An integro-differential equation is written down that contains terms responsible for nonlinear absorption, visco-heat-conducting dissipation, and relaxation processes in a medium. A general integral expression is obtained for calculating energy losses of the wave with arbitrary characteristics—intensity, profile (frequency spectrum), and kernel describing the internal dynamics of the medium. It is shown that for weak waves, the general integral leads to well-known results of a linear approximation. Profiles of stationary solutions are constructed both for an exponential relaxation kernel and for other types of kernels. Energy losses at the front of week shock waves are calculated. General integral formulas are obtained for energy losses of intense noise, which are determined by the form of the kernel, the structure of the noise correlation function, and the mean square of the derivative of realization of a random process.     

Cluster kinetics for glassforming materials

We propose that cluster kinetics based on monomer-cluster, addition-dissociation and cluster fission-fusion provide a way to understand glassforming behavior. The four rate coefficients for the transformations are Arrhenius functions of temperature. Local equilibrium relations for the reversible rates provide expressions for monomer and amorphous cluster concentrations that are related to the Kauzmann temperature and supercooled liquid viscosity. Cluster dynamics are related to observable vitrification kinetics through free volume theory. With a single fitting parameter representing phase transformation energies, the double exponential (nonArrhenius) temperature dependence for viscosity and dielectric relaxation time compares well with experimental data. The one energy parameter alone provides an interpretation for strong and fragile glass materials.     

Glauber dynamics of the asymmetric SK-model

We solve the Glauber dynamics of the fully asymmetric SK-model and some of its extensions: Incorporation of random external fields and additional storage of a finite number of patterns. An analytical expression for theT=0 autocorrelation function of the equilibrium dynamics is derived and compared with the asymptotic exponential decay. The dynamical behaviour of the fully asymmetric SK-model is also investigated in a situation with nonequilibrium initial conditions. For weak correlations between the synaptic couplings a first order perturbation expansion is made yielding a finite relaxation time for the response function and a vanishing EA-order parameter if the asymmetry is high enough.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

An extended chain Ising model and its Glauber dynamics

It was first proposed that an extended chain Ising (ECI) model contains the Ising chain model, single spin double-well potentials and a pure phonon heat bath of a specific energy exchange with the spins. The extension method is easy to apply to high dimensional cases. Then the single spin-flip probability (rate) of the ECI model is deduced based on the Boltzmann principle and general statistical principles of independent events and the model is simplified to an extended chain Glauber-Ising (ECGI) model. Moreover, the relaxation dynamics of the ECGI model were simulated by the Monte Carlo method and a comparison with the predictions of the special chain Glauber-Ising (SCGI) model was presented. It was found that the results of the two models are consistent with each other when the Ising chain length is large enough and temperature is relative low, which is the most valuable case of the model applications. These show that the ECI model will provide a firm physical base for the widely used single spin-flip rate proposed by Glauber and a possible route to obtain the single spin-flip rate of other form and even the multi-spin-flip rate.     

Changing shapes in the nanoworld

What are the mechanisms leading to the shape relaxation of three-dimensional crystallites? Kinetic Monte Carlo simulations of fcc clusters show that the usual theories of equilibration, via atomic surface diffusion driven by curvature, are verified only at high temperatures. Below the roughening temperature, the relaxation is much slower, kinetics being governed by the nucleation of a critical germ on a facet. We show that the energy barrier for this step linearly increases with the size of the crystallite, leading to an exponential dependence of the relaxation time.     

A stochastic theory of non-ideal behaviour

An approximate theory is proposed for the thermal relaxation of condensed binary [A, B] systems. During the relaxation the composition varies, or is kept constant. (The former case corresponds to an Ising lattice, the latter to a binary alloy, for example.) A variation of the composition is described with the help of a stochastic exchange A?B (as in the Glauber Model), but the microscopic transition probabilities are replaced by smoothed probabilities which depend on the system's composition and internal energy, as proposed in Part I. The variation of the internal energy is estimated with the help of a dynamic counter-part of the quasi-chemical approximation. The equations can be integrated numerically, describing the time dependence of the system's composition (if varying), internal energy, entropy and of intensive parameters corresponding to transient ‘quasi-equilibria’. The theory is applied to an Ising lattice undergoing heating or cooling through the critical temperature and compared with Monte Carlo simulation of identical processes. The agreement is judged to be very good, especially since it involves no parameter fitting. The relaxation of an [A, B] system of fixed composition, undergoing cooling to inside the co-existence curve is also studied. Three alternative lines for the evolution of internal energy are estimated: (i) unstable, with no decomposition into two phases; (ii) metastable, with spinodal decomposition and (iii) stable, with binodal decomposition. Comparison with a computer simulation of the process indicates that the initial relaxation (up to order of 100 trial exchanges per particle, depending on the system's size) fits the metastable line, the initial stage being followed by an extremely slow crossover to the stable line.     

Long-Time Behavior for the 1-D Stochastic Ising Model with Unbounded Random Couplings

We consider the ferromagnetic Ising model with Glauber spin flip dynamics in one dimension. The external magnetic field vanishes and the couplings are i.i.d. random variables. If their distribution has compact support, the disorder averaged spin auto-correlation function has an exponential decay in time. We prove that, if the couplings are unbounded, the decay switches to either a power law or a stretched exponential, in general.     

Simulations of the flipping images and microparameters of molecular orientations in liquids according to the molecule string model

The relaxation dynamics of liquids is one of the fundamental problems in liquid physics,and it is also one of the key issues to understand the glass transition mechanism.It will undoubtedly provide enlightenment on understanding and calculating the relaxation dynamics if the molecular orientation flipping images and relevant microparameters of liquids are studied.In this paper,we first give five microparameters to describe the individual molecular string(MS) relaxation based on the dynamical Hamiltonian of the MS model,and then simulate the images of individual MS ensemble,and at the same time calculate the parameters of the equilibrium state.The results show that the main molecular orientation flipping image in liquids(including supercooled liquid) is similar to the random walk.In addition,two pairs of the parameters are equal,and one can be ignored compared with the other.This conclusion will effectively reduce the difficulties in calculating the individual MS relaxation based on the single-molecule orientation flipping rate of the general Glauber type,and the computer simulation time of interaction MS relaxation.Moreover,the conclusion is of reference significance for solving and simulating the multi-state MS model.     

Simulations of the flipping images and microparameters of molecular orientations in liquids according to molecule string model

The relaxation dynamics of liquids is one of the fundamental problems in liquid physics, and it is also one of the key issues to understand the glass transition mechanism. It will undoubtedly give enlightenments on understanding and calculating the relaxation dynamics if the molecular orientation flipping images and relevant microparameters of liquids are studied. In this paper, we first give five microparameters to describe the individual molecular string (MS) relaxation based on the dynamical Hamiltonian of the MS model, and then simulate the images of individual MS ensemble, at the same time calculate the parameters of the equilibrium state. The results show that the main molecular orientation flipping image in liquids (including supercooled liquid) is similar to the random walk. In addition, two pairs of the parameters are equal, and one can be ignored compared with the other. This conclusion will effectively reduce the difficulties in calculating the individual MS relaxation based on the single-molecule orientation flipping rate of general Glauber type, and the computer simulation time of interaction MS relaxation. Moreover, the conclusion has no doubt of the reference significance for solving and simulating the multi-state MS model.     

Potential energy landscape and long-time dynamics in a simple model glass

We analyze the properties of a Lennard-Jones system at the level of the potential energy landscape. After an exhaustive investigation of the topological features of the landscape of the systems, obtained by studying small size samples, we describe the dynamics of the systems in multidimensional configurational space by means of a simple model. This considers the configurational space as a connected network of minima where the dynamics proceeds by jumps described by an appropriate master equation. Using this model we are able to reproduce the long-time dynamics and the low temperature regime. We investigate both the equilibrium regime and the off-equilibrium one, finding those typical glassy behaviors usually observed in the experiments such as (i) a stretched exponential relaxation, (ii) a temperature-dependent stretching parameter, (iii) a breakdown of the Stokes-Einstein relation, and (iv) the appearance of a critical temperature below which one observes a deviation from the fluctuation-dissipation relation as a consequence of the lack of equilibrium in the system.     

The expanded triangular Kitaev–Heisenberg model in the full parameter space

The classical Kitaev–Heisenberg model on the triangular lattice is investigated by simulation in its full parameter space together with the next-nearest neighboring Heisenberg interaction or the single-ion anisotropy. The variation of the system is demonstrated directly by the joint density of states (DOS) depending on energy and magnetization obtained from Wang–Landau algorithm. The Metropolis Monte Carlo simulation and the zero-temperature Glauber dynamics are performed to show the internal energy, the correlation functions and spin configurations at zero temperature. It is revealed that two types of DOS (U and inverse U) divide the whole parameter range into two main parts with antiferromagnetic and ferromagnetic features respectively. In the parameter range of U type DOS, the mixed frustration from the triangular geometry and the Kitaev interaction produces rich phases, which are influenced in different ways by the next-nearest neighboring Heisenberg interaction and the single-ion anisotropy.     

Measurement of the relaxation rate of the magnetization in Mn12O12-acetate using proton NMR echo

We present a novel method to measure the relaxation rate W of the magnetization of Mn 12O (12)-acetate (Mn12) magnetic molecular cluster in its S = 10 ground state at low T. It is based on the observation of an exponential growth in time of the proton NMR signal during the thermal equilibration of the magnetization of the molecules. We can explain the novel effect with a simple model which relates the intensity of the proton echo signal to the microscopic reversal of the magnetization of each individual Mn12 molecule during the equilibration process. The method should find wide application in the study of magnetic molecular clusters in off-equilibrium conditions.     

Spectrum of relaxation times for ising spin clusters in random fields

Summary Exact results on the single-spin-flip Glauber dynamics of six-coupled random field Ising spins with the coordination number of four are presented. Two distributions of random fields (RF), binary (BD) and Gaussian (GD) ones, are investigated. The effects of the static magnetic field are discussed. In the zero-magnetic-field case, the number of diverging relaxation times is equal to the number of energy minima minus one. This rule breaks in the presence of a magnetic field. The longest relaxation times in the absence of the field verify the Arrhenius law with the energy barrier determined by the energy needed to invert the ground-state spin configuration. At low temperature, according to the Arrhenius law, the spectrum of relaxation times shows a two-peaked distribution on a logarithmic scale. In the GD case of RF, the energy barrier distribution is continuous, while it is quasi-discrete in the BD case.     

Universal Long-Time Relaxation on Lattices of Classical Spins: Markovian Behavior on Non-Markovian Timescales

The long-time behavior of certain fast-decaying infinite temperature correlation functions on one-, two-, and three-dimensional lattices of classical spins with various kinds of nearest-neighbor interactions is studied numerically, and evidence is presented that the functional form of this behavior is either simple exponential or exponential multiplied by cosine. Due to the fast characteristic timescale of the long-time decay, such a universality cannot be explained on the basis of conventional Markovian assumptions. It is suggested that this behavior is related to the chaotic properties of the spin dynamics.     

Modelling Debye Dielectric Relaxation in Monohydroxy Alcohols

The Debye relaxation of dielectric spectroscopy exists extensively in monohydroxy alcohols.We model the relaxation based on the infinite-pseudospin-chain Ising model and the Glauber dynamics,and the corresponding complex permittivity is obtained.The model results are in good agreement with the experimental data of 3,7-dimethyl-1-octanol,2-ethyl-1-hexanol and 5-methyl-2-hexanol in a wide temperature range.Moreover,in the model parameters,the sum of the mean-field interaction energy and two times the orientation is nearly twice the hydrogen bond energy,which further states the rationality of this model.     

Parallel cluster labeling for large-scale Monte Carlo simulations

We present an optimized version of a cluster labeling algorithm previously introduced by the authors. This algorithm is well suited for large-scale Monte Carlo simulations of spin models using cluster dynamics on parallel computers with large numbers of processors. The algorithm divides physical space into rectangular cells which are assigned to processors and combines a serial local labeling procedure with a relaxation process across nearest-neighbor processors. By controlling overhead and reducing inter-processor communication this method attains good computational speed-up and efficiency. Large systems of up to 65536² spins have been simulated at updating speeds of 11 nanosecs/site (90.7 × 10⁶ spin updates/sec) using state-of-the-art supercomputers. In the second part of the article we use the cluster algorithm to study the relaxation of magnetization and energy on large Ising models using Swendsen-Wang dynamics. We found evidence that exponential and power law factors are present in the relaxation process as has been proposed by Hackl et al. The variation of the power-law exponent λ_M taken at face value indicates that the value of Z_M falls in the interval 0.31–0.49 for the time interval analysed and appears to vanish asymptotically.     

Strong decay to equilibrium in one-dimensional random spin systems

We consider a spin system on a lattice with finite-range, possibly unbounded random interactions. We show that for such systems the Glauber dynamics cannot decay to equilibrium exponentially fast inL ₂ even at high temperatures. Additionally, for one-dimensional systems with unbounded random couplings we prove that with probability one the corresponding Glauber dynamics has a fast (subexponential) decay to equilibrium in the uniform norm, provided that the distribution of random couplings satisfies some exponential bound.     

Dielectric relaxation in nanopillar NiFe–silicon structures in high magnetic fields

We explore the dielectric relaxation properties of NiFe nanowires in a nanoporous silicon template. Dielectric data of the NiFe–silicon structure show a strong relaxation resonance near 30 K. This system shows Arrhenius type of behavior in the temperature dependence of dissipation peaks vs. frequency. We report magnetic field dependence of dipolar relaxation rate and the appearance of structure in the dielectric spectrum related to multiple relaxation rates. A magnetic field affects both the exponential prefactor in the Arrhenius formula and the activation energy. From this field dependence we derive a simple exponential field dependence for the prefactor and linear field approximation for the activation energy which describes the data. We find a significant angular dependence of the dielectric relaxation spectrum for regular silicon and nanostructured silicon vs. magnetic field direction, and describe a simple sum rule that describes this dependence. We find that although similar behavior is observed in both template and nanostructured materials, the NiFe–silicon shows a more complex, magnetic field dependent relaxation spectrum.     

Thermodynamic picture of the glassy state gained from exactly solvable models

A picture for thermodynamics of the glassy state was introduced recently by us [Phys. Rev. Lett. 79, 1317 (1997); 80, 5580 (1998)]. It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, nonequilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper, the underlying arguments are discussed in greater length. The main part of the paper involves details of the exact dynamical solution of two simple models introduced recently: uncoupled harmonic oscillators subject to parallel Monte Carlo dynamics, and independent spherical spins in a random field with such dynamics. At low temperature, the relaxation time of both models diverges as an Arrhenius law, which causes glassy behavior in typical situations. In the glassy regime, we are able to verify the above-mentioned relations for the thermodynamics of the glassy state. In the course of the analysis, it is argued that stretched exponential behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime.