Flip Dynamics Research Articles

Cluster formulation of spin glasses and the frustrated percolation model: statics and dynamics

We study the properties of the q-state frustrated bond percolation model by a Monte Carlo "bond flip" dynamics, using an algorithm originally devised by Sweeny and suitably modified to treat the presence of frustration. We analyze the percolation transition of the model, and find that it falls in the universality class of the q/2-state ferromagnetic Potts model. We then investigate the bond flip dynamics of the model, and find that for temperatures lower than the percolation transition Tp the relaxation functions show a two step decay, reminiscent of the relaxation of glass forming liquids. The long time decay (alpha-relaxation) is well fitted for T<Tp by a stretched exponential function, showing that in this model the relevant mechanism for the appearing of stretched exponentials is the percolation transition. At very low temperatures the relaxation functions develop a long plateau, as observed in glass forming liquids.

Ergodicity properties of energy conserving single spin flip dynamics in the XY model

A single spin flip stochastic energy conserving dynamics for the XY model is considered. We study the ergodicity properties of the dynamics. It is shown that phase space trajectories densely fill the geometrically connected parts of the energy surface. We also show that while the dynamics is discrete and the phase point jumps around, it cannot make transitions between closed disconnected parts of the energy surface. Thus the number of distinct sectors depends on the number of geometrically disconnected parts of the energy surface. Information on the connectivity of the surfaces is obtained by studying the critical points of the energy function. We study in detail the case of two spins and find that the number of sectors can be either one or two, depending on the external fields and the energy. For a periodic lattice in d dimensions, we find regions in phase space where the dynamics is non-ergodic and obtain a lower bound on the number of disconnected sectors. We provide some numerical evidence which suggests that such regions might be of small measure so that the dynamics is effectively ergodic.

Zero-temperature dynamic transition in the random field Ising model: a Monte Carlo study

The dynamics of a random (quenched) field Ising model (in two dimensions) at zero temperature in the presence of an additional sinusoidally oscillating homogeneous (in space) magnetic field has been studied by Monte Carlo simulation using the Metropolis single spin flip dynamics. The instantaneous magnetisation is found to be periodic with the same periodicity of the oscillating magnetic field. For very low values of amplitude of oscillating field and the width of randomly quenched magnetic field, the magnetisation oscillates asymmetrically about a nonzero value and the oscillation becomes symmetric about a zero value for higher values of amplitude of oscillating field and the width of the quenched disorder. The time-averaged magnetisation over a full cycle of the oscillating magnetic field defines the dynamic order parameter. This dynamic order parameter is nonzero for very low values of amplitude of oscillating magnetic field and the width of randomly quenched field. A phase boundary line is drawn in the plane formed by the amplitude of the oscillating magnetic field and the width of the randomly quenched magnetic field. A tricritical point has been located, on the phase boundary line, which separates the nature (discontinuous/continuous) of the dynamic transition.

A New Approach to Front Propagation Problems: Theory and Applications

In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics.

Monte Carlo study of the evolution of diffuse scattering and coherent modulation during h.c.p. to f.c.c. martensitic transition—II. Domain growth kinetics by spinodal process

Monte Carlo techniques are employed to study the kinetics of h.c.p. (2 H) to f.c.c. (3 C) martensitic transition by spin flip dynamics using a 1-D (one-dimensional) Ising model. The characteristic length scale is shown to exhibit t 0.5 dependence as confirmed by direct domain counting as well as dynamical scaling of the pair correlation functions. The diffuse scattering at late stages can be predicted using a scaling function for the pair correlations. The predicted evolution of the diffuse scattering pattern for the spinodal regime is in qualitative agreement with the experimentally observed patterns for ZnS single crystals.

Survival Probability of a Gaussian Non-Markovian Process: Application to theT=0Dynamics of the Ising Model

We study the decay of the probability for a non-Markovian stationary Gaussian walker not to cross the origin up to time $t$. This result is then used to evaluate the fraction of spins that do not flip up to time $t$ in the zero temperature Monte Carlo spin flip dynamics of the Ising model. Our results are compared to extensive numerical simulations.

Free volume microstructure of tetramethylpolycarbonate at low temperatures studied by positron annihilation lifetime spectroscopy: a comparison with polycarbonate

The annihilation of positrons in amorphous tetramethylpolycarbonate (TMPC) has been investigated in the temperature range from 30 to 300 K. The dependences of the mean lifetime of the ortho-positronium τ3 and its relative intensity I3 on temperature can be described by the empirical relations: τ3 = 2.08(1 + 4.8 × 10−4 T) (ns) and I3 = 20.2(1 + 7.9 × 10−4T) (%). These dependencies are interpreted within the framework of the microstructural free volume concept. The comparison with the polycarbonate (PC) reveals that both values τ3 and I3 are larger for TMPC, indicating larger free volume realized through higher number (concentration) of larger free volume entities. The temperature dependencies of τ3 and I3 for TMPC exhibit a different character and a weaker dependence on the temperature. This different behaviour of the free volume is due to a different packing efficiency of chains and a different dominant mechanism of the free volume generation, as well as a decrease in the flip dynamics of the phenyl rings. The implications of different free volume microstructures on some mechanical properties (e.g. relaxation and fracture) as well as on the transport properties of gases in TMPC and PC are discussed.

Dynamic exponent of the two-dimensional Ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix.

We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to $15\ifmmode\times\else\texttimes\fi{}15$, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent $z$ is determined as $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.1665(12)$.

Percolation techniques in disordered spin flip dynamics: Relation to the unique invariant measure

We consider lattice spin systems with short range but random and unbounded interactions. We give criteria for ergodicity of spin flip dynamics and estimate the speed of convergence to the unique invariant measure. We find for this convergence a stretched exponential in time for a class of “directed” dynamics (such as in the disordered Toom or Stavskaya model). For the general case, we show that the relaxation is faster than any power in time. No assumptions of reversibility are made. The methods are based on relating the problem to an oriented percolation problem (contact process) and (for the general case) using a slightly modified version of the multiscale analysis of e.g. Klein (1993).

EXTENDED APPLICATION OF CONSTRUCTIVE CRITERIA FOR ERGODICITY OF INTERACTING PARTICLE SYSTEMS

We discuss computational aspects of verifying constructive criteria for ergodicity of interacting particle systems. Both discrete time (probabilistic cellular automata) and continuous time spin flip dynamics are considered. We also investigate how the criteria have to be adapted if stirring is added to the dynamics.

When is an interacting particle system ergodic?

We consider stochastic flip dynamics for an infinite number of Ising spins on the lattice Έά. We find a sequence of constructive criteria for the system to be exponentially ergodic. The main idea is to approximate the continuous time process with discrete time processes (its Euler polygon) and to use an improved version of previous results (MS) about constructive ergodicity of discrete time processes.

Kinetics of a monomer-monomer model of heterogeneous catalysis

The kinetics of an irreversible monomer-monomer model of heterogeneous catalysis is examined. In this model, two reactive species, A and B, absorb and stick to single sites of a catalytic substrate. Surface reactions are assumed to occur only between dissimilar species which are nearest neighbours on the substrate. The kinetics of the process is studied in the reaction-controlled limit, where the adsorption occurs readily so that the process is limited by the reaction rate. The author maps the monomer-monomer model of heterogeneous catalysis on to a kinetic Ising model and solves the model exactly in one dimension, where the dynamics turns out to be a superposition of zero-temperature Glauber spin flip dynamics and infinite-temperature Kawasaki spin exchange dynamics. Finally, the author discusses the monomer-monomer processes with desorption.

Critical dynamics of the alternating bond kinetic Ising model

The critical dynamics of the alternating bond kinetic Ising model on a chain is examined. It is shown that previous nonuniversal results for the dynamical critical exponent z for single spin flip dynamics should be reinterpreted. The critical slowing down is due to two distinct contributions which, when separated, indicate that the value of z is universal. The exponent remains universal even when two spin flip dynamics are considered.

Crossover from Ising to mean-field critical behavior in a kinetic Ising model with competing flip and exchange dynamics

A two-dimensional kinetic Ising model evolving by a combination of spin flips at temperature T and spin exchanges which are random and of arbitrary range is investigated. We present a mean-field theory and compare its predictions with Monte Carlo simulations. The results indicate that the equilibrium Ising transition that is present without the spin exchanges turns into a mean-field transition as soon as the spin-exchange rate is different from zero. The crossover from Ising to mean-field critical behavior is characterized by an exponent, Φ ≈ 2, equal to the analogous crossover exponent in equilibrium systems.

Analysis of the aromatic 1H-NMR spectrum of the kringle 5 domain from human plasminogen. Evidence for a conserved kringle fold.

A kringle 5 domain fragment from human plasminogen has been investigated by 1H-NMR spectroscopy at 300 MHz and 620 MHz. The study focuses on the kringle 5 aromatic spectrum as aromatic side chains appear to mediate the binding of benzamidine. Spin-echo experiments and acid/base-titration studies in conjunction with two-dimensional double-quantum and chemical-shift-correlated spectroscopies were used to identify individual spin systems. Sequence-specific assignments of aromatic resonances are derived from direct comparison of the kringle 5 spectrum with spectra of the homologous kringle 1 and kringle 4 domains of plasminogen. As previously observed for kringles 1 and 4, the pattern we detect for Tyr9 in kringle 5 reflects a slow conformational exchange between two states in equilibrium, one in which the Tyr9 ring is freely mobile and one in which its flip dynamics are constrained. Proton Overhauser experiments in 1H2O and in 2H2O have been used to probe aromatic ring interactions and to identify residues which are part of the hydrophobic core centered at the Leu46 side chain. Overall, the data indicate a strong structural homology among the three plasminogen kringles.

Critical dynamics of the one-dimensional BEG model

We apply domain wall motion arguments to study the critical relaxation of the 1-d BEG model with single spin flip dynamics. Non-universal critical dynamical behaviour is found for the two order parameter dynamical exponents. These are shown to depend both on the rates chosen and on the system's couplings.

Reaction-diffusion equations for interacting particle systems

We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ɛ−2 the macroscopic density, defined on spatial scale ɛ−1, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.