Slow Dynamics of Ising Models with Energy Barriers

Using Monte Carlo simulations we study the dynamics of three-dimensional Ising models with nearest-, next-nearest-, and four-spin (plaquette) interactions. During coarsening, such models develop growing energy barriers, which leads to very slow dynamics at low temperature. As already reported, the model with only the plaquette interaction exhibits some of the features characteristic of ordinary glasses: strong metastability of the supercooled liquid, a weak increase of the characteristic length under cooling, stretched-exponential relaxation, and aging. The addition of two-spin interactions, in general, destroys such behavior: the liquid phase loses metastability and the slow-dynamics regime terminates well below the melting transition, which is presumably related with a certain corner-rounding transition. However, for a particular choice of interaction constants, when the ground state is strongly degenerate, our simulations suggest that the slow-dynamics regime extends up to the melting transition. The analysis of these models leads us to the conjecture that in the four-spin Ising model domain walls lose their tension at the glassy transition and that they are basically tensionless in the glassy phase.

Solvable dynamics in a system of interacting random tops

A new solvable model of synchronization dynamics is introduced. It consists of a system of long range interacting tops or magnetic moments with random precession frequencies. The model allows for an explicit study of orientational effects in synchronization phenomena as well as nonlinear processes in resonance phenomena in strongly coupled magnetic systems. A stability analysis of the incoherent solution is performed for different types of orientational disorder. A system with orientational disorder always synchronizes in the absence of noise.

Exactly solvable phase oscillator models with syncrhonization dynamics

Populations of phase oscillators interacting globally through a general coupling function f(x) have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, f(x) = sgn(x), have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved.

Critical clusters and efficient dynamics for frustrated spin models

A general method to find, in a systematic way, efficient Monte Carlo cluster dynamics among the avast class of dynamics introduced by Kandel et al. [Phys. Rev. Lett. 65, 941 (1990)] is proposed. The method is successfully applied to a class of frustrated two-dimensional Ising systems. In the case of the fully frustrated model, we also find the intriguing result that critical clusters consist of self-avoiding walk at the theta point.

Potts fully frustrated model: Thermodynamics, percolation and dynamics in two dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore, this transition can be mapped onto a percolation transition. We numerically study the phase diagram in two dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of (i) the model with frustration and disorder and (ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

Percolation and cluster Monte Carlo dynamics for spin models

A general scheme for devising efficient cluster dynamics proposed in a previous paper [Phys. Rev. Lett. 72, 1541 (1994)] is extensively discussed. In particular, the strong connection among equilibrium properties of clusters and dynamic properties as the correlation time for magnetization is emphasized. The general scheme is applied to a number of frustrated spin models and the results discussed.

Relaxation dynamics in suspensions of ferromagnetic particles

We have studied the relaxation dynamics of a dilute assembly of ferromagnetic particles in suspension. A formalism based on the Smoluchowski equation, describing the evolution of the probability density for the directions of the magnetic moment and of the axis of easy magnetization of the particles, has been developed. We compute the rotational viscosity from a Green-Kubo formula and give an expression for the relaxation time of the particles which comes from the dynamic equations of the correlation functions. Concerning the relaxation time for the particles, our results agree quite well with experiments performed on different samples of ferromagnetic particles for which the magnetic energy, associated with the interaction between the magnetic moments and the external field, or the energy of anisotropy plays a dominant role.

Noise and dynamics of self-organized critical phenomena

Different microscopic models exhibiting self-organized criticality are studied numerically and analytically. Numerical simulations are performed to compute critical exponents, mainly the dynamical exponent, and to check universality classes. We find that various models lead to the same exponent, but this universality class is sensitive to disorder. From the dynamic microscopic rules we obtain continuum equations with different sources of noise, which we call internal and external. Different correlations of the noise give rise to different critical behavior. A model for external noise is proposed that makes the upper critical dimensionality equal to 4 and leads to the possible existence of a phase transition above d=4. Limitations of the approach of these models by a simple nonlinear equation are discussed.

Invaded cluster dynamics for frustrated models

The invaded cluster (IC) dynamics introduced by Machta et al. [Phys. Rev. Lett. 75, 2792 (1995)] is extended to the fully frustrated Ising model on a square lattice. The properties of the dynamics that exhibits numerical evidence of self-organized criticality are studied. The fluctuations in the IC dynamics are shown to be intrinsic of the algorithm and the fluctuation-dissipation theorem is no longer valid. The relaxation time is found to be very short and does not present a critical size dependence.

Evaporation and coarsening dynamics with open boundaries

We present a study of the evaporation dynamics of a substance undergoing a coarsening process. The system is modeled by the Cahn-Hilliard equation with absorbing boundaries. We have found that the dynamics, although of a diffusive nature, is much slower than the usual one without coarsening. Analytical and simulation results are in reasonable agreement.

Excitability transitions and wave dynamics under spatiotemporal structured noise

We present an analytic and numerical study of the effects of external fluctuations in active media. Our analytical methodology transforms the initial stochastic partial differential equations into an effective set of deterministic reaction-diffusion equations. As a result we are able to explain and make quantitative predictions on the systematic and constructive effects of the noise, for example, target patterns created out of noise and traveling or spiral waves sustained by noise. Our study includes the case of realistic noises with temporal and spatial structures.

Dynamics of a paramagnetic colloidal particle driven on a magnetic-bubble lattice

We present a theoretical study of the recently observed dynamical regimes of paramagnetic colloidal particles externally driven above a regular lattice of magnetic bubbles [P. Tierno, T. H. Johansen, and T. M. Fischer, Phys. Rev. Lett. 99, 038303 (2007)]. An external precessing magnetic field alters the potential generated by the surface of the film in such a way to either drive the particle circularly around one bubble, ballistically through the array, or in triangular orbits on the interstitial regions between the bubbles. In the ballistic regime, we observe different trajectories performed by the particles phase locked with the external driving. Superdiffusive motion, which was experimentally found bridging the localized and delocalized dynamics, emerge only by introducing a certain degree of randomness into the bubbles size distribution.

Dynamics of orientational fluctuations in a homeotropic Fréedericksz transition under a rotating magnetic field

We study the problem of the Fréedericksz transition under a rotating magnetic field by using a dynamical model which incorporates thermal fluctuations into the whole set of nematodynamic equations. In contrast to other geometries, nonuniform textures in the plane of the sample do not appear favored. The proper consideration of thermal noise enables us to describe the dynamics of orientational fluctuations both below and above the shifted instability.

Dynamics of transient domain structures in nematic liquid crystals: The splay Fréedericksz transition

We discuss the dynamics of the transient pattern formation process corresponding to the splay Fréedericksz transition. The emergence and subsequent evolution of the spatial periodicity is here described in terms of the temporal dependence of the wave numbers corresponding to the maxima of the structure factor. Situations of perpendicular as well as oblique field-induced stripes relative to the initial orientation of the director are both examined with explicit indications of the time scales needed for their appearance and posterior development.