Two classical subjects in statistical physics are kinetic theory and phase transitions. The latter has been traditionally studied in the equilibrium framework, where the goal is to characterize the ordered phases that arise below the critical temperature in systems with short-range interactions. A more recent development has been to investigate how this order is formed dynamically . In the present and following chapter, we focus on this dynamics. Phenomenology of coarsening The theory of phase transitions was originally motivated by the goal of understanding ferromagnetism. The Ising model played a central role in this effort and provided a useful framework for investigating dynamics. The basic entity in the Ising model is a spin variable that can take two possible values, s = ±1, at each site of a lattice. A local ferromagnetic interaction between spins promotes their alignment, while thermal noise tends to randomize their orientations. The outcome of this competition is a disordered state for sufficiently high temperature, while below a critical temperature the tendency for alignment prevails and an ordered state arises in which the order parameter – the average magnetization – is non-zero. Suppose that we start with an Ising model in an equilibrium disordered phase and lower the temperature. To understand how the system evolves toward the final state, we must endow this model with a dynamics and we also need to specify the quenching procedure. Quenching is usually implemented as follows: • Start at a high initial temperature T i > T c , where spins are disordered; here T c is the critical temperature. • Instantaneously cool the system to a lower temperature T f .
The ordinary kinetic versions of the Ising model may be modified to exhibit steady nonequilibrium states. This is illustrated in chapter 4 where a conflict between two canonical mechanisms ( diffusion and reaction ) drives the configuration away from equilibrium. A more systematic investigation of this possibility, when the conflict is between different reaction processes only, is described here. We focus on spin systems evolving by a superposition of independent local processes of the kind variously known as spin flips, birth/death or creation/annihilation. The restriction to spin flip dynamics does not prevent the systems in this class from exhibiting a variety of nonequilibrium phase transitions and critical phenomena. Their consideration may therefore help in developing nonequilibrium theory. In addition, they have some practical interest, e.g., conflicting dynamics may occur in disordered materials such as dilute magnetic systems, and some of these situations can be implemented in the laboratory. The present chapter describes some exact, mean-field and MC results that together render an intriguing picture encouraging further study. It is argued in §7.1 that some of the peculiar, emergent, macroscopic behavior of microscopically disordered materials may be related to diffusion of disorder. This provides a physical motivation for the nonequilibrium random-field Ising model (NRFM).
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.
Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions
d=1 and
d=2 with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale
sc grows faster than the aging length scale
ℓ as a power law relation
sc∼ℓ2.58. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.
PACS. 05.10.Cc – Renormalization group methods. PACS. 64.60.-i – General studies of phase transitions. PACS. 75.10.Hk – Classical spin models. Abstract. – A study of the influence of different types of disorder on systems in the Ising universality class by employing both a dynamical field theory approach and extensive Monte Carlo simulations is presented. We reproduce some well-known results for the case of quenched disorder (random temperature and random field), and analyze the effect of four different types of time-dependent disorder scarcely studied so far in the literature. Some of them are of obvious experimental and theoretical relevance (as, for example, globally fluctuating temperatures or random fields). All the predictions coming from our field-theoretical analysis are fully confirmed by extensive simulations in two and three dimensions, and novel qualitatively different, non-Ising transitions are reported. Possible experimental setups designed to explore the described phenomenologies are also briefly discussed. The Ising model has played a central role in the development of modern statistical mechanics. Its universality class includes a broad variety of models and real materials, as magnetic
Abstract We study the early time dynamics of the 2 d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.