Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model.
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Several early-warning signals have been reported in time series representing systems near catastrophic shifts.In this paper we analyze this universal behavior for a system that is a paradigm of phase transitions, the Ising model.We study the behavior of the early-warning signals and the way the temporal correlations of the system increase when the system is near the critical point.
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Affiliation: Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México, D.F., México; Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, México, D.F., México; Laboratorio Nacional de Ciencias de la Complejidad, D.F., México.
ABSTRACT
Among the properties that are common to complex systems, the presence of critical thresholds in the dynamics of the system is one of the most important. Recently, there has been interest in the universalities that occur in the behavior of systems near critical points. These universal properties make it possible to estimate how far a system is from a critical threshold. Several early-warning signals have been reported in time series representing systems near catastrophic shifts. The proper understanding of these early-warnings may allow the prediction and perhaps control of these dramatic shifts in a wide variety of systems. In this paper we analyze this universal behavior for a system that is a paradigm of phase transitions, the Ising model. We study the behavior of the early-warning signals and the way the temporal correlations of the system increase when the system is near the critical point. No MeSH data available. |
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Mentions: The Ising model is a statistical physics model for ferromagnetism. It is paradigmatic both for systems in which cooperative phenomena play an important role and for the study of physical phase transitions. The definition of the system is very simple. Consider a lattice of N sites with a spin state σ defined on each site. We let each of the spins take one of two possible orientation values, denoted by σ = ±1. There are thus 2N possible configurations of the system. Each of these spin sites interact with its nearest neighbors with an interaction energy given byH(σ)=-∑i,jJijσiσj-μ∑i=1NBiσi(1)where the first summation runs only through neighboring spins and Jij represents the coupling strength between spins i and j. If this coupling is positive then the neighboring spins will tend to align parallel to each other, since this minimizes the energy. Bi represents the external magnetic field acting on site i and μ is the magnetic moment. In this work we focus only in the case where Jij = constant and Bi = 0 (no external magnetic field). The probability that the system is in a given configuration depends on the energy of the configuration and the value of the parameter T, which is identified as the temperature of the system. This probability is given by the Boltzmann distributionPβ(σ)=e-βH(σ)Zβ(2)where β = (kT)−1, k is the Boltzmann constant and the normalization Zβ is the partition function. It is possible to measure the order present in the system through the total magnetization, defined asM=1N∑i=1Nσi(3)A well-known fact is that if it is defined on a 1-dimensional lattice, in which each spin has only two nearest neighbors, the system will have no phase transition. However, for lattices in 2 or more dimensions the system goes through a phase transition when T is equal to a critical value Tc. Below the critical value, the system undergoes spontaneous magnetization and all the spins tend to align towards either the +1 state or the −1 state. For temperatures higher than Tc, the system becomes paramagnetic, where the total magnetization of the system is zero on average. The presence and size of clusters of equally aligned spins is also markedly different in these two regimes: when T is lower than Tc, large resilient clusters form, while above Tc only small clusters can survive momentarily. If the temperature is high enough, all the clusters are completely destroyed. In the critical point (T = Tc), however, clusters are continually formed and destroyed in a wide range of scales, with the distribution of cluster sizes following a power law. Fig 1 shows a typical spatial configuration for each of the three regimes of temperature for a 2-dimensional system. Black squares represent spins with σ = +1 and white ones represent those with σ = −1. |
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Affiliation: Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México, D.F., México; Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, México, D.F., México; Laboratorio Nacional de Ciencias de la Complejidad, D.F., México.
No MeSH data available.