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Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model.

Morales IO, Landa E, Angeles CC, Toledo JC, Rivera AL, Temis JM, Frank A - PLoS ONE (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

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.


Power Spectral Density as a function of temperature.Ensemble behavior of the Power Spectral Density as a function of temperature. Panel (a) shows the behavior of the PSD for temperatures T ≤ Tc. Temperature increases from bottom to top, with Tc corresponding to the topmost curve. Panel (b) shows the behavior of the PSD for temperatures T ≥ Tc. Temperature increases from top to bottom, with Tc corresponding to the topmost curve. The crossover frequency for each temperature is shown as a red dot.
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pone.0130751.g008: Power Spectral Density as a function of temperature.Ensemble behavior of the Power Spectral Density as a function of temperature. Panel (a) shows the behavior of the PSD for temperatures T ≤ Tc. Temperature increases from bottom to top, with Tc corresponding to the topmost curve. Panel (b) shows the behavior of the PSD for temperatures T ≥ Tc. Temperature increases from top to bottom, with Tc corresponding to the topmost curve. The crossover frequency for each temperature is shown as a red dot.

Mentions: When a system is critical, all scales are important for the system dynamics. This is reflected in the fact that the system becomes scale invariant. Because of this, an accurate representation of the system should include all the scales available. According to theory [5] and our previous results, it is evident that Autocorrelation(τ = 1) signal is enough in order to catch the dynamical shift. However an important question arises: How the presence of the different scales available to the system are modified when a system approaches criticality? This can be explored through the behavior of the whole autocorrelation function and the Power Spectral density (PSD). The autocorrelation function is related with the PSD through the Wiener-Khinchin theorem, provided that the time series is a stationary random process [42]. In order to explore the behavior of the long range correlations we analyzed the PSD of the system. Fig 8 shows the memory effect for the whole range of scales. In panel (a) we can observe the evolution of the PSD for temperatures smaller than the critical value. The temperature increases from bottom to top, with the critical value corresponding to the topmost curve. Panel (b) shows the corresponding PSD evolution for the high temperature regime. Again, the topmost curve corresponds to the critical temperature, and this time temperature increases from top to bottom. In both panels the PSD curves for the different temperatures have vertically shifted for clarity. It is clear that a power law appears in the PSD at the critical point. Power laws have been previously connected to criticality, specially with temporal scale invariance [7, 8]. As we have mentioned, at the critical point the Ising model exhibits spatial scale invariance as well as fractal structure in the sizes of the magnetization clusters formed by the system. It is remarkable that the system also displays temporal scale invariance because as far as the authors know it is not well understood whether temporal scale invariance implies spatial scale invariance or vice versa. The spatial scale invariant properties of the critical Ising model are well studied and reported in the literature. However, it is our personal opinion that the temporal properties of the critical state are less known. Temporal scale invariance means that the time series is statistically the same at all temporal scales. This property is related to long range correlations and long range memory in the system [8]. As soon as the system’s temperature departs from the critical value, either to lower or higher temperatures, the low frequency part of the PSD flattens out. A flat PSD is characteristic of an uncorrelated system, where the fluctuations are white noise. We observe that the flat region of the PSD becomes wider as temperature gets further away from the critical value. For temperatures that are very far from the critical one, we can expect that the PSD will flatten out for all frequencies.


Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model.

Morales IO, Landa E, Angeles CC, Toledo JC, Rivera AL, Temis JM, Frank A - PLoS ONE (2015)

Power Spectral Density as a function of temperature.Ensemble behavior of the Power Spectral Density as a function of temperature. Panel (a) shows the behavior of the PSD for temperatures T ≤ Tc. Temperature increases from bottom to top, with Tc corresponding to the topmost curve. Panel (b) shows the behavior of the PSD for temperatures T ≥ Tc. Temperature increases from top to bottom, with Tc corresponding to the topmost curve. The crossover frequency for each temperature is shown as a red dot.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4477971&req=5

pone.0130751.g008: Power Spectral Density as a function of temperature.Ensemble behavior of the Power Spectral Density as a function of temperature. Panel (a) shows the behavior of the PSD for temperatures T ≤ Tc. Temperature increases from bottom to top, with Tc corresponding to the topmost curve. Panel (b) shows the behavior of the PSD for temperatures T ≥ Tc. Temperature increases from top to bottom, with Tc corresponding to the topmost curve. The crossover frequency for each temperature is shown as a red dot.
Mentions: When a system is critical, all scales are important for the system dynamics. This is reflected in the fact that the system becomes scale invariant. Because of this, an accurate representation of the system should include all the scales available. According to theory [5] and our previous results, it is evident that Autocorrelation(τ = 1) signal is enough in order to catch the dynamical shift. However an important question arises: How the presence of the different scales available to the system are modified when a system approaches criticality? This can be explored through the behavior of the whole autocorrelation function and the Power Spectral density (PSD). The autocorrelation function is related with the PSD through the Wiener-Khinchin theorem, provided that the time series is a stationary random process [42]. In order to explore the behavior of the long range correlations we analyzed the PSD of the system. Fig 8 shows the memory effect for the whole range of scales. In panel (a) we can observe the evolution of the PSD for temperatures smaller than the critical value. The temperature increases from bottom to top, with the critical value corresponding to the topmost curve. Panel (b) shows the corresponding PSD evolution for the high temperature regime. Again, the topmost curve corresponds to the critical temperature, and this time temperature increases from top to bottom. In both panels the PSD curves for the different temperatures have vertically shifted for clarity. It is clear that a power law appears in the PSD at the critical point. Power laws have been previously connected to criticality, specially with temporal scale invariance [7, 8]. As we have mentioned, at the critical point the Ising model exhibits spatial scale invariance as well as fractal structure in the sizes of the magnetization clusters formed by the system. It is remarkable that the system also displays temporal scale invariance because as far as the authors know it is not well understood whether temporal scale invariance implies spatial scale invariance or vice versa. The spatial scale invariant properties of the critical Ising model are well studied and reported in the literature. However, it is our personal opinion that the temporal properties of the critical state are less known. Temporal scale invariance means that the time series is statistically the same at all temporal scales. This property is related to long range correlations and long range memory in the system [8]. As soon as the system’s temperature departs from the critical value, either to lower or higher temperatures, the low frequency part of the PSD flattens out. A flat PSD is characteristic of an uncorrelated system, where the fluctuations are white noise. We observe that the flat region of the PSD becomes wider as temperature gets further away from the critical value. For temperatures that are very far from the critical one, we can expect that the PSD will flatten out for all frequencies.

Bottom Line: 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.

View Article: PubMed Central - PubMed

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.