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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.


Absolute values of temporal skewness as a function of temperature.Ensemble behavior of the skewness as a function of temperature. Three regimes are shown, T < Tc, T ≈ Tc and T > Tc.
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pone.0130751.g005: Absolute values of temporal skewness as a function of temperature.Ensemble behavior of the skewness as a function of temperature. Three regimes are shown, T < Tc, T ≈ Tc and T > Tc.

Mentions: The third moment of the distribution, the skewness, is shown in Fig 5. This moment is related to the asymmetry of events in the time series, and we expect this asymmetry to appear only in the low temperature regime. The reason for this is that in this regime the system has the possibility of becoming trapped in a meta-stable state where the average magnetization is lower than the expected value. These meta-stable states are produced when instead of one large cluster of magnetization, two stable clusters with opposite alignments are formed. A large perturbation is required for the system to spring out of this state. If the temperature is too low, the stochastic perturbations are too weak compared to the neighbor interactions and the system can spend a long time in meta-stabilty. However, near the critical point the stochastic perturbations are large enough to carry the system away from such meta-stability. This effect increases the skewness of the probability distribution. We thus conclude that skewness is an effective early-warning signal if the system is going towards the critical point from below. When approaching Tc from higher temperatures it is not possible to produce asymmetries in the distribution and thus the skewness is not sensitive to the approaching criticality. It is possible to understand, in terms of the potential, why the skewness of the distribution appears only for the T < Tc regime. When T > Tc the potential is symmetric, therefore we expect a symmetric distribution. On the other hand, when T < Tc the system should be on one of the two stable minima. The boundaries of the potential around the chosen minimum are not symmetric. On one side lies the wall of the potential, on the other side lies the barrier made by the unstable maximum. Because of this asymmetry the rate of change due to fluctuations is lower on the barrier side. This is reflected as a less step slope in that side of the potential [5].


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)

Absolute values of temporal skewness as a function of temperature.Ensemble behavior of the skewness as a function of temperature. Three regimes are shown, T < Tc, T ≈ Tc and T > Tc.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0130751.g005: Absolute values of temporal skewness as a function of temperature.Ensemble behavior of the skewness as a function of temperature. Three regimes are shown, T < Tc, T ≈ Tc and T > Tc.
Mentions: The third moment of the distribution, the skewness, is shown in Fig 5. This moment is related to the asymmetry of events in the time series, and we expect this asymmetry to appear only in the low temperature regime. The reason for this is that in this regime the system has the possibility of becoming trapped in a meta-stable state where the average magnetization is lower than the expected value. These meta-stable states are produced when instead of one large cluster of magnetization, two stable clusters with opposite alignments are formed. A large perturbation is required for the system to spring out of this state. If the temperature is too low, the stochastic perturbations are too weak compared to the neighbor interactions and the system can spend a long time in meta-stabilty. However, near the critical point the stochastic perturbations are large enough to carry the system away from such meta-stability. This effect increases the skewness of the probability distribution. We thus conclude that skewness is an effective early-warning signal if the system is going towards the critical point from below. When approaching Tc from higher temperatures it is not possible to produce asymmetries in the distribution and thus the skewness is not sensitive to the approaching criticality. It is possible to understand, in terms of the potential, why the skewness of the distribution appears only for the T < Tc regime. When T > Tc the potential is symmetric, therefore we expect a symmetric distribution. On the other hand, when T < Tc the system should be on one of the two stable minima. The boundaries of the potential around the chosen minimum are not symmetric. On one side lies the wall of the potential, on the other side lies the barrier made by the unstable maximum. Because of this asymmetry the rate of change due to fluctuations is lower on the barrier side. This is reflected as a less step slope in that side of the potential [5].

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.