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Long-range correlation: Difference between revisions
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Here, <math>\langle . \rangle</math> refers to an average over ''t'' and the observables ''X'' and ''Y'' depend on the time ''t'', but an analogous expression can be written down for spatial dependence. | Here, <math>\langle . \rangle</math> refers to an average over ''t'' and the observables ''X'' and ''Y'' depend on the time ''t'', but an analogous expression can be written down for spatial dependence. | ||
== Coherent states == | |||
Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=X''. | Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=X''. | ||
Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | ||
== Turbulence == | |||
More interesting is the typical behaviour of the correlation function for turbulent states. | More interesting is the typical behaviour of the correlation function for turbulent states. | ||
In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | ||
== Long range effects == | |||
When the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δ<sup>α</sup> (α > 0 but not too large, < 2), the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case, in which large values of Δ can be safely ignored). | When the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δ<sup>α</sup> (α > 0 but not too large, < 2), the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case, in which large values of Δ can be safely ignored). | ||
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These issues can be understood in the framework of [[Self-Organised Criticality]]. | These issues can be understood in the framework of [[Self-Organised Criticality]]. | ||
The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | ||
=== Experimental determination === | === Experimental determination === | ||
It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref>[[doi:10.1063/1.873192|B.A. Carreras, D.E. Newman, B.Ph. van Milligen, and C. Hidalgo, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''' (1999) 485]]</ref> | It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref>[[doi:10.1063/1.873192|B.A. Carreras, D.E. Newman, B.Ph. van Milligen, and C. Hidalgo, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''' (1999) 485]]</ref> | ||
Rather, techniques such as the [[:Wikipedia:Rescaled range|Rescaled Range]], [[:Wikipedia:Hurst exponent|Hurst]] analysis, or Structure functions should be used to determine long-range correlations in data series. | Rather, techniques such as the [[:Wikipedia:Rescaled range|Rescaled Range]], [[:Wikipedia:Hurst exponent|Hurst]] analysis, or Structure functions should be used to determine long-range correlations in data series. | ||
== See also == | |||
* [[:Wikipedia:Long-range dependency|Long-range dependency]] | |||
* [[Non-diffusive transport]] | |||
== References == | == References == | ||
<references /> | <references /> | ||