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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 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. | |||
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''. | |||
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). | ||