Long-range correlation

The expression 'long-range correlation' specifically refers to the slow decay of the (temporal or spatial) correlation function (covariance), defined as

$\gamma _{XY}(\Delta )=\langle X(t)Y(t+\Delta )\rangle .$

Here, $\langle .\rangle$ 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. 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_XY 'inherits' the spectral properties of the original time series.

Tubulence

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_corr, 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^α (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).

This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity), which can be understood in the framework of Self-Organised Criticality. Also see Non-diffusive transport and Continuous Time Random Walk.

It can be shown that determining the long-range behaviour of the correlation function directly from γ_XY is not a good idea, due to its sensitivity to noise. Rather, techniques such as the Rescaled Range, Hurst analysis, or Structure functions should be used to determine long-range correlations in data series.