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=sin(ωt+φ). The cross phase φ can be determined from the delay Δ of the maxima of the cross correlation γ (modulo 2π): φ = -Δ_max/ω.

Note also that the correlation function is a convolution, hence by the convolution theorem 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.

Turbulence

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. Δ_corr constitutes the typical scale length for turbulent dynamics (turbulent transport).

Long range effects

However, often it is observed that 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). In this case, the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case). The particular choice of the power law as the main alternative of the exponential decay is not casual: it is motivated by the fact that power law distributions are self-similar. Particularly, an algebraic decay of the mentioned type implies that no particular scale length can be assigned to the turbulent dynamics, and all scales (up to the system size) will participate in the global description of system behaviour.

This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity). A 'memory effect' refers to the fact that the evolution of the system is affected by previous system states over times (much) longer than the turbulence decorrelation time. An analogous interpretation is possible for 'non-local' behaviour, in which the system state at remote points affects the local evolution of the system. 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.

Experimental determination

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 to determine the Hurst exponent, such as the Rescaled Range or Structure Functions should be used to determine long-range correlations in data series.

See also

• Long-range dependency
• Non-diffusive transport