# Long-range correlation

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The expression 'long-range correlation' specifically refers to the slow decay of the (temporal or spatial) correlation function (covariance), defined as [1]

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

Here, ${\displaystyle \langle .\rangle }$ 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 = -ωΔmax.

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 γXY '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) Δcorr, calculated as the distance at which the correlation has dropped from its maximum value by a factor 1/e. Δcorr 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/Δα (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. [2]

### Experimental determination

It can be shown that determining the long-range behaviour of the correlation function directly from γXY is not a good idea, due to its sensitivity to noise.[3] Rather, techniques to determine the Hurst exponent, such as the Rescaled Range[4][5] or Structure Functions[6] should be used to determine long-range correlations in data series.

In practice, long-range correlations may have various origins, and proper techniques are required to distinguish between those. [7] This is particularly important when trying to separate Zonal Flow contributions to the long-range correlation from other effects. [8]

## References

1. Cross-Correlation (MathWorld)
2. R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations, Phys. Rev. E 71 (2005) 011111
3. 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
4. B.A. Carreras, B. van Milligen, M.A. Pedrosa, et al., Long-Range Time Correlations in Plasma Edge Turbulence, Phys. Rev. Lett. 80 (1998) 4438
5. B.A. Carreras, B.Ph. van Milligen, M.A. Pedrosa, et al., Experimental evidence of long-range correlations and self-similarity in plasma fluctuations, Phys. Plasmas 6 (1999) 1885
6. M. Gilmore, C.X. Yu, T.L. Rhodes, and W.A. Peebles, Investigation of rescaled range analysis, the Hurst exponent, and long-time correlations in plasma turbulence, Phys. Plasmas 9 (2002) 1312
7. B.Ph. van Milligen et al, Parallel and perpendicular turbulence correlation length in the TJ-II Stellarator, Nucl. Fusion 53 (2013) 093025
8. B.Ph. van Milligen, E. Sánchez, A. Alonso, M.A. Pedrosa, C. Hidalgo, A. Martín de Aguilera, A. López Fraguas, The use of the Biorthogonal Decomposition for the identification of zonal flows at TJ-II, Plasma Phys. Control. Fusion 57, 2 (2015) 025005