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Computational Complexity pp 1131–1154Cite as

Fluctuations ,Importance of:Complexity in the View of Stochastic Processes

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Article Outline

Glossary

Definition of the Subject

Introduction

Stochastic Processes

Stochastic Time Series Analysis

Applications: Processes in Time

Applications: Processes in Scale

Future Directions

Further Reading

Acknowledgment

Bibliography

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Notes

  1. 1.

    Here it should be noted that the term “intermittency” is used frequently in physics for different phenomena, and may cause confusions. This turbulent intermittency is not equal to the intermittency of chaos. There are also different intermittency phenomena introduced for turbulence. There is this intermittency due to the nonlinear scaling, there is the intermittency of switches between turbulent and laminar flows for non local isotropic fully developed turbulent flows, there is the intermittency due to the statistics of small scale turbulence which we discuss here as heavy tailed statistics.

Abbreviations

Complexity in time:

Complex structures may be characterized by non regular time behavior of a describing variable \( { q \in \mathbf{R}^d } \). Thus the challenge is to understand or to model the time dependence of \( { q(t) } \), which may be achieved by a differential equation \( { \frac{\text{d}} {\text{d} t} q(t) = \dots } \) or by the discrete dynamics \( { q(t+\tau) = f(q(t), \dots) } \) fixing the evolution in the future. Of special interest are not only nonlinear equations leading to chaotic dynamics but also those which include general noise terms, too.

Complexity in space:

Complex structures may be characterized by their spatial disorder. The disorder on a selected scale l may be measured at the location x by some scale dependent quantities, \( { q(l,x) } \), like wavelets, increments and so on. The challenge is to understand or to model the features of the disorder variable \( { q(l,x) } \) on different scales l. If the moments of q show power behavior \( { \langle q(l,x)^n\rangle \propto l^{\zeta(n)} } \) the complex structures are called fractals. Well known examples of spatial complex structures are turbulence or financial market data. In the first case the complexity of velocity fluctuations over different distances l are investigated, in the second case the complexity of price changes over different time steps (time scale) are of interest.

Fokker–Planck equation:

The evolution of a variable \( { \mathbf{x}(t) } \) from \( { \mathbf{x^{\prime}} } \) at t′ to \( { \mathbf{x} } \) at t, with \( { t^{\prime}>t } \), is described in a statistical manner by the conditional probability distribution \( { p(\mathbf{x},t|\mathbf{x}^{\prime},t^{\prime}) } \). The conditional probability is subject to a Fokker–Planck equation, also known as second Kolmogorov equation, if

$$ \begin{aligned} &\frac{\partial}{\partial t}\, p(\mathbf{x},t \,|\,\mathbf{x}^{\prime}, t^{\prime}) =\\ &\quad-\sum\limits_{i=1}^{d}\frac{\partial}{\partial x_{i}} D_{i}^{(1)} (\mathbf{x},t) p(\mathbf{x},t \,|\,\mathbf{x}^{\prime},t^{\prime})\\ &\quad+\frac{1}{2}\sum\limits_{i,j=1}^{d} \frac{{\partial}^{2}}{\partial x_{i}\partial x_{j}} D_{ij}^{(2)}(\mathbf{x},t) p(\mathbf{x},t\,|\,\mathbf{x}^{\prime},t^{\prime}) \: , \end{aligned} $$

holds. Here \( { \mathbf{D}^{(1)} } \) and \( { \mathbf{D}^{(2)} } \) are the drift vector and the diffusion matrix, respectively.

Kramers–Moyal coefficients:

Knowing for a stochastic process the conditional probability distribution \( { p(\mathbf{x(t)},t|\mathbf{x}^{\prime},t^{\prime}) } \), for all t and t′ the Kamers–Moyal coefficients can be estimated as nth order moments of the conditional probability distribution. In this way also the drift and diffusion coefficient of the Fokker–Planck equation can be obtained form the empirically measured conditional probability distributions.

Langevin equation:

The time evolution of a variable \( { \mathbf{x}(t) } \) is described by Langevin equation if for \( { \mathbf{x}(t) } \) it holds:

$$ \frac{\text{d}} {\text{d} t} \mathbf{x}(\mathbf{t}) =\mathbf{D}^{(1)}(\mathbf{x},\mathbf{t}) \cdot \tau + \sqrt{\mathbf{D}^{(2)} (\mathbf{x},\mathbf{t})} \cdot \boldsymbol{\Gamma}(\mathbf{t}_i) \: .$$

Using Itô's interpretation the deterministic part of the differential equation is equal to the drift term, the noise amplitude is equal to the square root of the diffusion term of a corresponding Fokker–Planck equation. Note, for vanishing noise a purely deterministic dynamics is included in this description.

Stochastic process in scale:

For the description of complex system with spatial or scale disorder usually a measure of disorder on different scales \( { q(l,x) } \) is used. A stochastic process in scale is now a description of the l evolution of \( { q(l,x) } \) by means of stochastic equations. As a special case the single event \( { q(l,x) } \) follows a Langevin equation, whereas the probability \( { p(q(l)) } \) follows a Fokker–Planck equation.

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Acknowledgment

The scientific results reported in this review have been worked out in close collaboration with many colleagues and students.We mention St. Barth,F. Böttcher, F. Ghasemi, I. Grabec, J. Gradisek, M. Haase, A. Kittel, D. Kleinhans, St. Lück, A. Nawroth, Chr. Renner,M. Siefert, and S. Siegert.

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Authors and Affiliations

  1. Institute for Theoretical Physics, University of Münster, Münster, Germany

    Rudolf Friedrich

  2. Institute of Physics, Carl-von-Ossietzky University Oldenburg, Oldenburg, Germany

    Joachim Peinke

  3. Dept. of Physics, Sharif University of Technology, Theran, Iran

    M. Reza Rahimi Tabar

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  1. Rudolf Friedrich
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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Friedrich, R., Peinke, J., Reza Rahimi Tabar, M. (2012). Fluctuations ,Importance of:Complexity in the View of Stochastic Processes . In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_71

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