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Correlations in Complex Systems

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Book cover Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Correlation and Memory in Discrete Non-Markov Stochastic Processes

Correlation and Memory in Discrete Non‐Markov Stochastic Processes Generated by Random Events

Information Measures of Memory in Complex Systems

Manifestation of Strong Memory in Complex Systems

Some Perspectives on the Studies of Memory in Complex Systems

Bibliography

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Abbreviations

Correlation :

A correlation describes the degree of relationship between two or more variables. The correlations are viewed due to the impact of random factors and can be characterized by the methods of probability theory.

Correlation function :

The correlation function (abbreviated, as CF) represents the quantitative measure for the compact description of the wide classes of correlation in the complex systems (CS). The correlation function of two variables in statistical mechanics provides a measure of the mutual order existing between them. It quantifies the way random variables at different positions are correlated. For example in a spin system, it is the thermal average of the scalar product of the spins at two lattice points over all possible orderings.

Memory effects in stochastic processes through correlations:

Memory effects (abbreviated, as ME) appear at a more detailed level of statistical description of correlation in the hierarchical manner. ME reflect the complicated or hidden character of creation, the propagation and the decay of correlation. ME are produced by inherent interactions and statistical after‐effects in CS. For the statistical systems ME are induced by contracted description of the evolution of the dynamic variables of a CS.

Memory functions :

Memory functions describe mutual interrelations between the rates of change of random variables on different levels of the statistical description. The role of memory has its roots in the natural sciences since 1906 when the famous Russian mathematician Markov wrote his first paper in the theory of Markov Random Processes. The theory is based on the notion of the instant loss of memory from the prehistory (memoryless property) of random processes.

Information measures of statistical memory in complexsystems:

From the physical point of view time scales of correlation and memory cannot be treated as arbitrary. Therefore, one can introduce some statistical quantifiers for the quantitative comparison of these time scales. They are dimensionless and possess the statistical spectra on the different levels of the statistical description.

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Author information

Authors and Affiliations

  1. Department of Physics, Kazan State University, Kazan, Russia

    Renat M. Yulmetyev

  2. Tatar State University of Pedagogical and Humanities Sciences, Kazan, Russia

    Renat M. Yulmetyev

  3. University of Augsburg, Augsburg, Germany

    Peter Hänggi

Authors
  1. Renat M. Yulmetyev
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  2. Peter Hänggi
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

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

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Yulmetyev, R.M., Hänggi, P. (2012). Correlations in Complex Systems. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_46

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