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Synergetics: Basic Concepts

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Encyclopedia of Complexity and Systems Science

The Role of Synergetics in Science

In science, we may essentially distinguish between two trends:

  1. 1.

    The accumulation of knowledge

  2. 2.

    Information reduction in the sense of finding general principles, common features.

In physics, such unifying approaches are well known: the unification of magnetism, electricity and, later on, weak and other interactions leading eventually to a unified field theory. General relativity unifies concepts of space, time and gravitation. While these unifications take place at a fundamental level, one may ask whether it is worthwhile to look also for unifications at say more macroscopic or phenomenological levels. One example is thermodynamics, another the theory of phase transitions of systems in thermal equilibrium by means of the renormalization group approach, or the concept of fractals, etc.

The main goal of Synergetics is the search for unifying principles for systems that arecomposed of many individual parts or components, and that may...

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Abbreviations

Synergetics :

Science of cooperation.

Pattern :

A pattern is essentially an arrangement. It is characterized by the order of the elements of which it is made rather than by the intrinsic nature of these elements (Norbert Wiener).

Self‐organization :

Formation of spatio‐temporal patterns (structures) and/or performance of functions without an “ordering hand”.

State vector :

Set of time- or time‐independent variables that characterize the state of a system.

Evolution equations :

Determine the temporal evolution of the state vector. May be deterministic, stochastic or both.

Control parameter :

One or a set of (mostly externally) fixed parameters in the evolution equations.

Spectrum:

Set of eigenvalues belonging to linear stability equations with boundary conditions.

Stability of a system:

System returns after a (small) perturbation of its state vector into original state.

Instability :

Loss of stability.

Order parameters :

Collective variables that determine the macroscopic behavior of systems.

Slaving principle :

A general theorem that allows the reduction of the variables of a system to order parameters (close to instability).

Trajectory:

Smooth curve q(t) of solution of evolution equation in q‑space.

Attractor :

Region in the state vector space (“q‑space”) to which all neighboring states are attracted in the course of time.

Fixed point , stable:

Point in q space to which all neighboring trajectories converge in course of time.

Limit cycle , stable:

A closed trajectory to which all neighboring trajectories converge.

Probability distribution function:

Function that determines the probability of a random variable r to have fixed value \( { r=r_0 } \).

Fokker Planck equation :

Evolution equation for probability density function, based on drift and diffusion.

Normal form :

Especially simple polynomial expression that still captures the essential features, e. g. of the right hand side of deterministic evolution equations.

Schrödinger picture of quantum mechanics:

In it operators are time‐independent, while the wave‐function (“state vector”) is time‐dependent and determined by the Schrödinger equation.

Heisenberg picture in quantum mechanics:

The state vector is time‐independent, while the operators are time‐dependent and determined by Heisenberg equations of motion.

Fluctuating forces :

Stochastic (random) forces appearing in evolution equations.

Quantum classical correspondence :

Establishes relation between quantum mechanical density matrix and classical quasi‐probability distribution.

Symmetry :

Invariance of a system against specific transformations (e. g. mirror symmetry).

Group:

Set of elements with specific multiplication rules (axioms).

Dynamical system:

System whose state vector changes in the course of time deterministically.

Langevin equation :

Originally: evolution equation for velocity of a Brownian particle subject to damping and fluctuating force.

Generalized Langevin equation:

General evolution equations that contain both a deterministic and a stochastic part (“fluctuating forces”).

Hamilton operator:

Classical Hamilton function, in which variables, e. g. position x and momentum p, are replaced by quantum mechanical operators.

Spatial coordinate (vector x):

in one, two or three dimensions.

$$ \begin{aligned} \Delta &\quad\text{Laplace operator (in 1,2 or 3 dimensions)}\:. \\ \nabla &\quad\text{Vector } \left(\frac{\text{d}} {\text{d} x_1},\frac{\text{d}} {\text{d} x_2},\frac{\text{d}} {\text{d} x_3}\right) \text{ in 1,2 or 3 dimensions}\:. \end{aligned} $$

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  1. Institut für Theoretische Physik, Universität Stuttgart, Stuttgart, Germany

    Hermann Haken

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

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Haken, H. (2009). Synergetics: Basic Concepts. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_533

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