Table Of ContentSpringer Series in Synergetics
Editor: Hermann Haken
Synergetics, an interdisciplinary field of research, is concerned with the cooper
ation of individual parts of a system that produces macroscopic spatial, temporal
or functional structures. It deals with deterministic as well as stochastic processes.
1 Synergetics An Introduction 3rd Edition 17 Evolution of Order and Chaos in Physics,
ByH. Haken Chemistry, and Biology Editor H. Haken
2 Synergetics A Workshop 18 The Fokker-Planck Eqnation
Editor: H. Haken ByH. Risken
3 Synergetics Far from Equilibrium 19 Chemical Osdliations, Waves, and Tnrbnlence
Editors: A. Pacault, C. Vidal By Y. Kuramoto
20 Advanced Synergetics
4 Structural Stability in Physics
ByH. Haken
Editors: W. Giittinger, H. Eikemeier
21 Stochastic Phenomena and Chaotic Behavionr
5 Pattern Formation by Dynamic Systems and
in Complex Systems
Pattern Recognition
Editor: P. Schuster
Editor: H. Haken
22 Synergetics - From Microscopic to Macroscopic
6 Dynamics of Synergetic Systems Order Editor: E. Frehland
Editor: H. Haken
23 Synergetics of the Brain
7 Problems of Biological Physics Editors: E. Ba§ar, H. Flohr, H. Haken,
By L. A. Blumenfeld A. J. Mandell
8 Stochastic Nonlinear Systems 24 Chaos and Statistical Methods
in Physics, Chemistry, and Biology Editor: Y. Kuramoto
Editors: L. Arnold, R. Lefever 25 Dynamics of Hierarchical Systems
9 Numerical Methods in the Stndy of Critical An Evolutionary Approach By J. S. Nicolis
Phenomena 26 Self-Organization and Management of
Editors: J. Della Dora, J. Demongeot, Social Systems
B. Lacolle Editors: H. Ulrich, G.J.B. Probst
10 The Kinetic Theory of Electromagnetic 27 Non-Equilibrium Dynamics in Chemical
Processes By Yu. L. Klimontovich Systems Editors: C. Vidal, A. Pacault
28 Self-Organization. Autowaves and Structures
11 Chaos and Order in Natnre
Far from Equilibrium
Editor: H. Haken
Editor: V.!. Krinsky
12 Nonlinear Phenomena in Chemical Dynamics
29 Temporal Order
Editors: C. Vidal, A. Pacault
Editors: L. Rensing, N. 1. Jaeger
13 Handbook of Stochastic Methods
30 Dynamical Problems in Soliton Systenis
for Physics, Chemistry, and the Natural Sciences
Editor: S. Takeno
2nd Edition By C. W. Gardiner
31 Complex Systems - Operational Approaches
14 Concepts and Models of a Quantitative in Neurobiology, Physics, and Computers
Sociology The'Dynamics of Interacting Editor: H. Haken
Populations By W. Weidlich, G. Haag
32 Dimensions and Entropies in Chaotic Systems
15 Noise-Induced Transitions Theory and Quantification of Complex Behavior
Applications in Physics, Chemistry, and Biology Editor: G. Mayer-Kress
By W. Horsthemke, R. Lefever 33 Selfo rganization by Nonlinear Irreversible
16 Physics of Bioenergetic Processes Processes
By L. A. Blumenfeld Editors: W. Ebeling, H. Ulbricht
Selforganization
by Nonlinear
Irreversible Processes
Proceedings of the Third International Conference
Kiihlungsbom, GDR, March 18-22, 1985
Editors: W. Ebeling and H. Ulbricht
With 78 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo
Professor Dr. Werner Ebeling
Sektion Physik, Humboldt-Universitat Berlin, InvalidenstraBe 42,
DDR-I040 Berlin, GDR
Professor Dr. Heinz Ulbricht
Sektion Physik, Wilhelm-Pieck-Universitat, Universitatsplatz 3,
DDR-2500 Rostock, GDR
Series Editor:
Professor Dr. Dr. h. c. Hermann Haken
Institut fUr Theoretische Physik der Universitat Stuttgart, Pfaffenwaldring 57 !IV ,
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13: 978-3-642-71006-3 e-ISBN-13: 978-3-642-71004-9
DOl: 10.1007/978-3-642-71004-9
Library of Congress Cataloging-in-Publication Data. Selforganization by nonlinear irreversible processes. (Springer
series in synergetics ; 33) Includes index. 1. Self-organizing systems--Congresses. 2. Irreversible process-Congresses.
3. Nonequilibrium thermodynamics-Congresses. 4. Chaotic behavior in systems-Congresses. 5. Stochastic proces
ses-Congresses. I. Ebeling, Werner, 1936-. II. Ulbricht, H. (Heinz) III. International Conference on Irreversible
Processes and Dissipative Structures (3rd : 1985 : Ostseebad Kithlungsborn, Germany) IV. Series: Springer series in
synergetics ; v. 33. Q325.S47 1986 001.53'3 86-20384
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically those of translation, reprinting, reuse of illustrations, Broadcasting, reproduction by photocopying
machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986
Softcover reprint of the hardcover I st edition 1986
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant protective laws and regulations and therefore free for
general use.
Offset printing: Druckbaus Beltz, Hemsbach, Bookbinding: J. Schaffer OHG, Griinstadt
2153/3150-543210
Preface
These Proceedings contain invited lectures presented at the third Interna
tional Conference on "Irreversible Processes and Dissipative Structures" in
Kiihlungsborn (German Democratic Republic) in March, 1985. These con
ferences, the first of which was held in Rostock in 1977 and the second in
Berlin in 1982, are devoted to the study of irreversible processes far from
thermal equilibrium and to the phenomena of selforganization. The meet
ing in Kiihlungsborn brought together some 160 mathematicians, physicists,
chemists and biologists from 10 countries, who are all interested in the inter
disciplinary field of synergetics.
The main topics of the conference were basic concepts of selforganization
and evolution, such as entropy, instabilities, nucleation, dissipative struc
tures, chaos and turbulence. The contributions cover methods from ther
modynamics, the theory of dynamic systems, stochastic and statistic theory,
the method of Green's functions, the fluctuation-dissipation theorem, etc.
Several problems are studied in more detail, e.g., the kinetics of nucleation
especially in finite systems, the dynamics of interfaces, reaction-diffusion sys
tems, chemical and biochemical pattern formation and information process
ing. Further, several contributions are devoted to the development of the
concepts of chaos and turbulence.
The editors hope that the contributions collected in this volume will pro
vide some new information about the field of selforganization, which is in full
development now.
In the name of the participants, we express sincere thanks to the spon
soring organizations and especially to the Physical Society of the German
Democratic Republic and to the Wilhelm-Pieck-University at Rostock for
their support of the conference. Our special thanks go to R. Mahnke, H.
Engel-Herbert, F. Schweitzer and A. Budde for their help in the preparation
of the manuscripts. Last but not least, we thank H. Bahlo, who typed the
whole manuscript in a perfect way, and express our sincere thanks to Pro
fessor H. Hakenfor inviting us to include these Proceedings in the Springer
Series in Synergetics.
Berlin, Rostock, GDR w. Ebeling H. Ulbricht
July 1986
v
Contents
Part I Basic Concepts
Thermodynamic Aspects of Selforganization
By W. Ebeling, H. Engel-Herbert, and H. Herzel (With 6 Figures) 2
Entropy in Time Averaging. By Yu.L. Klimontovich 17
Nucleation Theory and Dynamics of First-Order
Phase Transitions in Finite System
By H. Ulbricht, F. Schweitzer, and R. Mahnke (With 11 Figures) 23
Dynamic Chaos in Ensembles of Structures and Spatial Development
of Turbulence in Unbounded Systems
By A.V. Gaponov-Grekhov and M.1. Rabinovich (With 5 Figures) 37
Part II Stochastic and Statistic Theory
Nonlinear Chemical Reactions in Diluted Solutions. By R. Feistel .... 48
Stochastic Thermodynamics. By C. Van den Broeck ................... 57
Stochastic M~asures as Critical Quantities in NonequiIibrium Phase
Transitions. By F. Schlogl ................................................ 62
On the Kinetics of Nucleation in Isochoric Gases
By L. Schimansky-Geier, F. Schweitzer, W. Ebeling, and H. Ulbricht
(With 2 Figures) .......................................................... 67
On the Properties of the Nonequilibrium Potential Near Bifurcation
Points. By A.V. Tolstopjatenko and L. Schimansky-Geier ............. 76
Nonlinear Fluctuation-Dissipation Models of Charge Transfer
Nonequilibrium Electrical Noise and I/f-Noise. By G.N. Bochkov 85
Application of Path Integrals to Stochastic Reaction-Diffusion
Equations. By A. Forster and A.S. Michailov .......................... 89
The Method of Green's Functions in Nonequilibrium Statistical
Mechanics. By D. Kremp, M. Schlanges, and Th. Bornath ............ 95
VII
Transport Phenomena in Many-Particle Systems and the Quantum
Statistical Approach to Nonequilibrium Thermodynamics
By G. Ropke (With 3 Figures) ........................................... 99
Part III Dissipative Structures
Dynamics of Interfaces in Random Media
By A. Engel, W. Ebeling, R. Feistel, and L. Schimansky-Geier
(With 4 Figures) .......................................................... 110
Autosolitons in Active Systems with Diffusion
By B.S. Kerner and V.V. Osipov (With 4 Figures) ..................... 118
Repressing of Chemical Waves by Photochemical Inhibitor Releasing
By 1. Kuhnert, 1. Pohlmann, H.-J. Krug, and G. Wessler
(With 3 Figures) .......................................................... 128
Dynamic Patterns in Interacting Chemical Cells and Effects of
External Periodic Forcing
By M. Marek, M. Dolnik, and I. Schreiber (With 1 Figure) ............ 133
Pattern in Marangoni-Instability Varied by Shear Stress Between
Interface and Bulk. By W. Waldhelm and H. Linde (With 7 Figures) 137
Ostwald Ripening in Viscoelastic Media
By J. Schmelzer and I. Gutzow (With 8 Figures) ....................... 144
Thermodynamic Parameters and Structural Criteria for Phase
Transformations in Amorphous Covalent Bound Materials
By W. Pompe and A. Richter (With 3 Figures) ........................ 149
Part IV Evolution Processes
Turing Structures in Cytokinesis, Mitosis and Early Embryogenesis
By A. Hunding (With 5 Figures) ........................................ 158
Qualitative Properties of Chains as Models of Growth and Binary
Coding. By M. Peschel and W. Mende (With 4 Figures) .............. 164
A Study Program of Chaotic Dynamics Applied to Information
Processing. By J.S. Nicolis (With 2 Figures) ........................... 174
The Evolutionary Triad. By M.V. Volkenstein .......................... 188
Part V Chaotic Processes
Fully Developed Chaos in One-Dimensional Discrete Processes
By G. Gyorgyi and P. Szepfalusy (With 3 Figures) ................... 196
VIII
Auto-Oscillatory Regimes in the Chain of Coupled Generators
By V.S. Anishchenko (With 3 Figures) ....................... ........ ... 198
Universal Scaling Properties of Nonlinear Dissipative Systems
By R.W. Leven and B. Pompe (With 3 Figures) ............. ... ... .... 203
New Aspects of Instabilities and Chaos in Laser Systems
By G. Wallis...... ......... ............................ .......... ... ...... 207
Complexity of Patterns Generated by One-Dimensional Cellular
Automata
By R. Mahnke and A. Budde (With 1 Figure) .......................... 213
Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
IX
Part I
Basic Concepts
Thermodynamic Aspects of Selforganization
w.
Ebeling, H. Engel-Herbert, .and H. Herzel
Humboldt-Universitat Berlin, Sektion Physik, Invalidenstra6e 42,
DDR-1040 Berlin, GDR
1. Introduction
Thermodynamics is a rather old discipline of physics, however, it
is not oldfashioned. On the contrary. such modern topics as the hot
big bang model, the theory of black holes, as well as the theory of
biological systems /1,2/, show that thermodynamics goes through a
renaissance. Thermodynamics is also intimately related to informa
tion theory, a key discipline for the study of selforganization and
evolution /3,4/. The very origin of this discipline is closely con
nected with thermodynamical reasoning, as shown in the fundamental
papers of SZILARD (1929), SHANNON (1948) and BRILLOUIN (1956). Thus
S"TRAToNoVICH, one of the pioneers of several branches of modern in
formation theory, writes that thermodynamics and statistical phy
sics are the cement which hold together the disciplines forming
modern information theory /5/. Besides the informational aspects
also the direct consideration of thermodynamic functions and of the
entropy production is of much interest for the study of selforgani
zation processes /6,7/. The structures created in the process of
selforgdnization are often called "dissipative structures" /6-8/.
Besides this term, which underlines the aspect of dissipation, we
shall also use the term "autostructures", which underlines the as
pect autonomy /9/. The term autostructure is a generalization of
well-known terms as "autooscillations" and "autowaves" /10/.
2. The Concept of Entropy
One of the central concepts of thermodynamics is entropy. There is
a close correspondence between entr6py and order /11-14/. Let us
discuss now several entropy concepts.
(i) Phenomenological thermodynamic entropy (Clausius entropy)
As well known, a system in thermodynamic equilibrium is characteri
zed by a unique state function Seq(E,X) depending on energy and the
other extensive parameters of the system. The thermodynamic entropy
may also be defined for a certain class of nonequilibrium states
/6/. For example, the notion of nonequilibrium thermodynamic en
tropy has a clear meaning in the field of hydrodynamics, however,
the equilibrium parameters change throughout space-time according to
the solutions of the hydrodynamic equations. According to BOLTZMANN
the equilibrium entropy is maximal
Seq(E,X) = sup S(E,X) . (2.1)
Therefore the lowering of entropy with respect to the equilibrium
value taken at the same energy and the other extensive parameters
may be taken as a measure of order /12,14/. Let us describe now
measuring procedures which would allow us to determine experimen-
2
