10. 



H.HAKEN. 1978. SYNERGETICS, an introduction. Nonequilibrium phase transi- 

 tions and self-organization in physics, chemistry and biology. 2nd enlarged 

 edit. 



Springer, Berlin, etc. Springer Series in Synergetics vol.1. XII, 355 pp., 

 152 figs., subject index. DM 66.00, $ 33.00 



This book has been well received and in fact this is the second, enlarged 

 edition, necessitated because the first edition (1977) is already sold out. 



The subject matter of the book is almost entirely mathematical. We an- 

 nounce it here because it is similar in spirit to a book by Nicolis and 

 Prigogine we reviewed last year (vol.17, part 2, review no. 9). The approach, 

 however, is entirely different. The term Synergetics was coined by the au- 

 thor when he realised that "the co-operation of many subsystems of a system 

 is governed by the same principles irrespective of the nature of the subsys- 

 tems " The author is a German theoretical physicist who, among other 



things, has published a book on laser theory. The book encompasses many do- 

 mains, from Brownian movement to sociology. The mathematics is kept at an 

 elementary level as far as possible. 



The book is perhaps best characterised by a quotation from the Preface: 

 "In recent years it has becom.e more and more evident that there exist 

 numerous examples in physical and chemical systems where well organized 

 spatial, temporal, or spatio-temporal structures arise out of chaotic 

 states. Furthermore, as in living organisms, the functioning of these 

 systems can be maintained only by a flux of energy (and matter) through 

 them. In contrast to man-made machines, which are devised to exhibit 

 special structures and functionings , these structures develop sponta- 

 neously - they are self organizing. It came as a surprise to many 

 scientists that numerous such systems show striking similarities in 

 their behavior when passing from the disordered to the ordered state. 

 This strongly indicates that the functioning of such systems obeys 

 the same basic principles. In our book we wish to explain such basic 

 principles and underlying conceptions and to present the mathematical 

 tools to cope with them." 



In chapter one, which sets the stage for the whole book, the problems it 

 considers are defined as follows: "It will turn out that equations governing 

 self-organization are intrinsically nonlinear. From those equations we shall 

 find in the following that "modes" may either compete, so that only one 

 "survives", or coexist by stabilizing each other. Apparently the mode con- 

 cept has an enormous advantage over the microscopic description. Instead of 

 the need to know all "atomic" coordinates of very many degrees of freedom 

 we need to know only a single or very few parameters, e.g., the mode ampli- 

 tude. As we will see later, the mode amplitudes determine the kind and de- 

 gree of order. We will thus call them order parameters and establish a con- 

 nection with the idea of order parameters in phase transition theory 



The amazing thing in self-organizing systems [is] this. Though energy is fed 

 into the system in a completely random fashion, the system forms a well-de- 

 fined macroscopic mode The systems we shall investigate organize them- 

 selves coherently" (author's italics). 



One difference with the approach of the Prigogine group is that the pre- 

 sent approach "investigates what happens at the instability point and it de- 

 termines the new structure beyond it. Some of these problems can be dealt 

 with by the mathematical theory of bifurcation, or, more generally, by a 

 mathematical discipline called dynamic systems theory. In many cases pre- 

 sented in this book we had to treat still more complex problems, however. 

 For instance, we had to take into account fluctuations, small band excita- 

 tions and other features. Thus synergetics has established links between dy- 



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