Synergetics, the book

Hermann Haken: Synergetics. Introduction and Advanced Topics. 

[Disclaimer: There are many things in this book that I do not understand, although hopefully I have grasped the big picture.]

Under the term Synergetics, Haken collects a number of approaches that can be useful in a variety of scientific disciplines ranging from physics, chemistry and biology, to economics and even sociology. Synergetics is presented as its own discipline with its characteristic concepts and methods. Yet this discipline draws on related fields such as thermodynamics, statistical dynamics, information theory, dynamic systems, control theory, bifurcations and catastrophe theory. Synergetics proposes to shed light on self-organized phenomena in various areas and to treat them within a unified apparatus. In particular, the slaving principle is the one trick that is used again and again. The slaving principle can be thought of in terms of a dynamic system where some variables change fast and others slowly, but there is also a separation into stable and unstable modes. The stable modes can be eliminated and treated as parameters, resulting in great simplifications.

This tome contains two classic volumes in one. Volume one (Introduction) begins gently with tutorial chapters on basic probability theory, ordinary differential equations, and their combination in stochastic differential equations. After the theoretical background has been presented, there is a chapter on self-organization followed by several chapters devoted to applications in various domains. First, the chapter on physics deals mainly with lasers. Then, as the chapters turn to chemistry, biology and economics in turn, the treatment becomes more and more accessible to the non-specialist. However, at the same time the models seem to become increasingly simplistic. Already the examples from biology and population dynamics are sketchy, and the discussion of applications to economics and sociology do not introduce many useful ideas. Nonetheless, one should remember that Haken was among the pioneers who brought a physicist's tool kit to these fields. In particular,

[...] synergetics has established links between dynamic systems theory and statistical physics. Undoubtedly, the marriage between these two disciplines has started. (p. 364 of the double volume) 

Further, regarding the connections of physics, chemistry, biology and even softer sciences:

It thus appears that we are presently from two different sides digging a tunnel under a big mountain which has so far separated different disciplines, in particular the “soft” from the “hard” sciences. (p. 364-5) 

We see the results of this excavation in numerous papers today, where physicists have begun to address such problems as the motion of crowds at concerts or the opinion formation before elections. However, there are obvious dangers involved in attacking problems that lie far beyond one's sphere of specialization. In the words of Buckminster Fuller (who also wrote a two volume book called Synergetics, otherwise bearing little resemblance to Haken's):

The word generalization in literature usually means covering too much territory too thinly to be persuasive, let alone convincing. In science, however, a generalization means a principle that has been found to hold true in every special case.

Apparently both kinds of generalization are involved in Hakens work; the applicability seems to decrease the further away from physics one gets, till it begins to look suspicious when applied to the social sciences. Meanwhile, the single finding that unites all chapters, the slaving principle, exemplifies the kind of generalization that holds true in several special cases, if not in all conceivable scenarios. It is the method of finding solutions that survives generalizations, not necessarily so with the modelling of systems in different fields.

Volume two (Advanced Topics) starts over with a long expository chapter on the application domains followed by the introduction of the theory. There are short sections on deterministic chaos, but Haken is not the best source on this. Quasi-periodicity is treated extensively. Although the exposition is clear to begin with, soon enough matters get complicated. If you ever wondered what makes a system of differential equations with quasi-periodic coefficients stable or unstable, this is the text to read.

Matters of style

The first chapters of each volume are tutorial in character and cover material that most readers probably already know. The manner of exposition changes as Haken begins to introduce his own findings—one can sense a shifting of gears when his enthusiasm sets in. Unfortunately, these parts involve solutions that stretch over sections or entire chapters, sometimes using idiosyncratic notation. It is often hard to tell whether a variable is supposed to be real, complex, or a vector, even though one may be able to figure it out from the context.

The writing has the appearance of a stream of consciousness layed out at the blackboard, rather than elaborated at the typing machine. Throughout the book, variable substitutions are profusely employed; so much, in fact, that one almost inevitably loses track of the variables' meaning. The derivations are decidedly informal, with almost no theorems and proofs. (There are a handful of theorems that rely on a long list of assumptions with long, unwieldy proofs.) Instead there are long chains of “simplifications” or “abbreviations”, often resulting in expressions that are longer than the one they replace, truncations of higher order terms in series expansions and other sorts of approximations. All these tricks are of course what physicists are usually good at, but for readers without the proper background, they may appear as incomprehensible as pulling rabbits out of a hat.

If synergetics has to do with self-organization of complex systems, it must be said that Haken is quite terse on the topic of self-organization as such. This is where some conceptual analysis is lacking. On the other hand, the cyberneticians have already contributed much hand-waving philosophizing on self-organization, without necessarily having contributed much to its understanding. Here, at least, one has a class of problems and an approach to their solution, but there is more to self-organization than what is covered in this book.