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PHYSICS REPORTS (Review Section of Physics Letters) 204, No. 1(1991)1—163. North-Holland

PHYSICS AND SOCIAL SCIENCE - THE APPROACH OF SYNERGETICS

Wolfgang WEIDLICH Inst itut für Theoretische Physik, Universitbt Stuttgart, Germany

Editor: B. Muhlschlegel Received September 1990

Contents:

Introduction 3 4.5. Probability distribution and stochastic trajectories 51

1 . Comparison of structures 7 5. Collective opinion formation 531.1. The level structure of nature and society 7 5.1. The two-opinion case 54

1.2. The interaction between levels: the “reductionist” 5.2. Dynamics of party images and voters opinions in the

versus the holistic” view 9 democratic system 691.3. Quantitative modelling in social science 12 6. Migration of populations 77

2. Interaction of macrovariables — semi-quantitative consid- 6.1. Comparison of migration and opinion formation 77

erations 14 6.2. The migratory equations of motion 78

2.1. The macroscopic approach 14 6.3. The case of two interacting populations 81

2.2. An abstract metamodel describing stability and cy- 6.4. Deterministic chaos in migratory systems 92clicity 14 6.5. Empirical evaluation of interregional migration 95

2.3. Selected examples of model interpretation 23 7. Formation of settlements 10 9

3. The framework of microbehaviour and macrostructures 25 7.1. Master equation description of urban evolution on the3.1. The space of social structures— aspects, issues and microscale 11 0

attitudes 25 7.2. Migration and agglomeration on the macroscale 12 03.2. The variables: socioconfiguration, associated variables 7.3. Settlement formation on the mesoscale: an integrated

and trend configuration 27 economic and migratory model 13 1

3.3. The elements of sociodynamics: dynamic utilities and 7.4. Alternative approaches to urban dynamics 14 1

probability transition rates 30 8. Master equation approach to nonlinear nonequilibrium

4. Constitutive equations of motion 34 economics 14 6

4.1. The master equation for the socio an d trend configu- 8.1. Introductory remarks 14 6ration 36 8.2. Modelling concepts: the economic configuration and

4.2. Mean-value equations for the components of the socio the elementary dynamic processes 14 8

and trend configuration 38 8.3. Master equation and mean-value equations for the

4.3. Stationary solutions of the mean-value and master economic evolution 15 2

equations 41 8.4. Analysis of market instabilities 15 4

4.4. A special time-dependent solution to the master References 16 2

equation 47

Abs tra ct:

Universally applicable methods originating in statistical physics an d synergetics are combined with concepts from social science in order to set upand to apply a model construction concept for the quantitative description of a broad class of collective dynamical phenomena within society.

Starting from the decisions of individuals and introducing the concept of dynamical utilities, probabilistic transition rates between attitudes and

actions can be constructed. The latter enter the central equation of motion, i.e. the master equation, for the probability distribution over the

possible macroconfiguration s of society. From the master equation the equations of motion for the expectation values of the macrovariables of

society can be derived. These equations are in general nonlinear. Their solutions may include stationary solutions, limit cycles and strange

attractors, and with varying trend parameters also phase transitions between different modes of social behaviour can be described.The general model construction approach is subsequently applied to characteristic examples from different social sciences, such a s sociology,

demography, regional science and economics. These examples refer to collective political opinion formation, to interregional migration of

interactive populations, to settlement formation o n the micro-, meso- and macroscale, and to nonlinear nonequilibrium economics, including market

instabilities.

0 370-1573/91/$57.05 © 1991 — Elsevier Science Publishers B.V. (North-Holland)


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PHYSICS AND SOCIAL SCIENCE -THE APPROACH OF SYNERGETICS

Wolfgang WEIDLICH

Insti tu t für Theoretische Physik, Universitàt Stuttgart, Germany

INORTH-HOLLAND


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W . Weidlich, Physics and social science — the approach of synergetics 3

Introduction

In the last 20 years great progress has been made in understanding complex systems in physics,

chemistry and biology. The new concepts necessary for this understanding were primarily developed inthe statistical physics of open systems far from thermodynamic equilibrium and in fields like quantum

optics and the theory of chemical reactions, where new types of “nonequilibrium dissipative structures”(using the nomenclature of I. Prigogine) appear.The conceptual framework for the mathematical treatment of such systems finds a rather general

formulation in “synergetics”, by definition, the science of the macroscopic space—time structures of

multi-component systems with cooperative interactions between their units. The framework of synergetics has been set up and worked out since 1970, mainly by H. Haken [1, 2 1 .

It is the purpose of this article to give an account — in view of these developments in naturalscience — of new approaches to make methods of statistical physics and synergetics available andtransferable to a quantifiable description of dynamic processes in human society.

In doing so, we are not aiming to present an uncoordinated collection of quantitative treatments of sectors of society. Instead, our intention is more systematic. W e will try to set up a general “model

construction strategy” for a quantitative — or at least semi-quantitative — treatment of a whole class of macrodynamic evolutions in society, making use of those concepts of statistical physics, whoserelevance is not only particular to physics but is much more universal!

However, since the article is primarily written for physicists, we will also make side remarkswhenever structural analogies between physical and social systems appear and seem worthy of consideration. It is our hope that the interdisciplinary relevance of some concepts of synergetics can bedemonstrated in this way, and that the approaches developed here contribute to the idea of the “unity

of sciences”.For two decades the author of this article, being a theoretical physicist, has been engaged with

increasing intensity in the problem of a quantitative description of the dynamics of social systems. Someof the main results of this research, obtained in cooperation with a small but highly motivated group of physicists, are summarized here (cf. the acknowledgements).

The work in the new field has also led, due to their gratefully acknowledged open-minded attitude,to new contacts and even to close cooperation with social scientists. In this context a special observationunder the perspective of a physicist should be made:

Physicists are used to a thoroughgoing mathematical quantitative fonnulation of their theories andconsider qualitative argumentations only as a preliminary stage of theorizing about phenomena. On theother hand, the mainstream of social science — with the exception of economics — concentrates on andprefers a qualitative argumentation, perhaps arguing that social systems are too complex for quantification (we shall come back to this problem). But even dwelling on a purely qualitative description, socialscientists have developed a high-level intuition and ingenuity of introducing adequate concepts whichcharacterize the complex social system on its micro- and macrolevels. Such characteristics of societycorrespond to the microvariables as well as to the macrovariables or order parameters in physical

systems. Therefore, no attempt at a correct quantified description can dispense with these qualitativecharacteristics of social systems investigated by social scientists. That means that a close cooperationbetween scientists arguing qualitatively and scientists constructing quantitative models is desirable andalmost indispensable.

However, before going into more detail, in chapter 1 in comparing the structures of systems ofnatural and social sciences and in considering the possible transfer of concepts, we must throw a glance


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4 W . Weidlich, Physics and social science — the approach of synergetics

on the past, where the mainstreams of natural sciences on the one hand and of the arts and socialsciences on the other appeared to diverge, separated frequently by a gap, denoted by Snow in hisfamous lecture [31as “the gap between the two cultures”.

As this divergent evolution took place despite the ever present ideal of the unity of sciences, there

must have existed immanent reasons for the failure of a closer cooperation between natural and socialsciences. Without going into historical detail we must briefly analyse the causes of this dichotomy andalso the newly arising reasons which lead to better prospects for a renewed convergence of scientific

approaches.On the side of natural science, first attempts emerged in the last century to describe social systems in

terms of a “social physics”. These approaches consisted of a more or less direct comparison of physicalsystems and their equations with social behaviour. However, the shortcomings of this “physicalistic”approach soon became clear: Generally, on the fundamental level there does not exist any directstructural isomorphism. Indeed, such an isomorphism would only exist if the states and the interactions

of the elementary units of a physical system could be formally and uniquely mapped to the states and

interactions of the units (individuals) of the social system. Evidently such an isomorphism, if possible atall, will only be an exceptional case.

On the other hand, also a direct comparison of physical and social systems on the phenomenologicallevel (for instance, comparing the equation of state of a gas and concepts like pressure, temperature andenergy with the behaviour of a society) can only lead to a superficial, short-breathed analogy lackingstructural depth. Therefore, physicalistic approaches of this type to explain social behaviour have beencriticized by social scientists for good reasons [41.But, along with this criticism, the whole idea of athoroughgoing quantitative treatment of the dynamics of social systems is perhaps in danger to bediscarded prematurely.

On the side of the arts and the social sciences there also existed developments widening the gapbetween these and the natural sciences. Thus, in the last century the arts and the social sciences wereseeking to establish their foundations independently of the natural sciences. There existed good reasonsfor this endeavour.

Firstly, the high complexity of the human individual and of the human society required methods of research adequate for the systems under consideration. Evidently it was not possible to wait for areductionist explanation in terms of biology, neurology or even genetics concerning the human being

and his behaviour within society to be found.Secondly, and more importantly, it was even claimed that the humanities require descriptive

categories which are completely independent of and irreducible to concepts of the natural sciences. (Weshall shortly discuss this problem of reductionism versus holism in chapter 1.)

Indeed, the level of human existence comprises essential features not adequately tractable byanalysis and theory in the terminology of natural science. These essentials include the uniqueness of thehuman individual and his history, the ultimate meaning and purpose of life, the problems of ethics,values and norms.

In connection with these new qualities emerging at the level of man and his society, the social

scientist is in a different situation to that of the natural scientist. Whereas the latter has no problem inobserving and analysing the objects of his science in an objective manner “sine ira et studio”, the socialscientist is, simultaneously, the participant and the observer of the system under discussion.

Being participant of the socio-political system he makes valuations by distinguishing, for instance,between its desirable and nondesirable evolutions according to his social, political, ethical and


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ideological attitudes and norms. But in his role as scientist he must observe the same system in a

remote, objective manner with complete impartiality, as if he were not affected by its fate. It seems notalways easy to separate the valuating and the objective perspectives, to be taken by one and the sameperson.

However, it is evident that any transfer of analytical and mathematical methods from natural tosocial science can only contribute to the descriptive and objective side of social science, but not to the“metatheoretical” problems of values and choice of norms.

Having discussed some reasons which may have given rise to a divergent evolution of social andnatural sciences, we now come back to the question of why a new convergence between sciences seems

more promising at the moment than previously.A first reason i s that natural science has since matured. More and more, complex systems are within

the focus of interest in physics, chemistry and biology. The problems of dynamic phase transitions inquantum optics, fluid dynamics (turbulence) and solid state physics; chemical reactions and neural

systems and the deterministic chaos in a variety of nonlinear systems provide a manifold of examples.Thus the overall experience in finding methods to treat complexity has grown in natural science.

Secondly, and most importantly in our context, our approach towards a quantified description ofsocial systems has a differing structure to the physicalistic approaches mentioned above. The lattermade direct use of physical models in order to interpret them in sociological terms. Instead, our

fundamental concepts refer to social systems from the very beginning. In the further construction of thequantitative formulation, we make use of mathematical concepts to describe the dynamics of statisticalmulti-component systems, which are universal and therefore applicable to social systems as well as to

physical systems. However, whereas these methods have already found widespread use in physics, theiruse in the social sciences is as yet only in an initial state. In pursuing their consequences in the social

sciences we will indeed find some deep and rather universal structural analogies between social andphysical systems. But these analogies are not due to a direct similarity between physical and socialsystems. Instead they reflect the fact that, due to the universal applicability of certain mathematical

concepts to statistical multi-component systems, all such systems exhibit an indirect similarity on themacroscopic collective level, which is independent of their possible comparability on the microscopic

level. The formation of such indirect structural similarities will be shown on several occasions in theforthcoming sections. The structural differences between physicalistic and synergetic approaches tosocial science are schematically exhibited in table 1 .1 .

Our further procedure of explanation is organized as follows:

Since our subject is of an unusual nature, we will go into more detail in chapter 1 , comparing thestructure of the objects in the sciences involved. The contribution of synergetics to the general problemsof reductionism versus holism will also be discussed. And finally the potential achievements ofquantitative modelling i n social science are considered. In chapter 2 semi-quantitative considerations onthe basis of interacting macrovariables demonstrate the possibility of general insights using simple“metamodels”. In chapters 3 and 4 the central concepts are developed. Starting from the microlevel,the decisions of individuals are introduced along with probability transition rates and their effect on

certain classes of macrovariables in society. The constitutive equation of motion for the macro dynamicsof society then turns out to be a comprehensive master equation of the probability distribution over themacrovariables. M ean-value equations of in general nonlinear structure can be derived from this masterequation. Chapters 5 to 8 are accordingly devoted to applications to different sectors of the society,demonstrating that a unified treatment of social dynamics is feasible. The applications of the same


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Table 1.1a) The questionable direct application of models of natural science to social science. b) The indirect structural analogy between systems of nature

and society substantiated by universally valid synergetic concepts

(a )

Models and Theories ~estioMe Models and TheoriesDirect Modelof Natural Science A~iication of Social Science

ft ft

Str~ctures MatI~ematical Mathematical S~ctu~s 1

(b)Indirect Structural Analogies

Between Systems of

Nature and Society

Models and Theories Models and Theories

of Natural Science of Social Science

ft ____ f tAdequate __________ Universally ~ Adequate

System SystemConcept “. Synergetic Concepts ] / Concepts

formalism comprise as different fields as opinion formation, migration of populations, formation of settlements, and non-equilibrium economics. The interest focusses on the one hand on the theoreticaldescription and partial explanation of certain collective self-organizing processes within society, in termsof (semi-)quantitative models, and on the other hand, as far as it is possible, on the retrospective andprospective quantitative analysis of concrete processes.


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W. Weidlich, Physics arid social science — the approach of synergetics 7

1. Comparison of structures

The formulation of a framework for the quantitative treatment of social systems is not such anobvious procedure as setting up quantitative theories for physical systems. Thus initially it seems

adequate to have a broader view on the structural relations between sciences. Some principal questionsof a more philosophic nature must be raised in this context and this must be done i n terms of qualitativeformulations, since qualitative considerations always have to precede any possible quantification. Let us

first reflect on the level structure of reality.

1.1. Th e level structure of nature and society

It seems completely obvious that all of reality, including the inanimate and the animate world, is

stratified into sequences of organizational levels of varying complexity. The higher, more macroscopiclevels are composed of and rest upon the lower, more microscopic levels.

A level i s defined as a stratum of reality of a certain self-contained organization. Each stratum at thisorganizational level of complexity has various characteristic qualities emerging; however, these qualities

cannot be found in other levels, in particular lower levels. If quantitative formulations of the levelstructure exist, certain macrovariables (order parameters) then characterize the specific qualities of thatlevel. The relative self-contained property of a level is then expressed by the fact that i ts variables obeyan almost self-contained, autonomous subdynamics. The structure of lower levels, of which that level is

composed, and the structure of higher levels, into which that level is embedded, enter the quasi-autonomous dynamics of the level under consideration, but only in terms of internal or externalconstant constituents and boundary conditions.

Already physics — not to mention biology — provides so many examples of this hierarchical structurethat it is sufficient to conclude with this example: Take molecules as the organizational level underconsideration. Molecules are composed of nuclei and electrons. But the detailed structure of the lowerlevel of nuclei, i.e. their composition of protons and neutrons which are again composed of quarks andgluons etc., does not matter for the molecular level, apart from some global nuclear constants like

mass, charge and spin. And also the higher level into which the molecules may be embedded, namelythe gas or the molecular crystal, provides only certain conditions, such as temperature or the crystal

field, under which the molecular self-organized structure exists.Since the level structure of reality is such a dominant fact, it is only natural that the system of

sciences, where each science is defined as having a certain domain of objects, should also be seen asstructured according to the organizational levels of the inanimate and animate world. Tables 1.1 and 1.2depict — in a rather coarse-grained form — how the sequence of levels of reality imposes a sequence of sciences. The interaction between the levels — to be discussed now — leads to a corresponding overlap of the domains of sciences.

1.2. The interaction between levels: the “reductionist” versus the “holistic” view

The level structure which appears to be a dominant feature of nature, is now seen as a problemdemanding an explanation. We begin by formulating two extreme and opposite standpoints with respectto the nature of levels. Later both standpoints will be partially justified and also made partially relative.

The first standpoint i s named reductionism. According to reductionism, all properties including the


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newly emerging qualities of the level of higher complexity must and can be reduced to — and thereforeexplained by — the properties and qualities of the lower (microscopic) level forming the units of whichthe higher level is composed.

The second and opposing standpoint is named holism. According to holism, the properties andqualities of a complex, organizational level define the wholeness of this level. These qualities have an

existence of their own and it is neither necessary nor possible to derive them from lower levelstructures. Thus the separation of levels, each in its own specific wholeness, is considered as an absoluteontological structure.

It seems that physicists and most natural scientists are more inclined to adopt the reductioniststandpoint, whereas psychologists and scholars of the arts seem to prefer holism. At least theseattitudes appear understandable for the following reasons:

In physics there already exist successful examples of a procedure of reduction. The most prominent

example is the derivation of the laws of phenomenological thermodynamics from the microlevel, thefundamental laws of statistical mechanics.

On the other hand, psychologists, scholars and social scientists have to work with the mind and its

interactions. But the level of mind structure — including logic and the ability to act scientifically —

exhibits a self-quality of its own of almost perfect autonomy, whereas the reductionist postulate hereleads to the psycho-physic problem, which is, as yet, unsettled.

However, we can now see that the two standpoints of reductionism and holism, if formulatedproperly, are not as irreconcilable as it first appears. On the side of holism it becomes more and moreclear that the independence of the qualities of a complex level cannot be an absolute one but only arelative one. Indeed, since the complex organizational levels are composed of units belonging to a lowerlevel, all qualities of the higher level have to be the collective resultant and somehow the aggregatedeffect of interactions between the constituent elementary units, even if this macroeffect cannot beexplicitly decoded (see, e.g., ref. [5]).

On the other side, a close reductionist inspection of the interaction of units within multi-unit systemshas, perhaps surprisingly, led to rather universal insights into the manner of how levels of relative (not

absolute!) self-contained structure are organizing themselves and how these levels, once established,

mutually interact.In this respect, let us consider two principles originally formulated in physics but also applicable,

mutatis mutandis, to other systems like human society, namely the principle of self-consistency and theslaving principle.

It i s well known in physics that self-consistent calculations, like the Hartree—Fock theory, are one of

the backbones of multi-particle theory, particularly applicable to particles with long-range interactions,such as electrons with Coulomb interaction in the atom or in a solid. Here, we are interested in thatprinciple with respect to its potential transfer to more general multi-component systems.

It is the essence of the self-consistency principle, that each particle contributes to the generation of acollective field and that, conversely, each particle moves (according to an effective one-particleSchrödinger equation) under the influence of this collective field. Self-consistency expresses the

compatibility of the cyclic relation between causes and effects: The field is the collective effect ofindividual particles in certain modes of motion (wave functions), where the individual modes of motionare, in turn, determined by that collective field. Simultaneously the whole system is split into twointeracting levels. Instead of multiples of individually and directly interacting particles, we now have thelevel of the global field generated by all particles, and, on the other hand, the level of individual


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10 W . Weidlich, Physics a nd social science — t he approach of synergetics

particles moving within that field. Hence, the direct interaction between the particles has beensubstituted by an indirect interaction via the medium of the global field.

In this formulation it is now possible to transfer the self-consistency principle to social systems.

Originally and fundamentally, society (and its history) could also be viewed as a system of interactingindividuals (including the momentary interactions between living persons and the retarded interactionsof historical with living persons). However, this description would be extremely cumbersome andinconvenient and would disguise the relevant determinants of the sociological process. Instead thefollowing formulation was chosen and i s fully in accordance with the self-consistency principle:

The individual members of society contribute, via their cultural and economic activities, to the

generation of a general “field” of civilisation consisting of cultural, political, religious, social andeconomic components. In particular, all institutions of the state, religion, economy, jurisdiction and thepolitical ideology belong to this collective field, which determines the socio-political atmosphere as well

as the cultural and economic situation of a society. This field may therefore be considered as therelevant (multi-component) order parameter of society.

Conversely, this collective field strongly influences the range of possible activities of an individual, bygiving him orientation and incorporation into cultural traditions, by extending or narrowing the scope of information available and of thought and action, by partially relieving the individual from making

decisions about issues already predetermined by the structure of society, and by activating or

deactivating his latent aptitudes.Again, the two levels of individual behaviour and of the collective social field have been organized in

a self-consistent manner, and the direct interaction between individuals i s to a large extent substitutedby an indirect interaction mediated through the institutions and organizations of society.

The other principle relevant to the problem of level structure is the “slaving principle”, set up by

Haken [1]. It was his ingenious idea to explain the fact that on the macroscopic level only a few orderparameters dominate the dynamics of the system via consideration of the time scales of motion of a setof interacting variables. The essence of his algorithm can be described as follows: Start from thestationary solution of a set of nonlinear equations of motion for the system variables. After changing

some exogenous control parameters, it appears that in general only a few variables v~(t)become

unstable and slowly begin to grow, whereas the motion of the bulk of the variables v~(t)around theirstationary values is quickly damped out. It turns out that the further temporal evolution of the v4(t) is

“slaved” by the v~(t),since the fast variables v4(t) hastily adapt to the ruling variables v~(t).Hence thev5(t) can be expressed in terms of v~(t),and therefore “adiabatically” eliminated. As a result thedynamics of the whole system can be expressed solely by equations of the few “order parameters”v~(t)which dominate the whole system.

This algorithm not only explains the arisal of a quasi-self-contained dynamics for a few order

parameters growing to macroscopic size, but also how this fact is connected with — in physics frequentlywell-separated — characteristic timescales separating the slow, dominant order parameters from themajority of fast, but slaved variables.

The slaving principle can also be transferred to social systems, at least in i ts general spirit, however,making two natural reservations: In contrast to physics, social science has no fundamental sets of equations of motion available for the variables from which the order parameter dynamics can bederived; furthermore, due to the complexity of the social system, we must expect an overlap of tirnescales, preventing perhaps in most cases a clear-cut distinction between slow and fast, or unstable

and stable variables. Due to the lack of equations of motion for fundamental variables our starting


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point for quantification must therefore be a different one. We wi l l see, however, that our formalismimplicitly takes into account the slaving principle, too.

After having discussed some generally applicable principles for the stratification of reality, let us nowdiscuss some differences between physical and social systems. Although we cannot give an exhaustivedefinition of “complexity”, the following considerations lead to the conclusion that physical systems stillexhibit a relatively “well-ordered” kind of complexity, whereas in social systems we find a fullydeveloped “intertwined” kind of complexity.

In physics we have relatively few hierarchically ordered levels, ascending only a few steps from themicroscopic to the macroscopic structures. Each level is distinguished by only a few stable, qualitativecharacteristics. The levels are well separated, and the interaction between them takes place verticallybetween adjacent levels, as in the case of particles interacting with their self-consistent field. Forexample, taking the earlier sequence: (nuclei +electrons) — (molecules) — (gas or liquid or solid), weobserve that the qualities of the individual levels are well defined and well separated. Take, forexample, the complex level of a gas. The properties of this level are not greatly influenced by thedetailed structure of molecules and are influenced even less by the structure of the nuclei composing themolecules: Only the specific heat of the gas is somewhat modified due to certain energetically excitablestates of the molecules.

Therefore it appears that the lower level provides the constituent units for the next higher level only.This means that atoms or molecules are the constituent units of the gas, but the detailed composition of these atoms in terms of elementary particles is no longer important for the properties of the gas.Statistical mechanics and/or solid state theory then shows us the (vertical) interaction between the levelof the constituent units and the level of their collective.

The characteristic timescales on which the motion of the microvariables and of the macrovariablesoccurs, are usually well separated (as presumed for the slaving principle), so that the faster variablesnormally adapt to quasi-equilibrium in relation to the momentary values of the slow variables.

The social system in many respects exhibits a higher degree of complexity. Firstly there exist manyorganizational levels, such as the family, the school, the firm, the political party, the church, the “club”,the association, the university, the government etc. These levels are more densely spaced and are also

mutually overlapping, as the same individual can simultaneously be the elementary unit in severalorganizations, playing a specific role in each of them. The qualities and attributes characterizing eachlevel are more diverse than in physics. Also new types of qualities appear. For example, one candistinguish between structural and functional characteristics, which do not necessarily coincide since thesame function could be exerted by different structures and one structure can carry out several functions.

Since many overlapping organizational structures compete at the macroscopic collective level, we notonly have “verti~al”interaction between the micro- and macrolevel, but also “horizontal” interorgani-zational interaction on the macrolevel.

Even birth and death processes of whole organizations, such as the birth or death of a political party,must be taken as part of the effects of such interactions. Under critical conditions, i.e. revolutions,which signify the phase transitions of society, the whole intra- and interorganizational dynamics may

change dramatically, so that simultaneously many old order parameters (i.e. governments, institutionsetc.) may decay and new ones may be formed. Usually the dynamics of all the macrovariables whichcharacterize the organizations take place on around the same timescale. Thus a distinction between fastand slow variables, allowing the elimination of fast variables, is virtually impossible. Hence, in generalall suborganizations on the macrolevel show a fully dynamic interaction and thus it becomes difficult to


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12 W . Weidlich, Physics and social science — t he approach of synergetics

separate off the quasi-self-contained subdynamics of certain organizational sublevels. Only if i t can beproved that the intra-organizational interaction is much stronger than the inter-organizational inter-action, then the organizational substructures have a certain stability which may justify their treatment as

separate sectors of society.A further complication arises due to the fact that the elementary units at the microlevel of physical

systems are poorly structured, whereas the elements of society, i.e. the individuals, are themselvescomplex systems. The microsystems (atoms, molecules) of physics have at most only a few excitabledegrees of freedom; in contrast, the soc ial individual has a manifold of potential modes of behaviour. Itdepends on the state of all organizational macrolevels in which the individual simultaneously plays hisrole, which mode of his behaviour or action is excited and hence surfaces, and which on the other handremain latent and dormant. Obviously the relative flexibility of the transitions between the dormant and

active modes of individual behaviour, in sensitive interaction with various macrolevels, will furtherincrease the degree of complexity of the social system.

This high complexity, the causes of which have been enumerated here to some extent, may have ledsome social scientists to the conclusion that a quantitative treatment of social systems is inadequate,perhaps apart from some minor investigations. However, in the next section we will give argumentsfavouring quantitative mathematical modelling in social science.

1.3. Quantitative modelling in social science

We have already argued that in all sciences, and particularly in those rich of different qualities, acareful, qualitative analysis of the system must precede quantitative modelling. Now we will argue thatthe gap between qualitative and quantitative thinking in the social sciences is not a principal one, butthat there exists a considerable, although at times tacit overlap and even an intense joining of both linesof thought.

On the one hand the analysis of a — qualitatively arguing — social scientist or of a statesman will be

excellent if he selects at most a few dozen relevant macrovariables of the social systems underconsideration and if he makes, in his qualitative model, an assessment of the interactions of these

variables. If he also has information or intuition enabling him to make estimates concerning the order of magnitude of the variables and their mutual influences (that means already quantitative estimates!),then he can come to conclusions about the dynamic behaviour of the system. This dynamics can be

interpreted in relation to changes of variables and changes in the strengths of their mutual influences.

(The latter are again estimates concerning the evolution of magnitudes and of degrees of interactions.)Thus qualitative modelling tacitly implies estimates about the quantitative amount or magnitude,

strength or intensity of the relevant qualities.On the other hand, the explanatory potential of quantitative (dynamic) models would be very much

underestimated if their value would only be seen i n making simple statements about the output of some

quantities in some process. Instead, the most important questions answerable by a quantitative modelare of a qualitative nature!

Let us now ask some more refined qualitative questions about system behaviour, which can only beposed with some exactitude if an adequate quantitative model can be developed. Which variables aredominant and which are negligible under the control of given exogenous parameters and in a givenendogeneous stage of the process? When is the process accelerating and when i s it saturating? Underwhich circumstances do the effects sustain the causes, and when do they counteract the causes?


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Other questions to be answered by quantitative models are of an even more qualitative nature andrefer to the global dynamic behaviour of the system. For example: Will the system exhibit cyclicbehaviour or stability? Will there exist phase transitions between different dynamic modes, at thecritical values of certain exogenous parameters, for instance transitions between stability, periodic (limitcycle) behaviour and chaotic (strange attractor) behaviour? What i s the uncertainty of a given path, thatmeans, the variance of deviations from the given system trajectory due to fluctuations?

With respect to such structural insights, quantitative mathematical models have an operational valuesuperior to that of purely qualitative models and explanations, only orientated towards the given realsociety: Once set up, a mathematical model not only attempts to describe the given realized society, butby variation of its system parameters, it yields insights concerning the behaviour of a manifold of

fictitious, unreal societies and thus sharpens the view on the structure of the given society.Qualitative concepts are indispensable prerequisites for setting up a mathematical quantitative

model, but, on the other hand, quantitative models may contribute in making qualitative concepts

unique and measurable and in providing insights into a manifold of qualitative structures. Thereforeone may speak of a “feedback loop between qualitative and quantitative thought”.

In relation to the value of quantitative models within such a mutually supporting and reinforcingfeedback loop, it is not necessary, and in general also not attainable, that the models reach exactitudein the literal sense of exact numerical values, for a ll variables and system parameters involved. Instead,

it is better that we introduce the concept of “semi-quantitative modelling”.Thus, semi-quantitative models should be generic and robust. That means, they should describe the

essentials of the dynamic interrelations between the variables of a sector of the society, typical for aclass of phenomena in that sector. Such models can neglect minor and incidental items, however, evenat the expense of exactitude concerning numerical detail, if the essentials are robust, that means if theyremain stable under such slight neglect.

Semi-quantitative modelling therefore implies the philosophy of keeping the model as transparent aspossible by selecting a small (but still quasi-self-contained) set of variables of high explanatory valueand by introducing understandable exogenous parameters, instead of using an inflated multi-variablemodel, and thus simultaneously losing the interpretative reasoning.

Having the potential advantages of quantitative methods in mind, we will now formulate somegeneral requirements for quantitative modelling, in relation to social science:

1 . The models should be generic and robust, and simultaneously parsimonious in the number ofvariables and parameters used, so that their interpretative transparency is still preserved.

2. Since the level structure is so fundamental, the models should relate in some transparent way to atleast the main levels, namely to the level of individual behaviour (i.e. decision making) and to the level

of collective macrovariables (i.e. the global dynamics).3 . Since the elementary decision making can only enter into the theory via a probabilistic description,

the theoretical framework should allow for formulations including stochastics, and, on the other hand,by neglecting fluctuations, also allow for a quasi-deterministic description.

4. Starting with certain “phenomenological” considerations, the models should have a “horizontally”

and “vertically” open structure. That means that “horizontally” the model structure should in principlebe flexible enough to separate off sectors (for more specialized treatments) or, conversely, to combineseveral sectors of society (for more comprehensive treatments). “Vertically” speaking, the modelsshould be open to the addition of deeper, theoretical explanations and laws of the dynamics of anyparameters entering the models on the individual’s level.


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5. In cases where empirical, numerical data are available, the models should allow regression

analysis to determine their parameters. By solving their equations of motion, the models then shouldgive rise to short-term and medium-term forecasts.

2. Interaction of macrovariables — semi-quantitative considerations

2.1. Th e macroscopic approach

In this chapter we pursue a purely macroscopic approach by directly setting up equations of motionfor the dynamics of macrovariables. Such a macroscopic approach has, up to now, mainly been appliedin macroeconomics, where plausible model equations are set up for aggregate variables like the global

production, the gross national product and for global factors like total investment, total capital and thenumber of employed workers i n a national economy.

However, for each macroscopic theory there remains to be solved the nontrivial (reductionistic)“aggregation problem” of how the macrovariables, their interactions and dynamics can be defined andderived in terms of microvariables and their interactions. (This relation between micro- and macrolevel

will be treated in detail in chapters 3 and 4.)Here we will consider the interaction of macrovariables, in order to give a strongly idealized, hence

semi-quantitative explanation for the phenomenon, that, at the level of the group, the organization orthe whole society, in social systems certain processes approach either a stationary state or a quasi-cyclic

motion.

Our procedure will be the following: Initially, simple “generic” forms of interactions betweenquantified socio-economic macrovariables are introduced, including in particular “cooperative” and“antagonistic” interactions. Secondly, a dynamic model is set up implying these kinds of interactionsbetween its variables. All variants of its simplest version, namely the two-variable model, can be solvedexplicitly. According to the choice of the interaction type, the trajectories approach either fixed pointsor a quasi-cyclical motion. The latter case includes trajectories spiralling towards a fixed point or a limit

cycle.Finally, the so defined abstract “metamodel” is then applied to the dynamics of concrete cases by

appropriate concrete interpretations of the variables and their dynamics. One could say that themetamodel represents the skeleton of the system by providing the underlying dynamics, and the flesh of the system can be seen as the concrete meaning and substantial interpretation of the variables andinteractions. Two cases from different sectors of society will be directly discussed: political systems (theinteraction between government and people) and economic long term cycles (the dynamics of theprosperity, recession, depression and recovery phases). More examples can be found i n ref. [6].

2.2. An abstract metamodel describing stability and cyclicity

Let us now specify the possibilities of elementary influences between macrovariables. In doing so wewill at first completely disregard the concrete nature of the macrovariable considered. W e only assumethat the variables are quantifiable by whatever method and also assume real, continuous positive values.These values are a measure of the amount/degree/amplitude/intensity of the quality behind themacrovariable.


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Being reduced to their quantitative measures only, the influence of one variable, say x, upon anothervariable, say y, can only be expressed in quantitative terms, disregarding its concrete meaning. Onlyfew possibilities of such a quantitative influence are feasible, and will now be discussed:

(a) The variable x can support (enhance) the amplitude of y, or(b) the variable x can suppress (diminish) the amplitude of y.

If y coincides with x itself, we have in case (a) a self-supportive x variable enhancing its own amplitude,and in case (b) a self-suppressive x variable saturating its own amplitude.

Of course the degree of support or suppression of the passive variable y by the active variable x willdepend on the amplitude of both variables. In particular, the active variable x may — depending on~itsown amplitude — switch from support to suppression of the passive variable y, or vice versa.

Let us now introduce two kinds of active x variables with specific support/suppression behaviour inrelation to their passive partner variable y: The (active) variable x is denoted as cooperative with the

(passive) variable y, if it supports y for large x, but suppresses y for small x. (Hence the cooperativevariable x tends to assimilate the magnitude of y in relation to its own magnitude.)

On the other hand, the (active) variable x is denoted as antagonistic to the (passive) variable y, if itsuppresses y for large x, but supports y for small x . (Hence the antagonistic variable x tends to opposethe magnitude of y in relation to its own magnitude.)

This type of cooperative — or alternatively antagonistic — interaction of one (active) variable with

another (passive) variable seems to occur frequently in social systems. Therefore we will use thisinteraction type (appropriately idealized) as a design element in building a “generic metamodel of dynamic interaction”. W e will restrict the model to two variables, although generalizations are easilypossible. And of course, each variable can play the active as well as the passive role.

Furthermore, assuming that both variables x and y are self-saturating, the simplest type of equationsof motion, capable of including interactions of the cooperative or antagonistic type and of self-saturation seem to be generalized logistic equations of the form

dx/dT=x[a(y)s—x], 0


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these micro-equations (in this sense the aggregation problem can here be seen as solved): Let us think of two groups of individuals i = 1,2,. . . , N and j = 1,2,. . . ,M, whose individual activities can becharacterized by the variables x.(t), i = 1 , . . . , N and y.(t), j = 1 , . . . , M, respectively. Furthermore,let

x(t) = c~x1(t), (2.5)

y(t) = ~ d1y1(t) (2.6)

be collective variables formed with the x1(t) and y1(t), where c, and d . are coefficients describing therelative weight of x1(t) and y.(t) in x(t) and y(t), respectively.

Assuming now that x,(t) and y.(t) satisfy micro-equations of the kind

(d/dT)x1 = x,F(x, y), (2.7)

(dIdT)y1 = y1.G(x, y), (2.8)

one easily obtains, after multiplying (2.7) with c, and (2.8) with d and making sums, the exactmacro-equations,

(d/dT)x = xF(x, y), (2.9)

(dldr)y = yG(x, y). (2.10)

Obviously, (2.1), (2.2) and (2.3), (2.4) are special cases of (2.9), (2.10) making the following choices:

F ( x , y)=a(y)s—x, G ( x , y)=b(x)s—y, (2.11)

or

F(x,y)=a(y)c(y—x)s—x, G(x,y)=b(x)c(y—x)s—y. (2.12)

One should note, however, that the form of the assumed micro-equations (2.7), (2.8) is a very specialone: Each individual variable x,, y1 interacts with each other individual variable Xk, y1, only via thecollective variables x and y in the functions F(x, y) and G(x, y)! In practically all other cases, obtainingexact macro-equations from the set of micro-equations is not possible, although the macro-equationsstill may make sense, but only as an approximate description of the collective dynamics.

Now we must specify the influence functions a( y), b(x) and the twist function c( y — x). We havechosen idealized forms, namely step functions, for a(y) and b(x), showing either a cooperative or an

antagonistic influence of y on x, or vice versa of x on y. Also c( y — x) is chosen as a step function.Thus, the influence of y on x in (2.1) is cooperative, if

a(y)=a


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and i s antagonistic, if

a(y)=a~>0, for0


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18 W . Weidlich, Physics and social science — t he approach of synergetics

case - y the influence of x on y is cooperative [see (2.15)],

the influence of y on x is antagonistic [see (2.14)];

case ~ the influence of x on y is antagonistic [see (2.16)1the influence of y on x is cooperative [see (2.13)]. (2.20)

In each of the four variants of the model, a, 1 3 , - y , ~, each of the four quadrants I, II, III, IV of thevariable space assumes one of the types a, b, c, d . The following allocations are easily made, taking into

account the definitions (2.20) of the model variants, (2.18) of the quadrants, and (2.19) of the types.

Thus:

casea I~(a), II~(b), III2(c), IV~(d);

case13 I~(c), II~(d), III~(a), IV~(b);

(2.21)case - y I (d) , II (c) , II I (b) , IV~(a)

case ~ I (b) , II (a) , III (d) , IV~(c)

Figures 2.la to 2.Th illustrate the quadrant structure of the cases a to ~l.Now we will discuss the solutions of the model variants. The simple choice of the influence functions

has the advantage that the coupled equations of motion (2.1), (2.2) [and also (2.3), (2.4)] can be solvedexactly in each quadrant. This is the first step to obtaining the full trajectory within the space of variables. In each of the quadrant types a, b, c, d both equations of motion (2.1), (2.2) have certainconstant growth rates and can be solved explicitly. By eliminating the time one also obtains the explicit

form of the flux lines.The results are summarized in eqs. (2.22), (2.23), (2.24) through (2.31), (2.32), (2.33) for quadrants

of type a, b, c, d, respectively.

Quadrants of type a .Equations of motion:

dx/dr=x(a~s—x), dy/dr—y(b~s—y). (2.22)

Solutions:

x0a~sexp(a~sr) y0b~sexp(b~sr)

x(r)= , y(r)= . (2.23)x0exp(a÷sr)+(a~s—x0) y0exp(b~s’r)+(b÷s—y0)

Flux lines:

[y/(b+s — y)]a+ — [y0/(b~s — y0)]’~ 2 24

[x/(a+s — x)]b+ — [x0/(a~s — x0)]~ ( . )


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Y (a) Y ( 1 3 )

a(y)=a~>0 a(y)=a~>O a(y)=ci.Ob(x)=b.O b(x)=b~>0 b(x)=b_O a(y)=a~>Ob(x)=b_O b(x)=b~>O b(x)=bO a(y)=aO b(x)=b...


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Flux lines:

[y/(~b~s + y)j~[x/(a~s — x)]~= [y 0/(~b~s + y)]a+[/(as —x0)J~. (2.27)

Quadrants of type c.

Equations of motion:

d x / d T = x(—~ajs — x), dy/dT = y(—~bjs — y). (2.28)

Solutions:

— x0(ajs — y0~bjs

x(T)— —x0+(~as+x0)exp(~a~sr)’ y(r)_ —y0+(~bs+y0)exp(~b~sT)~ (2.29)

Flux lines:

[y/(~b~s + y~Q- — [y0!(~b~s+ y0)]Ia~~[b — . ( . )

[x/(~a~s + x ) 1 - [x~/(~a s +xe)]

Quadrants of type d .

Equations of motion:

d x / d r = x(—~a.js — x), dy/dT = y ( b + s — y). (2.31)

Solutions:

x0~ajs y0b~sexp(b~sr)x(T)= , y(T)= . (2.32)

—x0+(~as+x0)exp(~ajsr) y0exp(b÷sr)+(b~s—y0)

Flux lines:

[y/(b~s — y)1~ [x/(~a~s+ x)]b+ = [y0/(b~s — y0)~Ia][~0/(~~s + ~ (2.33)

In all equations (2.22) to (2.33), the x 0 and y0 are initial values for x(T) and y(T) at T = 0 . In the fluxline equations (x0, y0) can be considered as any predetermined point on the flux line.

The second step in finding the full trajectories in the variable space is the formulation of matching

conditions for the flux lines at the inner boundaries of adjacent quadrants.As an example for matching the flux lines between different quadrants, let us discuss the model

variant a and the matching at the boundary between quadrants I and II, that means at points (x12, y12),where x 1 2 = x~and y~


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W. Weidlich, Physics and social science — the approach of synergetics 21

[yI(b~s — y)]a+ — [y12/(b~s —y2)}a÷ 2 3

[x/(a÷s_x)]b+ — [xsI(a+s_xs)Ib+ ( 4)

and the flux line in quadrant II, which is to be matched at (x~,y12) obeys, according to eq. (2.27), theequation

[yI(~bjs + y)]a+[ I(as — = [y12!(~bjs + y )Ia+[xI(as —x~)]~. (2.35)

In this way all the matching conditions between the quadrants I, II, III and I V , can be set up for allvariants a, 1 3 , y , ~ of the model.

This somewhat cumbersome but straightforward analysis of a ll matching conditions (see ref. [6])nowleads to the following results, which seem to be generic. The analysis also seems usable for lessidealized forms of cooperative or antagonistic interactions.

1 . The cases a and 1 3 with a symmetric relationship between the variables x and y, and with bothvariables being cooperative or both being antagonistic, leads to stability. In these cases the flux lines of

each quadrant cross either one or the other boundary to an adjacent quadrant. The two types of flux

lines in each quadrant are separated by separatrices which end at the center of the quadrant system,namely at point (x~,y~).

In case a, with both variables being cooperative, the flux lines end in one of two stable fixed points,

either in point (0, 0) or in point (a~s;b~s),where both variables simultaneously reach their saturationlevel. In case 1 3 , with both variables being antagonistic, the flux lines also end in one of two stable fixed

points, either in point (0, b÷s),where x is zero and y has reached its saturation level, or in point(a÷s,0), where y is zero and x has reached its saturation level.

The cases a and 1 3 are illustrated in figs. 2.2 and 2.3.2. The cases - y and ~ with an asymmetric relationship between the variables x and y, that is one

variable being cooperative and the other antagonistic, lead to cyclicity. In these cases the flux lines

__ ~

/ /// -. ///“A\\\ \I J 17 _-‘~ ~—.-.-_ / I / ,“ Z \\\ \

I ~ .~ .,~ —.----—-.-—-.-—— I I f , ~ \‘\ ‘ ‘

~Er~—~I J 7 ~ \~ ~

li//V V\\\\\ \

////‘7~7’ \~\ \ III,,,- ~ ~ ~ ~ \ 0 ~

0 z $ 0 S x x

Fig. 2 . 2 . Case a. Flux lines and separatrices, if both variables x and y Fig. 2.3. Case 1 3 . Flux lines and separatrices, ifboth variables x and y

are cooperative. Parameters: x~= y~= z; s 2z; a~= b, = 1; a_ = are antagonistic. Parameters: x~= y~= 1~s = 2z; a~= b~= 1; a_ =

b =—1. = — 1.


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y x

Fig. 2.4. Case ~ywith cooperative x , antagonistic y , and contracting twist function. The flux lines spiral into the stable focus (z, z). Parameters:

c1 = 0.5; c 2 = 1.0; a~= b~= 2; a = b = — 1; z = 1 ; s 50 .

(a) (b)

y z

Fig. 2.5. Case ~ywith cooperative x, antagonistic y, and amplifying twist function. The flux lines spiral outward from the unstable focus (z, z) and

approach a limit cycle. Parameters: c 1 = 2; c2 = 1; a~= b~= 1; a_ = b_ = — 1 ; z = 1; s = 50 .

within a quadrant connect both inner boundaries of that quadrant. Hence the trajectories sequentiallytraverse the quadrants, which is only possible if they encircle the centre (x5, y~)of the quadrants. No

separatrices exist in these cases.The twist function c( y — x) can now either be chosen such that contraction of the variables prevails

and the trajectory spirals into the stable focus (x~, y~),or such that amplification of the variablesprevails. In the latter case, the focus (x5, y~)is unstable and the trajectory must spiral towards a finitelimit cycle, due to the saturation terms in the equation of motion.

Figures 2.4 and 2.5a,b illustrate cases - y with either contracting or amplifying twist function.


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2.3. Selected examples of model interpretation

Proceeding as indicated in section 2.1 we will now give different interpretations of the model, leadingto a semi-quantitative understanding of the social dynamics.

At first we apply the model to the relation between people and their government, to obtain an,admittedly vague, understanding of possible dynamic modes.

We begin by identifying the variables x and y. Thus let x represent the degree of influence anddemocratic participation of the people, and let y represent the degree of power and authority of the

government. (Both variables can in principle be quantified by forming weighted means over values of appropriately chosen indicators.)

Next, the interaction between the variables must be specified. It is obvious, that the kind of interaction must be assessed from the historical experience of governments and people. However, the

model can only show certain dynamic consequences of the presumed interactions. An important casefound in the analysis of attitudes within political systems seems to be that of “cooperative” people (x)

and a “cooperative” government (y). This means, in the sense of the abstract definition of cooperationas given above, that the following concrete interactions can be seen:

Influ en ce of x o n y: If the people are influential (large x), they tend to affirm the activities of the

government; however, if the people have little or no possibility of participation (small x), they tend toobstruct the measures of the government.

Influ en ce of y on x: If the government has power (large y), then i t is efficient in supporting the peopleand their participation; if, however, the government is afraid to lose its authority (decreasing y), it triesto suppress the influence of the people.

Evidently this kind of interaction can be identified with case a of the metamodel. Accordingly, weexpect that such a political system will eventually evolve into one of two possible states of stability:

either the state of “cooperative democracy” (where x is large and y is large), where the peoplerespect and cooperate with the government, and the government supports and acts for the people,

or the state of “frustrated democracy” (where x is small and y is small), where the people obstructgovernment policy and the government in its turn represses the participation of the people.

It depends on the momentary state (x, y) of the system as to which of the two stable states will bereached. The trajectories ending in “cooperative democracy” are separated from those ending in“frustrated democracy” by a critical separatrix in the system space.

The question now arises whether a momentary state of the system could be prevented fromapproaching the end state of “frustrated democracy” by changing some parameters of the system. Thisis indeed possible by changing the switching points x~and y~of the variables x and y: If the people beginto support the government, even while having small influence (lowering of x,), and if the governmentdoes not become repressive, even after having to renounce some power (lowering of y,), the quadrantsand separatrices in the system space are shifted. A system point originally doomed to end in “frustrateddemocracy” may now, under new system parameters, evolve towards the other stationary state of

“cooperative democracy”.

It is tempting to interpret the present endeavours in the Soviet Union as an attempt to shift theswitching point in the interaction between government and people, so that the government switchesfrom repression to the sharing of power, and the attitude of the people switches from obstruction tocooperation. By this shift of the separatrices the system point may now (hopefully) lie in the basin ofattraction of the fixed point “cooperative democracy” instead of being in the basin of attraction of theendpoint of the “frustrated democracy”.


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24 W . Weidlich. Physics and social science — the approach of .cynergetics

Secondly, we apply this metamodel to the cyclic nature of the long-term economic evolution. Manyeconomists consider the existence of economic long-term cycles as being very probable or even provedand as lasting for a period of about 50 to 60 years. They agree to discerning phases of recovery,prosperity, recession, and depression during one period. Several formal models of these cycles exist,differing in their assumptions about driving forces and relevant variables, but having partiallyoverlapping interpretations. If such models are reduced to the core of their assumptions, they consist of certain interactive schemes between macrovariables.

We can show that a plausible, rudimentary model of the economic long-term cycle can be given interms of two interacting variables, one of the cooperative and the other of the antagonistic type. Themodel sees the explanation of long-term cycles as the competition between young, innovative andmature or aging industries. In this sense the following variables are introduced:

x =volume of innovative young industries, y = volume of mature and aging industries.

It turns out that x acts as a cooperative variable, and y as an antagonistic one. The four quadrants I, II,III, and IV in the space of the variables, then correspond to the phases of prosperity, recession,depression, and recovery, respectively. Furthermore, by this choice of variables and their interaction,the model corresponds to case - y of the metamodel, exhibiting cyclical dynamics whose phases can nowbe interpreted.

Phase I (prosperity). The prospering, mature industry (volume y) is also supported by prospering,innovative industries (volume x). But y begins to suppress further innovations and developments,because their main profit comes from well-accepted mass products.

Phase II (recession). While the suppression of innovative developments (volume x) is still continuing,the lack of innovative drive begins to cause the deterioration of the aging industries (volume y) andleads to recession.

Phase III (depression). The aging industries (volume y) become obsolete and slide into crisis and thusdepression. However, their loss of repressive power leads to a revival of the strength of the innovativeindustries (volume x).

Phase IV (recovery). Since the crisis of the obsolete industries (y) has facilitated the rise of strong

innovative industries (x), the latter now also promote the recovery of the mature industries.We conclude this chapter by applying, somewhat jestingly, the metamodel to a quasi-cyclic

phenomenon from every-day life, well known to the keen observer: the restaurant cycle. Thus,gourmets living in a city will have detected the opening of a new restaurant; its reputation increases dueto a high quality of food, until some time later the reputation declines, so that the restaurant mustclose, perhaps to be reopened afterwards by a new owner.

We shall now explain this strange phenomenon in terms of two interacting variables. Let x be thequality of food (perunit of price) and let ybe the number of guests in the restaurant. It can be seen thatx interacts as a cooperative, but y as an antagonistic variable. If that can be confirmed, then one expectsa cyclic fate of the restaurant, according to case - y of the metamodel, whose phases in quadrants IV , I, IIand III can be interpreted as follows:

Phase IV (opening). The small number of guests (y) in the newly opened restaurant dictates theimprovement of the quality of food (x). This leads to an increase in the number of guests.

Phase I (prosperity). Due to the positive reputation of the food (x) the number of guests (y)

increases. However, this begins to have a negative effect on the quality of food, since the ownerbecomes negligent as he wants to derive more profit.

Phase II (decline). Since the number of guests y i s still large, the owner can afford to permit the


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W . Weidlich, Physics and social science — t he approach of synergetics 25

quality of food to deteriorate further; however, the majority of guests become aware of thisdeterioration and their number begins to decrease.

Phase Ill (crisis). The restaurant now has a bad reputation with a continued decrease in the number

of guests (y). Even an improvement in the quality of food (x) may now come too late. Perhaps therestaurant must be closed and be reopened by a new owner, beginning again with phase IV .

Professors having the duty to organize after-sessions for colloquia are advised to select restaurants at

the end of phase IV only, when the food is excellent and the dining room still relatively empty, butnever at the end of phase II, when the food is miserable, and the dining room still relatively crowded!

3. The framework of microbehaviour and macrostructures

In the preceding se