Wolfgang Weidlich

Physics and social science — The approach of synergetics

Abstract Universally applicable methods originating in statistical physics and synergetics are combined with concepts from social science in order to set up and 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 macroconfigurations 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 as sociology, demography, regional science and economics. These examples refer to collective political opinion formation, to interregional migration of interactive populations, to settlement formation on the micro-, meso- and macroscale, and to nonlinear nonequilibrium economics, including market instabilities.

Interregional Migration Dynamic Theory and Comparative Analysis

Migration processes in all their theoretical and empirical aspects are a constitutive part of regional science and demography: No demographic theory and no reliable prognosis of the regional evolution can be made without taking into consideration the migration of populations.

The master equation approach to nonlinear economics

A concept for modelling nonlinear economic dynamics is presented and exemplified by a concrete model. Generally, a configuration of macro-economic variables is considered whose probabilistic evolution is coupled to the decision making of agents and is described by a master equation. The transition rates in the master equation are modelled in terms of utility measures of the agents. Nonlinear dynamic meanvalue equations can be derived from the master equation.The concrete model describes firms producing substitutable durable commodities. They compete with respect to the quality of their products and a positive feedback between quality enhancement and customers reaction to quality is assumed. The case of two competing firms is treated explicitely. It is shown that beyond a critical value of a “competitivity parameter” a homogenous maret will develop into an inhomogenous one with a winner and a loser firm.

A phenomenological model for dynamic traffic flow in networks

A macroscopic model for dynamic traffic flow is presented. The main goal of the model is the real time simulation of large freeway networks with multiple sources and sinks. First, we introduce the model in its discrete formulation and consider some of its properties. It turns out, that our non-hydrodynamical ansatz for the flows results in a very advantageous behavior of the model. Next the fitting conditions at junctions of a traffic network are discussed. In the following sections we carry out a continuous approximation of our discrete model in order to derive stationary solutions and to consider the stability of the homogeneous one. It turns out, that for certain conditions unstable traffic flow occurs. In a subsequent section, we compare the stability of the discrete model and the corresponding continuous approximation. This confirms in retrospection the close similarities of both model versions. Finally we compare the results of our model with the results of another macroscopic model, that was recently suggested by Kerner and Konhauser [Phys. Rev. E 48, 2335-2338 (1993)].

How to control a chaotic economy

An economic system which exhibits chaotic behaviour has been stabilized on various periodic orbits by use of the Ott-Grebogi-Yorke method. This procedure has been recently applied to controlling chaotic phenomena in physical, chemical and biological systems. We adopt this method successfully for Feichtingers generic model of two competing firms with asymmetrical investment strategies. We show that the application of this control method to the particular economic process considered brings a substantial advantage: one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the systems economic properties. Numerical simulations are presented in order to illustrate the effectiveness of the whole procedure.

Synergetic modelling concepts for sociodynamics with application to collective political opinion formation

The article has three purposes. Firstly it introduces into synergetics, an interdisciplinary field of research which is concerned with “cooperative” interactions of individual parts of a system producing macroscopic spatial, temporal or functional structures. Secondly, a general modelling strategy for building mathematical models for dynamic processes at the macrolevel of the society is presented and its relation to the general concepts of synergetics is explained. Thirdly, the modelling procedure is concretely applied to a simple dynamical model of collective political opinion formation: After establishing the relation between the microlevel of individual decision making and the macrolevel of collective opinion variables, the master equation for the probability distribution over the “sodoconfiguration” is set up and dynamic equations for the quasi‐mean values of the relevant macrovariables are derived. The model comprises the cases of liberal and totalitarian political systems and enables the mathematica...

Settlement formation. Part I: a dynamic theory.

The dynamic process of settlement formation is a fundamental issue in regional science. Our proposed model integrates the economic and migratory sectors in terms of endogenous variables in order to describe the evolution of continuous population distributions as a self-organising process. The model has been designed as follows: an ensemble of populations is described by the respective population densities over the plane, with the latter being tessellated into equivalent unit cells. Populations produce commodities and the net incomes of the individuals depend on local production costs, including fixed costs and the transportation costs from production to consumption place. Therefore, these local incomes depend on the population densities. The economy is assumed to be in a momentary quasi-equilibrium with the population distribution, and the evolution of the population distribution is described by non-linear migratory equations of motion. The driving forces within these equations are local differences in individual incomes. These motivate the individuals to migrate to locations of optimal income. This process leads to the evolution of spatially heterogeneous population distributions forming the settlements.

Quasiadiabatic solutions of Fokker Planck equations with time-dependent drift and fluctuations coefficients

A systematic algorithm for finding solutions of Fokker Planck systems with time-dependent drift and fluctuation coefficients is developed. The solutions lag behind the adiabatic quasistationary distribution. This delay effect vanishes for slowly varying control parameters. In a first example the evolution of the phase transition from a monostable to a bistable system is treated. There is a fluctuation enhancement and the delay of the probability distribution leads to a transient central peak. In a second example a linear model with time-dependently shifted drift is considered. The introduced approximation scheme reproduces the exact solution already in first order.

From fast to slow processes in the evolution of urban and regional settlement structures

Complex systems consist of many intertwined organizational levels starting from micro-structures and ending with macrostructures. Their evolution takes place on different time scales: Micropatterns exhibit a fast dynamics whereas macropatterns develop slowly. Urban and regional science can make use of this fact by constructing a hierarchy of models on different spatio-temporal scales.

Stochastic migration theory and migratory phase transitions

Migration processes are an example of socio-economic dynamics of particular interest for quantitative research. The underlying motivations for the migration of individuals are relatively well defined, and must always result in a clear individual decision to maintain or to change the location in a given interval of time. In contrast, the population numbers and migratory fluxes can be measured and compared with theory. As such it is the objective of a quantitative migration theory to understand the migratory dynamics by connecting quantitatively the micro-level of motivations and decisions with the macro-level of a global migration process.