Why do we need models?

For most systems of interest a linear systems approach does not adequately reflect the complexity of reality. For the general case, the state equations are non-linear in terms of the state variables. Already for systems of a rather limited complexity, this leads to much more complex behavior, e.g. oscillations that are far less regular than the sinusoidal ones appearing in the linear systems. However, for the non-linear region an additional complication kicks in.

The systems may become infinitely sensible to the exact initial values of the state variables.

Small differences in the initial conditions can lead to very large differences after a time that is characteristic for the system. These characteristic times may be short compared to the time horizon the observer has in mind for his predictions. The future of the system is no longer contained in the present value of the state variables that are a reflection of the past of the system. The future cannot be predicted from the present and the past, no matter how good a job of macroscopic modeling we do. This problem exists even if we have a perfect microscopic model involving the whole complexity of the system at the microscopic level. It becomes even more prominent if we avail of only a macroscopic reduced information picture of the system. We analyze these complications later in this work. This involves the chaos theories of complex systems, also termed the science of complexity (Gleick (1988)). These theories introduce the so-called “butterfly-effect” where the flap of the wing of butterfly in the US today may lead to a hurricane in China next week. The problem is also present in the famous three-body problem, a relatively simple system. It shows that we cannot accurately predict the evolution of a system in which three bodies move subject to the laws of Newton for an indefinite period of time. We also cannot prove its long-term stability. The complexity of reality escapes our modeling and theorizing possibilities.

2.5. Why do we need models?

The foregoing highlights some of the methodology and the problems of modeling. Here we revisit the question of the why of the use of models. The main reason is that we want to understand reality. However, this shifts the question to the query why we want to understand reality. The reasons are manifold but an important consideration is that relying on

Fig. 2.4. Dynamics of a linear system involving two state variables.

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experimentation is often not economically feasible or to dangerous and unethical. We want to predict future behavior and to derive and implement measures that shape the future to our needs. A few examples are in order. Let us assume that we want to build an oil refinery at a cost of billions of dollars. Obviously, we cannot simply build one, see whether it works well and if it does not work properly, scrap it, build a second one, and repeat this process until we get something that shows reasonable performance. The amount of information we need is too large to obtain it by successive modification and selection of the proper one. Still this is the main approach on which progress by evolution relies before the era of the brain. It resulted in the extinction of most of the species that spawned from evolution. The same holds for experiments with e.g. the socioeconomic system. Random experiments may have unacceptable consequences. Hence, we need theories, laws and models to predict the future and to develop a future fitting our needs. This has a profound evolutionary significance. The

“invention” of the brain in evolution makes building mental models of reality possible. This gradually results in an increasingly sophisticated ability to construct (mathematical) models that fit our need for prediction of the future. This aspect of so-called exogenous evolution is a new driver behind evolution. It complements the possibilities of molecular evolution at the level of DNA and RNA and is the main driver behind the further evolution of the competiveness of the species Homo. It enables us to exploit the economic potential the solar radiation provides, more extensively. The energy contained in solar radiation is the ultimate driver of evolution on earth. This brings us back to one of the main themes of this book.

It seems worthwhile to reemphasize a significant problem concerning the modeling of systems in which rational, or at least partly rational, actors exist. An important assumption underlying the philosophy of modeling is that the model and the system are independent. The existence of the model should not influence the outcome of the processes taking place in the system. This is a highly questionable assumption in systems containing actors aware of the predictions of the model. This influences their behavior, certainly if the outcome of the modeling exercise is an important factor guiding the actors. In that case, the assumption of independence of model and system is no longer valid. This results in an important philosophical problem that affects all modeling efforts in systems with actors that behave partially rational and are aware of the existence of the model. A potential solution is to take this part of human behavior into account in the modeling exercise. Apart from this solution being a difficult one, we get into the problem of infinite recursion as the model and reality start a process of co-evolution. The direction of this evolution is unpredictable from a modeling perspective. To date the author has no clues on approaches that avoid these problems and he probably never will arrive at such clues.

2.6. Conclusion.

In this chapter, we analyze the methodology of modeling of complex systems. We state that in almost all cases of practical interest a full description of real systems at the microscopic level, i.e. taking into account the properties of the many individual entities in the system, is unpractical and in most cases impossible. In contemporary approaches to the mathematical modeling of complex system, this leads to an averaging strategy resulting in reduced complexity macroscopic models involving macroscopic variables such as temperature. We show that two types of macroscopic variables exist: Intensive quantities not depending on the system’s size and extensive quantities proportional to the system’s size. We explore the macroscopic balancing or accounting approach for the amounts of extensive quantities that leads to state equations that in principle describe the evolution of a system in time.

We also show that differences in intensive macroscopic quantities define thermodynamic forces that determine the direction in which natural processes proceed.

23 The averaging process involved in macroscopic modeling leads to penalties of two natures.

Firstly, part of the value intrinsically present in the system is no longer available to do useful work due to the limitations of our macroscopic information. Secondly, for systems not strictly linear, the evolution of the system may become infinitely sensible to the initial conditions. In those cases, a macroscopic model cannot predict the future evolution of the system.

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