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Mathematical Modelling Techniques
By Rutherford Aris
Dover Publications, Inc.Copyright © 1994 Rutherford Aris
All rights reserved.
What is a model?
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altr. to Aristophanes of Byzantium Syrianus in Hermog. (Rabe ij 23) (0 Menader, o life, which of you has imitated the other?)
1.1 The idea of a mathematical model and its relationship to other uses of of the word.
In these notes the term 'mathematical model'--usually abbreviated to 'model'-will be used for any complete and consistent set of mathematical equations which is thought to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model, though in detailed examples we shall be concerned with a few physico-chemical systems.
Being derived from 'modus' (a measure) the word 'model' implies a change of scale in its representation and only later in its history did it acquire the meaning of a type of design, as in Cromwell's New Model Army (1645). Still later (1788) came the complacent overtones of the exemplar that Gilbert was to use so effectively for his modern major general, while it is the first years of this century before fashion became so self-conscious as to claim its own models and make possible Kaplan's double entendre (see quotation at head of Ch. 5). In the sense that we are seeking a different scale of thought or mode of understanding we are using the word in its older meaning. However, the word model (without the adjective 'mathematical') has been and is used in a number of senses both by philosophers and scientists as merely glancing through the titles of the bibliography will suggest. Thus Apostel distinguishes nine motivations underlying the use of models ranging from the replacement of a theory-less domain of facts by another for which a theory is known (e.g., network theory as a model for neurological phenomena) to the use of a model as a bridge between theory and observation. Suppes in the same volume maintains that the logicians concept of a model is the same in the empirical sciences as in mathematics though the use to which they are put is different. The logician's definition he takes from Tarski as: "a possible realization in which all valid sentences of a theory T are satisfied is called a model of T". This is a non-linguistic entity in which a theory is satisfied and Suppes draws attention to the confusion that can arise when model is used for the set of assumptions underlying a theory, i.e. the linguistic structure which is axiomatized. In our context this suggests that we might usefully distinguish between the prototype (i.e. the physical entity or system being modelled), the precursive assumptions or what the logicians call the theory of the model (i.e. the precise statement of the assumptions of axioms) and the model itself (i.e. the scheme of equations).
The idea of a change of scale which inheres in the notion of a model through its etymology can be variously interpreted. In so far as the prototype is a physical or natural object, the mathematical model represents a change on the scale of abstraction. Certain particularities will have been removed and simplifications made in obtaining the model. For this reason some hard-headed, practical-minded folk seem to regard the model as less "real" than the prototype. However from the logical point of view the prototype is in fact a realization in which the valid sentences of the mathematical model are to some degree satisfied. One could say that the prototype is a model of equations and the two enjoy the happy reciprocality of Menander and life.
The purpose for which a model is constructed should not be taken for granted but, at any rate initially, needs to be made explicit. Apostel (loc. cit.) recognizes this in his formalization of the modelling relationship R(S,P,M,T), which he describes as the subject S taking, in view of a purpose P, the entity M as a model for the prototype T. J. Maynard Smith uses the notion of purpose to distinguish mathematical descriptions of ecological systems made for practical purposes from those whose purpose is theoretical. The former he calls 'simulations' and points out that their value increases with the amount of particular detail that they incorporate. Thus in trying to predict the population of a pest the peculiarities of its propagation and predilections of its predators would be incorporated in the model with all the specific detail that could be mustered. But ecological theory also seeks to make general statements about the population growth that will discern the broad influence of the several factors that come into play. The mathematical descriptions that serve such theoretical purposes should include as little detail as possible but preserve the broad outline of the problem. These descriptions are called 'models' by Smith, who also comments on a remark of Levins that the valuable results from such models are the indications, not of what is common to all species or systems, but of the differences between species of systems.
Hesse in her excellent little monograph "Models and Analogies in Science" distinguished two basic meanings of the word 'model' as it is used in physics and Leatherdale in a very comprehensive discussion of "The Role of Analogy, Model and Metaphor in Science" has at least four. They stem from the methods of "physical analogy" introduced by Kelvin and Maxwell who used the partial resemblance between the laws of two sciences to make one serve as illustrator of the other. In the hands of 19th century English physicists these often took the form of the mechanical analogues that evoked Duhem's famous passage of Gallic ire and irony. Duhem had in mind that a physical theory should be a purely deductive structure from a small number of rather general hypotheses, but Campbell claimed that this logical consistency was not enough and that links to or analogies with already established laws must be maintained. Leatherdale's four types are the formal and informal variants of Hesse's two. Her "model1" is a copy, albeit imperfect, with certain features that are positively analogous and certain which are neutral but shorn of all features which are known to be negatively analogous, i.e. definitely dissimilar to the prototype. Her "model2" is the copy with all its features, good, bad and indifferent. Thus billiard balls in motion, colored and shiny, are a model2for kinetic theory, whilst billiard balls in motion obeying perfectly the laws of mechanics but bereft of their colour, shine and all other non-molecular properties constitute a model1. It is the natural analogies (i.e. the features as yet of unknown relevance) that are regarded by Campbell as the growing points of a theory. In these terms a mathematical model would presumably be a formal model1.
Brodbeck, in the context of the social sciences, stresses the aspect of isomorphism and reciprocality when she defines a model by saying that if the laws of one theory have the same form as the laws of another theory, then one may be said to be a model for the other. There remains, of course, the problem of determining whether the two sets of laws are isomorphic. Brodbeck further distinguishes between two empirical theories as models one of the other and the situation when one theory is an "arithmetical structure". She then goes on to describe three meanings of the term mathematical model according as the modelling theory is (a) any quantified empirical theory, (b) an arithmetic structure or (c) a mere formalization in which descriptive terms are given symbols in the attempt to lay bare the axioms or otherwise to examine the structure of the theory. If arithmetical is interpreted with suitable breadth we are clearly concerned in these notes with sense (b).
It is obviously inappropriate in the present context to try to survey all the senses in which the word has been used, among which there is no lack of confusion. A more formal version of the definition of a (mathematical) model that we started with might be as follows: a system of equations, [summation], is said to be a model of the prototypical system, S, if it is formulated to express the laws of S and its solution is intended to represent some aspect of the behavior of S. This is vague enough in all conscience, but the isomorphism is never exact and we deny the name of modelling to the less successful efforts of the game. Rather, we should try and find out what constitutes a good or bad model.
It scarcely needs to be added that we shall not raise the old red herring about the model being less "real" than the prototype. Tolkien has reminded us of the failure of the expression "real life" to live up to academic standards. "The notion", he remarks, "that motor cars are more 'alive' than, say, centaurs or dragons is curious; that they are more 'real' than, say, horses is pathetically absurd".
The mention of reality leads me to add that by far the most enlightening discussion of models I have found is in Harré's excellent introduction to the philosophy of science, "The principles of scientific thinking". He writes from a realist point of view which eschews simplifications and attempts to present a theory of science based on the actual complexity of scientific theory and practice; he regards the alternative traditions of conventialism and positivism as vitiated by the attempt to force the description of scientific intuition and rationality into the deductivist mould. Model building becomes an essential step in the construction of a theory. I shall not attempt to summarize the argument of his second chapter, which demands careful and considered reading, but it may be useful to mention one or two of the distinctions he makes. He starts with the notion of a sentential model in which one set of sentences T is a model of (or with respect to) another set of sentences S if for each statement t of T there is a corresponding statement s of S such that s is true whenever t is acceptable and t is unacceptable whenever s is false. If T and S are descriptions of two systems M and N and T is a sentential model of S, then M is an iconic model of N. He recognizes that in mathematics the word is used in both ways: model theory is clearly a sentential model within mathematical logic, but we often conceive sets of objects, real or imaginary, which are described mathematically. The latter is an iconic model and the equations a sentential model of the sentences describing the set of objects. Harré goes on to distinguish between the subject and source of a model. The former is whatever the model is a model of, the latter what it is based on; for example elementary kinetic theory gives models of a gas (subject) based on the mechanics of particles (source) Homeomorphs are models in which the source and subject are the same as in a mechanical scale model. When source and subject are not the same, as with the English tubes and beads that amazed Duhem so much (cf. Sec. 2.1), Harré speaks of paramorphs. He goes on to discuss the taxonomy of models and to show how they are incorporated into the construction of theories first by the creation of a paramorph and then by supposing that it provides a hypothetical mechanism. This process evokes existential hypotheses and raises such questions as the degree of abstraction that can be tolerated leading into a full-scale discussion of the formation of scientific theories. Clearly mathematical modelling in the sense in which we are here discussing it is a small part of this much larger design.
1.2 Relations between models with respect to origins.
It seems well to use the term model for any set of equations that under certain conditions and for a certain purpose provide an adequate description of a physical system. But, if we do this, we must distinguish the kinds of relationships that can obtain between different models of the same process. (This approach seems more useful than to talk of models and sub-models, since the relations are more varied and mixed than can be compasses by this nomenclature). It is of the first moment to recognize that models do not exist in isolation and that, though they may at times be considered in their own terms, models are never fully understood except in relation to other members of the family to which they belong.
One type of relationship can be seen in the packed bed example, the full details of which are given in Appendix A. The physical system is that of a cylindrical tube packed with spherical particles and our purpose is to model the longitudinal dispersion phenomenon. By this we mean that if a sharp pulse of some tracer is put into the stream flowing through a packed bed it emerges as a broad peak at the far end of the bed, showing that some molecules of the tracer move faster through the bed than others and that the sharp peak of tracer is dispersed.
This is the physical system P and it is amenable to modelling in various ways. The most obvious one is to write down the equation of continuity, the Navier-Stokes equation and the diffusion equation with their several boundary conditions (eqns. Al–7). This model, which we will call Π1, is admirably complete and founded on the fewest and most impeccable assumptions, but, for two reasons, it is not a very useful model. In the first place, if the actual geometry of a given packed bed could be used the results would be peculiar to that bed, making it a good simulation but a bad model in the senses of Smith. Secondly, even if the geometry were standardized (and this presents its own difficulties) to, say, a cubic array of spheres the resulting equations would present ferocious difficulties to computation and the model would probably remain barren of results. If the models with standardized and peculiar geometries are denoted by Π1 and ?1 respectively they are clearly distinguishable but very closely related--in fact almost "non-identical twins".
The second way in which we might try to model P is to say that the same sort of dispersion is experienced in a much simpler system, namely that of plug or uniform flow through a tube with a longitudinal diffusion coefficient. If we call this modified prototype P2 we can easily derive a partial differential equation Π2, which is much simpler than those of Π1(see eqns. A8-12). There is no immediate connection between Π1 and Π2though we can imagine that some sort of averaging of the Navier-Stokes equations over the cross-section of the bed would lead to the plug flow approximation.
On the other hand, we might make something of the fact that in a packed bed the space between particles makes a natural cavity whilst the interstices narrow where the particles touch and the fluid can be thought of as jetting through into the cavity space. In this rather crude sense the packed bed as a sequence of little stirred tanks gives us a modified prototype, say P3, which can be modelled. To avoid Suppes' criticism, we do not say that P3 is a model of P, though recognizing that it is popularly called the "cell model" of the packed bed. The model of P3 (and therefore of P) consists N ordinary differential equations for the time-varying concentrations in the N stirred tanks of P3. This will be denoted by Π3 and the equations are numbered A13-16. There is no obvious connection between Π1 and Π3 or between Π2 and Π3.
A fourth way of modelling the system P would be to regard the system as a stochastic one in which a tracer molecule had at each step in time the options of either moving forward with the stream or of being caught in an eddy and remaining essentially in the same place. This modification of the prototype, say P4, leads to Π4 and the equations A25-26. Again there is no immediate or obvious connection between Π4 and the preceding models. The relationship of these models is expressed in the diagram.
The models Π1, ... Π4 are best described as cognate models since they appear to be siblings of the same parent system.
A rather different relationship obtains between the models [summation]1 ..., [summation]6 of the stirred tank S described in Appendix C. In these [summation]1 is the full set of ordinary and partial differential equations obtained by making mass balances for each of the S species and energy balances on the contents, the wall, and the cooling system of the reactor (see pp. 152164 for details and particularly pp. 153 and 154 for the hypotheses). This gives S+2 ordinary differential equations and a parabolic partial differential equation. This system, [summation]1, (eqns. C1-6) is again of considerable complexity, but less difficult of calculation than P1. [summation]2 is the steady state version of these equations obtained simply by deleting all time derivatives and with them the initial conditions. It thus consists of non-differential equations coupled to an elliptic differential equation. If the assumption is made that the wall of the reactor is thin, (hypothesis H6) the elliptic equation can be solved quite easily and [summation]3 then consists entirely of algebraic equations, C9 and 10. (We will call equations that are not differential equations 'algebraic' even though they may contain transcendental functions).
To reach the model [summation]4 we return to the full transient model [summation]1 and assume that the conductivity of the wall is very high (hypothesis H7). Then the parabolic partial differential equation can be replaced by an ordinary differential equation for the mean wall temperature and [summation]4 consists wholly of ordinary differential equations one for each reacting species and one for each of the reactor, wall and coolant temperatures. (eqns. C1, 2, 6, 11) If H6 and H8, the hypotheses that assert that the wall is thin and of negligible heat capacity, are imposed instead we have one fewer equation and the model [summation]5 (eqns. C1, 13, 14).
Excerpted from Mathematical Modelling Techniques by Rutherford Aris. Copyright © 1994 Rutherford Aris. Excerpted by permission of Dover Publications, Inc..
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