The microscopic and the macroscopic approach to modeling

A more common equivalent expression of this law is F = ma.

3. Third Law: Whenever a body exerts a force on a second body, that second body exerts a force on the first body equal in magnitude and opposite in direction. This is the

“action equals minus reaction” law.

Newton first proposes the laws of motion in Philosophiae Naturalis Principia Mathematica, published in 1687. Newton applies these laws to explain and investigate the motion of physical objects and systems. In the third volume of the text, Newton shows that these laws of motion, if combined with his law of universal gravitation, explain Kepler’s laws for the motion of planets in the solar system. The law of gravitation of Newton states that two bodies are subject to an attractive gravitational force proportional to the product of their masses and inversely proportional to the square of their distance.

We concentrate on mental models and mathematical models in particular. In addition, we always try to formulate the consequences of mathematical models in a verbal way to improve the reader’s understanding of the concepts.

Models, theories and laws derive from assumptions beyond the mere observations. Both the quality of the empirical material and of the inductive and deductive reasoning determine the validity and usefulness of the model. Therefore testing models by discriminating new experiments is necessary. This is an established part of the methodology of modern science. It relies on abstraction and logical thinking and comparing the results of the deductions and inductions with discriminating experiments on the real system or a close enough image of it.

This experimental verification is often not possible in socioeconomic systems or biological systems due to limitations of a financial, practical or ethical nature. This is an important hurdle in the development of predictive models of such systems. We return to this complication later.

Because of the fact that theories and laws rest on assumptions and a necessary reduction of the complexity of the real world, science does not represent an absolute truth. The assumptions and the reduction of complexity may be falsified when new conflicting experimental facts become available.

2.2. The microscopic and the macroscopic approach to modeling.

We usually study systems in which many interacting entities appear. Consider the number of water molecules in a glass containing 1 liter of pure water, the order of magnitude being 10²⁵, or the many actors involved in socioeconomic interactions. A full microscopic model has to

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take the behavior of all these entities into account. This requires specifying the state of the system in terms of the so-called state variables of all these entities. By definition, a full set of state variables, such as spatial coordinates, velocities and chemical or biological nature, fully specifies the properties of the entities deemed relevant in the modeling exercise. As three numbers specify the location of the molecules and three number their speed, we need of the order of 10²⁶ numbers to specify the system’s state if we consider the example of 1 liter of water. At least, if we consider these aspects to describe the state of the entities exhaustively (this is a significant simplifying assumption). This is clearly an impossible task. If we write down one number every second, it takes about 25 million times the estimated age of the universe to write down the state variables. We clearly need a more clever approach to avoid facing a dilemma. Fortunately, there is a way out, although, as we indicate earlier and substantiate further later in this book, avoiding complexity comes at a penalty. The penalty involves inability to harvest the full potential value in the system, be it in terms of capacity to perform useful work in a physical system or in terms of harvesting of economic value in socioeconomic systems. In addition, we lose in almost all cases relevant in practice, part of the ability to predict the time evolution of the system if it is not in equilibrium in the thermodynamic or economic sense. We lose the ability to predict future behavior in detail.

This becomes clear when we develop the systems theory of evolution in Chapter 6 and in discussing specific examples of evolving systems, e.g. the evolution of the cosmos and biological and socioeconomic evolution in Chapters 9-14.

We return to our glass of water. If we want to ascertain whether it is safe to drink the water from the perspective of danger of burning our lips, we do not need to consider the vast number of state variables that specify the detailed state of the system. We only need one state variable, albeit a state variable of a very different nature as we see in a while. In addition, this state variable is readily accessible. We only need to measure the temperature, e.g. using a thermometer. Temperature is a macroscopic state variable. It results from averaging the microscopic state variables of the objects in the system, i.e. the state variables of the water molecules in the glass. Temperature describes the movement of the many molecules by averaging their kinetic energy, the energy contained in the movement of the molecules. The temperature provides an adequate answer to the question if it is safe to drink the water or if it is too hot. This reduction of the number of state variables is the basis of the macroscopic approach in physics and chemistry.

Temperature is as said an example of a macroscopic state variable. We further use it to present a preview of a few other concepts of the methodology of macroscopic modeling. As it happens, macroscopic variables appear in two flavors. There are extensive macroscopic quantities that depend on the size of the system and intensive macroscopic quantities that do not depend on that size. Temperature does not change if we consider two equal glasses of water of the same temperature. This makes temperature an intensive quantity. The total volume of water doubles when we consider two glasses. This makes volume an extensive macroscopic quantity.

There is a second feature of the macroscopic approach or for that matter macroscopic thermodynamics that we illustrate using our elementary example. How do we make certain that the second glass of water that we add to the first indeed has the same temperature? To do this we simply measure the temperatures of the two quantities of water, e.g. using a mercury-based thermometer. We measure the temperature of our original water by putting the thermometer in en we allow the thermometer to exchange heat with the water in the glass until the temperature reading on the thermometer stops changing. We consider the final reading as the temperature of the water in the glass. In doing this we introduce a concept and an assumption. The assumption is that the heat exchange between the thermometer and the water in the glass does not significantly alter the temperature of the water in the glass, i.e. the

15 amount of mercury that heats or cools down must be very small compared to the amount of water in the glass. This is a rather obvious assumption and it generally is valid to a good

approximation. The concept is that of thermodynamic equilibrium. If the thermometer exchanges heat with the system, it reaches thermal equilibrium with the water in the glass. If we make the thermometer part of the system and isolate the system from the rest of the universe, we see that finally everywhere in the extended system the temperature is equal. We can extend this concept. An isolated system is in thermodynamic equilibrium if the extensive and intensive state variables do not change anymore. We now proceed by measuring the temperature of the second quantity of water with the thermometer and we see that the reading on the thermometer is the same. We think we can safely assume that the temperature of the two glasses of water is the same. Probably without noticing, we introduced the zero-th law of thermodynamics. It allows the specification of an unambiguous concept of temperature by noting that if system A has the same temperature as system B, B being the thermometer, and system C has the same temperature as system B, then the temperatures of A and C are equal.

This discussion formally introduced the concept of temperature. It turns out to be very important in thermodynamics. We use the symbol T for its value.

We proceed by using temperature to derive another important concept: The concept of a thermodynamic force. We consider the system depicted in Fig. 2.1. Two pieces of metal of different temperatures are in contact and isolated from the environment. The two pieces of metal exchange heat until their temperatures become equal. We know from experience that heat flows from the metal with the higher temperature to that with the lower temperature. This introduces the concept of a thermodynamic force. The difference in temperature, i.e. the difference in an intensive macroscopic quantity, provides a force that drives the flow of heat between the two pieces of metal. Later on, we see that to arrive at a consistent theoretical framework, we have to express the force in terms of the reciprocal of temperature but this does not matter at this stage. The discussion above leads to an important observation. Heat flows in a well-defined direction. The thermodynamic force sets an arrow of time. We always observe that heat flows in the direction of lower temperatures, never to higher temperatures.

Coffee cools down if left alone and never spontaneously extracts energy from the universe to increase in temperature, irrespective the fact that energy is abundantly available in the universe. In the next chapter, we see that we stumbled on one of the alternative formulations of the second law of thermodynamics: In an isolated system, heat will flow from high to low temperature regions. We never observe the reverse, at least from the macroscopic perspective.

This does, as we see later, not hold at the microscopic level as macroscopic laws only consider the average behavior of the microscopic entities. This becomes clearer when we study the statistical foundation of the macroscopic method (Chapter 4). In fact, to the

Fig.2.1. Contact between two pieces of metal of different temperature.

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thermodynamics purists the term heat flow that we used a few times in the last sentences is a little bit a loose and even dangerous term. Heat definitely is not a material substance (like e.g.

water) that flows. When we say that heat flows, we mean to say that a process takes place by which heat exchanges between two systems. To avoid lengthy phrasing we often use the term heat flow and the reader should note that we therewith indicate the process mentioned in the previous sentences.

We summarize the main features of the macroscopic approach introduced so far. Heat and temperature turn out to be a macroscopic reflection of the speed of the molecules, more precisely their kinetic energy, the energy contained in their movement. (For the time being we will be sloppy and ignore that at this stage, we are not completely sure what energy really is).

At the level of the individual molecules, the speed varies considerably and if we attribute the macroscopic concept of temperature at the level of a single molecule (something we cannot do) their “temperatures” vary largely. We resort to a concept from statistics: We invoke the so-called law of large numbers, i.e. if we consider a sufficiently large collection of molecules, the average speed of the molecules in two sufficiently large samples is almost equal for a system in thermodynamic equilibrium. In fact, if the samples are sufficiently large the difference becomes negligibly small. This is the essence of the macroscopic method: We simply average the behavior of individual entities to arrive at macroscopic properties that do not exhibit the large fluctuations taking place at the level of the individual entities. One macroscopic state variable takes the place of many microscopic state variables. The complexity of the microscopic level becomes manageable if we take a macroscopic perspective.

There is one further remark we need to make about the macroscopic method. The number of molecules we involve in the averaging has to be sufficiently large but the volume of the system involved in the averaging small compared to the total volume of the system. If that is not the case, problems arise if we consider systems that are not in equilibrium and spatial differences in the macroscopic variables are present. If we need to average over a significant part of the system’s volume, to arrive at consistent average quantities, the variables are no longer continuous in the spatial dimensions of the system and we get into mathematical difficulty. Although this may be a problem in some systems, we ignore this complication.