Teleology and extreme principles a tentative Fourth Law of Thermodynamics

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All is for the best in the best of all possible worlds.

Voltaire "Candide"

Teleology is a lady no biologist can live without, but whose company seems shameful in society. G. von Briikke

12.1. Introduction

It does not seem that we are living in an ordered world, and nature is not an exception to this, but we wish to live in such a world, which is close to the Hellenistic one. Its Aristotelian philosophy of life was teleological and deterministic in principle (a different form of philosophy was scarcely thinkable in the well-ordered Graeco-Roman society in its prime). On the other hand, our mentality had also been formed under the influence of Eastern philosophy, which considered Chance to be the motive (driving) force of development and evolution: "Again I saw that under the sun the race is not to the swift, nor the battle to the men of skill; but time and chance happen to them all" (The Bible. Ecclesiastes, 9). The fundamental duality of our "scientific" approach to the problems of the development and evolution of natural systems, which manifests itself in the form of a dual classification of dynamical systems (into deterministic and stochastic ones), is a result of the influence of these two basic paradigms. For instance, the modern synthetic theory of evolution is a typical compromise between these opposite paradigms. On the one hand, there is the Aristotelian paradigm, which was developed by Lucretius: "Survival of the fittest". He illustrated this principle by the following verse:

Multaque tum interisse animantum saecla necesses Nec potuisse propagando procudere prolem. Nam quaecumque vides vesci vitalibus auris, Aut dolus, aut vitrus, aut denique mobilitas est Ex ineunte aevo genus id tutata reservans

Lucretius De Rerum Natura. Liber 5, 855.

Towards a Thermodynamic Theory for Ecological Systems, pp. 301-323 © 2004 Elsevier Ltd. All Rights Reserved.

On the other hand, when the mutation process is considered as a driving force of evolution, this is an acceptance of the Ecclesiastesian "Chance" paradigm.

Nevertheless, determinism is more familiar for us. In the 1960s, in the USSR, a book under the title "The Foreseen Future" was published. Certainly, in 2003, many of its prognoses look very naive, but one very impressive statement it made was that "Mankind cannot live under conditions of an unpredictable future". We feel more comfortable if we believe that all living systems have a certain aim, and their being is meaningful.

Note that if we want to look for aims, goal functions, etc. in ecology, then our credo must be the Panglossism. Pangloss, Candide's teacher, believed that "All is for the best in the best of all possible worlds" (Voltaire's "Candide"). We believe that a certain aim exists, and a system attempts to maximise (or minimise) some goal function. Unfortunately, it seems to us, the number of such functions is a continuum. It is not a problem to construct an arbitrary number of similar functions; the problem is rather how to select and interpret them.

Although a trend to apply the various extreme (teleological) principles in sciences with sufficiently long histories (mechanics, physics) appeared a long time ago, in ecology it has started only recently. Note that ecology itself is a young science. A trend to formulate the basic regularities of population and ecosystem dynamics in the form of some extreme principles is a consequence of the perennial teleleologicity of our thinking on the one hand and, not least, of the quest for more concise and elegant description on the other. Though it is hardly probable that there is a goal that the population, community or ecosystem tries to attain, nevertheless our hypothesis, which is purely auxiliary in nature, is that such a goal exists and may put all the variety of ecological mechanisms in a proper perspective. We think that the first attempt in this line was probably the hypothesis of Lotka (1922), i.e. that in the process of its own evolution the ecosystem tends to increase the energy through-flow, reaching a maximum at equilibrium (of course with regard to different constraints). Lotka went so far as to suggest calling this statement "the Fourth Law of Thermodynamics", so that S.-E. J0rgensen was not the first to use this term when he has also named his exergy principle the Fourth Law.

It is interesting that thermodynamics with its one-directional time, with monotonous increasing entropy and with the monotonous decreasing function of internal entropy production, is a very favourable field for various teleological concepts and interpretations, where ecologists can meet each other without fear of being embarrassed by their teleological Ladies.

12.2. The maximum power principle

The balance of energy flows for ecosystems can be written as dE _ e _ e ^ Ii dt _ qout;

where E(t) is the energy storage in the form of biomass, different structures, etc., or captured energy, qfn and qout are energy in- and outflow. It is obvious that the derivative dE/dt is a power, the energy flow through the ecosystem is equal to qout, so that for growing systems, for which (dE/dt) > 0, q%ut < qfir Thus, in equilibrium q%ut reaches a maximum. We see that Lotka's principle holds, but its statement is trivial.

Let us rewrite the Gibbs equation in a form that is typical for "irreversible" thermodynamics:

Here Aus is the so-called "useful" work performed within the system, Q is the energy (heat) imported into the system from the environment and Airrev is the irreversible loss of energy (note that the expressions d'Aus, d'Q, and d'Airrev are not full differentials). Even if we neglect this term in Eq. (2.2), then the subdivision of dE/dt into two items is very arbitrary and depends on what we imply by the concept of "usefulness". The import of energy and the export of heat in ecosystems are non-spontaneous processes, which are realised by means of some special "pumps". Such active systems as ecosystems have their own internal pumps; therefore, they have to possess a high degree of internal organisation, i.e. their structure has to be rather complex. We assume, therefore, that some share of the useful work must be used for the creation of this structure and the maintenance of its functioning. Furthermore, a remaining share of the work is used for the growth of biomass and maintenance of metabolic and reproductive process. We shall consider the total work spent for the maintenance of biomass and structure and also the work performed against some generalised "forces of friction" as the total metabolism of the ecosystem. As a result, metabolic heat is produced, and it must be exported from the ecosystem.

As a rule, the characteristic time taken by an ecosystem to evolve is much longer than the mean generation time of species that form the ecosystem; therefore, their life cycles can be considered as fast processes. In other words, the new biomass decomposes and releases heat; the latter must also be exported.

Note that we have not yet used any such thermodynamic concepts as free energy and entropy, although when we talked above about the transformation of work to heat we implicitly used the Second Law.

Summing together everything mentioned above and comparing Eqs. (2.1) and (2.2), we can write e _ dQ e _ _ dAus d/Airrev o-, qin _ dt ' qout _ dt dt (23)

(in accordance with the definition both these works are negative). If we now assume additionally that irreversible processes within the ecosystem cannot, in principle, be controlled, then we can easily see that Lotka's principle, asserting that qout! max, is equivalent to the following statement: In the process of its own evolution the ecosystem tends to increase the useful power, reaching a maximum at equilibrium, i.e.

d/Aus dt

Thus, a useful work is exergy in accordance with its basic definition, so we can talk about maximisation of the exergy increment. Here the value of Q is considered only as the imported energy (heat), whereas in addition to energy the ecosystem also imports matter. In this case the Gibbs equation is written as (De Groot and Mazur, 1962)

k where a flux of the so-called "adduced" heat is defined as d*Q d0 Q tV de^k (26)

Here sk and hk = ¡k + Tsk are the specific molar entropy and enthalpy, (deNk /dt) are the exchange matter flows between the ecosystem and the environment, which can also be represented as a difference of in- and outflows, (deNk/dt) = (deNk/dt)in - (deNk/di)out. The following is obvious.

So, we have almost obtained Odum's formulation of maximum power principle (MPP). H.T. Odum used the principle to explain the structure and processes of ecosystems (Odum and Pinkerton, 1955).

H.T. Odum constructed his argument by analogy with the construction of the function of dissipation (see Sections 2.5 and 3.3):

k where Xk and Jk are generalised thermodynamic forces and fluxes. The kth flux— "ecoflux"—can be defined as Jk = (dNk/dt), where Nk is the biomass (or biomass density) of kth species (component) of an ecosystem (Odum et al., 1960). The organic matter accumulated in the biomass of kth species may be defined as the "ecoforce", Xk, equal to the Gibbs free energy (or "ecopotential") difference "released" by the process, DGk, per unit of biomass, Nk, expressed, for instance, in carbon units. Thus, the driving "ecoforce", Xk = DGk/Nk, is a function of the concentration of biomass and organic matter. Note that DGk > 0 for all forced, non-spontaneous processes, for photosynthesis, for instance. It is such processes that form the living matter. Then, formally, the function of power will be

where Dgk = AGk/Nk is the specific change of Gibbs free energy.

Power, as we have seen, is the increase in biomass density per time unit converted to free energy. Notice that the MPP focuses on a rate, in Eq. (2.8) indicated as (dNk/di), the ecoflow, multiplied with the fraction that is able to do useful work, i.e. DGk/Nk = Dgk. Maximum power thereby becomes equal to the rate of through-flow of useful energy.

Later on Odum (1983a,b) defined the MPP as a maximisation of useful power. Following this, Eq. (2.8) is applied to the ecosystem level by summing up all the contributions to the total power that are useful. This means that non-useful power is not included in the summation. The difference between the useful and non-useful power will be further discussed below, because the emphasis on useful power is perhaps the key to understanding Odum's principle and utilising it to interpret ecosystem properties.

Brown et al. (1993) and Brown (1995) have restated the MPP in more biological terms. According to the restatement, it is the transformation of energy into work (consistent with the term useful power) that determines success and fitness. Many ecologists have incorrectly assumed that natural selection tends to increase efficiency. If this were true endothermic organisms could never have evolved. Endothermic birds and mammals are extremely inefficient compared with reptiles and amphibians. They expend energy at high rates in order to maintain a high, constant body temperature, which, however, gives high levels of activities independent of environmental temperature (Turner, 1970). Brown (1995) defines fitness as reproductive power, the rate at which energy can be transformed into work to produce offspring. This interpretation of the MPP is more consistent with the maximum exergy principle (see Section 12.3) than with Lotka's and Odum's original idea.

In a recent book named "Maximum Power—The Ideas and Applications" by Odum and Hall (1995), a clear interpretation of the MPP has been presented, as has been applied in ecology by H.T. Odum. The principle claims that power or output of useful work is maximised—not the efficiency and the rate, but the trade-off between a high rate and high efficiency yielding "most useful energy = useful work". It is illustrated in Fig. 12.1.

Hall uses an interesting semi-natural experiment by Warren (1970) to illustrate the application of the principle in ecology. Streams were stocked with different levels of predatory cut-throat trout. When predator density was low, there was considerable invertebrate food per predator, and the fish used relatively little maintenance of food searching energy per unit of food obtained. With a higher fish-stocking rate, food became less available per fish, and each fish had to use more energy searching for it. Maximum production occurred at intermediate fish-stocking rates; i.e. at intermediate rates, the fish utilised their food.

Conflict Between Gibbsian Thermodynamics
Fig. 12.1. The maximum power principle claims that the development of an ecosystem is a trade-off (a compromise) between the rate and the efficiency, i.e. the maximum power output per unit of time.

Hall (Odum and Hall, 1995) also mentions another example. Deciduous forests in moist and wet climates tend to have a leaf area index (LAI) of about 6. Such an index is predicted from the maximum power hypothesis applied to the net energy derived from photosynthesis. Higher LAI values produce more photosynthate, but do it less efficiently because of the respiration demand by the additional leaf. Lower leaf area indices are more efficient per leaf, but draw less power than the observed intermediate values of roughly 6.

According to Gilliland (1982) and Andresen (1983), the same concept applies for regular fossil fuel power generation. The upper limit of efficiency for any thermal machine such as a turbine is determined by the Carnot efficiency. A steam turbine could run at 80% efficiency, but it would need to operate at a nearly infinitely slow rate. Obviously, we are not interested in a machine that generates revenues infinitely slowly, no matter how efficiently. Actual operating efficiencies for a modern steam powered generator are, therefore, closer to 40%, roughly half the Carnot efficiency.

The examples show that the MPP is embedded in the irreversibility of the World. The highest process efficiency can be obtained by endoreversible conditions, meaning that all irreversibilities are located in the coupling of the system to its surrounding; there are no internal irreversibilities. Such systems will, however, operate very slowly. Power is zero for any endoreversible system. If we want to increase the process rate, it will imply that we also increase the irreversibility and thereby decrease the efficiency. The maximum power is the compromise between endoreversible processes and very fast completely irreversible processes.

12.3. Hypothesis: a thermodynamic law of ecology

In previous chapters, we examined how much of a comprehensive ecosystem theory could be derived from the three laws of thermodynamics. The laws were cast as restrictions to contain growth and development, whose processes of course have to satisfy the conservation principle (the First Law) for applicable parameters, degrade energy (the Second Law) and evacuate effluent heat to surroundings. What we found can be summarised as follows: energy flow through a system, defining it as open or at least non-isolated, is necessary for continued existence (partly deduced from the Third Law), and a flow of usable energy is sufficient to form an ordered structure, called a dissipative system (Prigogine, 1980). Morowitz (1992) referred to this latter as a Fourth Law of Thermodynamics, but it would seem more appropriate if such a law could be expanded to state which ordered structure among possible ones will be selected. A hypothesis about this selection has been introduced more than two decades ago (J0rgensen and Mejer, 1977; Mejer and J0rgensen, 1979; see also J0rgensen, 1982, 1992a, 2002a).

Second-Law dissipation acts to tear down the structures and eliminate the gradients, but it cannot operate unless the gradients are established in the first place. Structure and organization can be expressed in different units, such as the number of state variables, number of connections in an interactive web and kJ of exergy that corresponds to distance from thermodynamic equilibrium. Biological systems, especially, have many possibilities for moving away from equilibrium, and it is important to know along which pathways among possible ones a system will develop. This leads to the following hypothesis (J0rgensen et al., 2000):

"If a system receives an input of exergy, it will utilise this exergy to perform work. The work performed is first applied to maintain the system far away from thermodynamic equilibrium whereby exergy is lost by transformation into heat at the temperature of the environment. If more exergy is available, the system is moved further away from thermodynamic equilibrium, as reflected in growth of gradients. If more than one pathway to depart from equilibrium is offered, the one yielding the most work under prevailing conditions, and ultimately moving the system the farthest from thermodynamic equilibrium under the prevailing conditions, providing the most ordered structure, tends to be selected. Or if differently expressed: among the many ways for ecosystems to move away from thermodynamic equilibrium, the one maximising of time derivative of exergy, d(Ex)/dt, under the prevailing conditions will be selected."

This is a restatement and expansion of J0rgensen and Mejer (1977, 1979). A paradox appears to exist in conflicting criteria, the joint maximization of two diametrically opposed properties—storage, which is build up, and dissipation, which is teardown. This chapter will try to resolve this paradox in an ecological context, and in the process, expose the complexity of the interplay between thermodynamics and the growth of order in ecosystems and the ecosphere.

However, we have to note that all these definitions contain one incorrectness: we assume implicitly that exergy is a function of state and extensive variable (namely, in this case we can talk about exergy storage, captured exergy, dissipated exergy, etc.), whereas exergy is a function of two values: current state and reference (initial) state. Therefore, the change of exergy, d0Ex, is not a full differential. Nevertheless, we can transform d0Ex to a total differential, i.e. consider exergy as a function of state, if we assume that different ecological systems have the same reference state. For instance, if such a reference state is "inorganic soup" or detritus, then any ecosystem evolution (short- or long-time) must start from this initial state. In other words, we define a single "zero" for all ecosystems, as we do in the classic thermodynamics, when we define "zeros" for energy and entropy.

If the exergy is an extensive state variable, then it can be represented as a bilinear form:

where Nk is the mass or concentration of kth component and exk is its specific exergy. Naturally, the amount of useful work that can be performed in the ecosystem is defined by both the total amount of biomass and the structure. The latter is determined by information coded in genomes of species that are joined in the ecosystem. Therefore, it is logical to consider the specific "genetic" exergy (see Section 5.7) as exk in Eq. (3.1).

In accordance with that formulated above, the exergy extreme principle is written as

In fact, the genome evolves very slowly; therefore, the derivatives d(exk)/dt are smaller by several orders of magnitude than dNk/dt. So we can neglect the first item if we consider microevolutionary dynamics, as well as the second item if macroevolu-tionary dynamics is interesting.

Unfortunately, today a universally adopted definition of exergy does not exist. We think that it is mainly explained by some indeterminacy in the concept of "useful work". The choice of either concept does very often depend on what kind of process takes place in the system. For instance, if the main processes are chemical, and they are taking place by constant temperature and volume, then the exergy is equivalent to both free energy and thermodynamic potential (Gibbs free energy). Of course, here we can talk about a partial case—the situation is more or less standard. However, different authors in different models define the meaning of "useful work" in different ways, and this choice is, as a rule, subjective. As a result, we have the whole spectrum of different definitions for exergy, and we often cannot establish among them a certain relation of equivalency. The situation can be described very well by a paraphrase of John von Neumann's motto to Chapter 4: "...nobody knows what exergy is in reality, that is why in the debate you will always have an advantage". That is why we have to be very careful when currently known ecological phenomena, conformities and principles are interpreted from the "exergical" point of view.

Just as it is not possible to prove the first three Laws of Thermodynamics by deductive methods, so the above hypothesis could only be "proved" inductively. In Section 12.4 a number of concrete cases, contributing generally to the support of the hypothesis, are presented. Models are used in this context to test the hypothesis. Consistency of the exergy-storage hypothesis with other theories (goal functions, orientors) describing ecosystem development will also be examined. Finally, it is discussed how these theories together form a pattern that can be considered a workable ecosystem theory.

The hypothetical Fourth Law of Thermodynamics is proposed to explain growth observed in ecological systems. Growth is defined as an increase in measurable quantity, often in ecology set to be biomass, but an ecosystem can grow in three different ways (J0rgensen et al., 2000):

1. The biomass or biological structure can grow.

2. The complexity of the structure can grow, i.e. the number of components, connections and thereby feedback's number in the trophic network are increasing. It implies that the mass and energy cycle and the goal system through-flow increase. See also Chapter 7 on the trophic chain.

3. The information can grow, which means that the level of organisation including the number of feedback mechanisms increases.

See also the relationship between Kullback's measure of information and exergy presented in Section 4.3.

In general, growth means an increase in a system's size, while development is an increase in organisation independent of a system's size. Growth is measured as mass or energy change per unit of time, for instance kg/day, while storage-specific growth is measured in 1/units of time, for instance 1/24 h. Development may take place without any change (growth) in biomass. In thermodynamic terms, a growing system is one moving away from thermodynamic equilibrium. At equilibrium, the system cannot do any work.

All its components are inorganic, have zero free energy (exergy) and all gradients are eliminated. Everywhere in the Universe there are structures and gradients resulting from growth and developmental processes cutting across all levels of organisation. A gradient is understood as a difference in an intensive thermodynamic variable, such as temperature, pressure or chemical potential.

12.4. Supporting evidence

Several case studies from J0rgensen et al. (2000) and J0rgensen (2002a) are presented below in which alternative energy-use pathways representing probably different gains in stored exergy are compared. More examples can be found in these references. However, Chapter 13 is devoted to showing a network between ecological observations, ecological rules and the pattern of an ecosystem theory, which is presented last in this chapter.

(1) Size of genomes. In general, biological evolution has been towards organisms with an increasing number of genes and diversity of cell types (Futuyma, 1986; compare also with Section 4.7). If a direct correspondence between free energy and genome size is assumed, this can be reasonably taken to reflect increasing exergy storage accompanying the increased information content and processing of "supreme" organisms.

(2) Le Chatelier's Principle. The exergy-storage hypothesis might be taken as a generalised version of Le Chatelier's Principle. Biomass synthesis can be expressed as a chemical reaction:

energy + nutrients = molecules with more free energy (exergy) and organisation + dissipated energy.

According to Le Chatelier's Principle, if energy is put into a reaction system at equilibrium the system will shift its equilibrium composition in a way to counteract the change. This means that more molecules with more free energy and organisation will be formed. If more pathways are offered, those giving the most relief from the disturbance (displacement from equilibrium) by using the most energy, and forming the most molecules with the most free energy, will be the ones followed in restoring equilibrium.

For example, the sequence of organic matter oxidation (e.g. Schlesinger, 1997) takes place in the following order: by oxygen, nitrate, manganese dioxide, iron (III), sulphate and carbon dioxide. This means that oxygen, if present, will always out-compete nitrate which will out-compete manganese dioxide, and so on. The amount of exergy stored as a result of an oxidation process is measured by the available kJ/mol of electrons, which determines the number of adenosine triphosphate (ATP) molecules formed. In this case exergy is the same as free energy. ATP represents an exergy storage of 42 kJ/mol. Usable energy as exergy in ATPs decreases in the same sequence as indicated above. This is as expected, if the exergy-storage hypothesis is valid (Table 12.1). If more oxidising agents are offered to a system, the one giving the highest storage of free energy will be selected. In Table 12.1, the first (aerobic) reaction will always out-compete the others because it gives the highest yield of stored exergy. The last (anaerobic) reaction

Table 12.1

Yields of kJ and ATPs per mole of electrons, corresponding to 0.25 mol of CH2O oxidised

Reaction kJ/mol e ATPs/mol e

Reaction kJ/mol e ATPs/mol e

Table 12.1

Yields of kJ and ATPs per mole of electrons, corresponding to 0.25 mol of CH2O oxidised

CH2O + O2 ! CO2 + h2o

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