# 22. The second maxim of ecology: the maxim of number

We must at some time factor in molecules. This is the knowledge that all terrestrial biological organisms are composed of DNA configured in different ways; and that they all sustain themselves by absorbing chemical components from the environment. We must therefore reckon their chemical bonds. But we also know, from our Maxim 1, Proviso 2 of M → 0, that any population’s mass flux has a net divergence, over a generation, of zero. By the first law of thermodynamics, all energies arise from a material source, and so therefore all energies, work, and heat arising from a population are a function not just of the mass of chemical components pertaining to each organism, but also of the chemical bonds binding those components together at a given rate per second per each entity. Our destructive harvesting of our Brassica rapa specimens specifically measured the population’s total chemical bond energy, H, in joules.

1. Maxim 2: Proviso 1

Since all energy must have a material source, then there must also be an energy flux emerging from the chemical components that are maintained by the entities, and in the divergence of the mass flux of M kilogrammes per second in and out of the population. We have also already determined that that associated mass flux resides in a set of point entities of determinate volume, n. Therefore—and again by the first law of thermodynamics—that volume for the mass flux is exactly the same population count, n, as pertains to the energy flux.

Again by the first law of thermodynamics, the chemical bond energies that bind the chemical components constituting any given population must fund the chemical behaviours that allow for all biological behaviours. DNA and its behaviour must therefore equally well depend upon the same divergence pattern in energy as is in the mass flux … which is itself based on n for by Maxim 1, Proviso 3, M = nm̅. And since the total chemical bond energy, H, must therefore depend on the same population count, n, as for the mass flux, then the divergences are over that count and we therefore have:

1. = H/n
1. Maxim 2: Proviso 2

Yet again by the first law of thermodynamics, a given biological entity is a discrete thermodynamic system with its given number of chemical components, and with the above chemical bond energy of for each entity being responsible for holding its components together. Its tendency for transformations, through its chemical components, is recognized by the partial differential equation (S/E)n,β (Encyclopaedia Britannica, 2002; Atkins, 1990). This implies that if the number of its constraints, β, and its moles of components or amount of substance, n, hold constant, while its energy, E, changes, then its entropy, S, must also change. So if the molecules in a first System A are more energetic than those in a second System B, then the energy and momentum will transfer over to B as their molecules interact. Such systems have a greater “escaping tendency”, a term coined by Gilbert Norton Lewis (Klotz & Rosenberg, 2008, p. 219). They will therefore have a one-sided and irreversible effect upon all systems whose escaping tendency is lower, for a lower escaping tendency is immediately a greater “capturing tendency”.

Figure 20: The Capturing and Escaping Tendencies

Granted the nature of these tendencies, then as in Figure 20, the net energy, , held by any given biological entity is the vector sum of its escaping and capturing tendencies relative to the environment … which is always and for an entire population their common and given system sink and source. These energies have a sense of direction in that the capturing tendency enters, while the escaping tendency leaves. They not only have direction … they also have magnitude.

Biological entities extract all needed energy and components from the environment with a capturing tendency; but they will eventually dissipate back into it with the escaping one. According to our Maxim 1 of dissipation, the escaping tendency will eventually and in all cases overcome the capturing one as in the proviso ∫dm < 0. Any given biological entity is therefore a given thermodynamic system composed of a specified number of chemical bonds and components, which amount of substance is at all times statable in moles.

Biological entities, in common with all other thermodynamic systems, are dependent on the first law of thermodynamics, δW = δQ - dU. It sets the quantity of work they can do with their chemical components retained. The infinitesimal value dU comes from their internal energy, U, which is a state function or variable. It is a property of the system. Its value depends upon the system’s current state. It is independent of the path … i.e. of how that state was acquired.

Since internal energy is a state function, then its infinitesimal change, dU, can always be inferred by comparing two different and neighbouring states such as U1 and U2. Its value gives exact information about the transition. For this reason, it is formally called an ‘exact differential’.

Although internal energy is a rigorous and invaluable scientific concept, it is unfortunately insufficient. For example, if I am standing beside a pickpocket; take careful note of his or her arms and hands; glance away for just a moment; and then look back again; it is entirely possible that he or she will still be in the same state. The net distance travelled by those arms and hands out to my shirt and my trouser pocket and then back again, with my possessions, is zero. That zero change in state inclines me to believe that nothing has happened. But when I later discover—which I do by some means external to the pickpocket—that my wallet and pocket watch are missing, I will correctly conclude that although the net paths or distances the arms and hands had travelled was certainly zero—which was why the observed state had remained the same—the total of the paths and distances they had travelled was not. It is always necessary to account for the interaction with the environment. In regard to biological entities, this means it is always necessary to determine whether the capturing tendency is or is not less than, or equal to, or greater than the escaping one.

Work and heat form a complete contrast to internal energy. Both are path or process functions or variables. As such, they are ‘inexact differentials’. As with the pickpocket’s arms and hands, their values depend entirely upon the specific path taken as a change in state occurs. So just like the pickpocket’s two arms can steal different things from different parts of my anatomy, it is entirely possible for two systems of identical chemical components with identical internal energies to pass between two identical initial and final states … but still to differ greatly in the work and heat they each absorb and/or emit. The difference lies in some external and measurable parameter that reflects their differences in work and heat. That difference is their process or path.

If a first system produces a given quantity of work, then its value for dU—its transition between its initial and final states—reflects that success. It is, however, possible to provide heat and/or work along a continuum, and in differing amounts. So should a second system fail to do the identical quantity of work, but instead only generates heat and friction, then it is entirely possible for it to move between the same initial and final states, even though its quantities of work and heat are vastly different. The work successfully done by the first but unsuccessfully done by the second have emerged from exactly the same states and transitions. Their difference is only evident externally to both—i.e. in the heat emitted and work done.

Work and heat, δW and δQ, are inexact or ‘imperfect’ differentials because their values cannot be inferred simply by examining a set of initial and final states. Since they are equivalent, a diminution of one can be compensated for by an increase of the other. Work and heat are not state functions but path functions or variables. They cannot be inferred simply by examining the initial and final states of any given system. Their sum is an exact differential by precisely matching the dU or change in state that led to them, as in the first law of thermodynamics … but they nevertheless each independently remain inexact differentials.

If a given system is unsuccessful in achieving the work intended, then it will have to expend additional energy. There will therefore be an additional change in dU from that second attempt. The system has therefore ended up with an increased number of changes in state, and entirely because of a difference in its ability to partition its tranches of its changes in internal energy between work and heat to produce more work and less heat.

Systems that are less efficient, and so that emit more heat and do less work, will always have to expend extra energy to make up any deficiency. Therefore, at each moment t, the work and the heat—which are path integrals generated only in that instant—are emitted as δW and δQ. If the mechanical work, δW, external to the system is still not achieved because attempts to execute it have dissipated as the heat δQ, then yet more work will have to be attempted. This requires yet more changes in state, and also emits more heat. By the second law of thermodynamics, these differences in the partitionings of work and heat will be cumulative, for entropy increases remorselessly and cannot be reversed by any one given entity.

Figure 21: Natural Selection & Competition

Enthalpy, H, is also a state property. With respect to biological systems, it is the sum, over the population, of the chemical bond energy holding the given chemical components together. The effect of enthalpy and internal energy, H and U, on Darwin’s proposals is made clearer in Figure 21. In Generation 1 are two plants that are identical in every way. They have the same numbers of chemical components and therefore internal energy, U; and those components are bound together with the same sets of chemical bonds, thus creating the same values for enthalpy, H. They now go through the cycle of the generations and produce the same numbers of progeny with the same components and bond energies as each other.

But as Generation 2 gets under way, let there be a change in the environment. A passing meteor showers down inhospitable rocks upon one of them; and a permanent pall of cloud now overhangs the sun above that same one. Since the chemical components and bonds are the same over both, then it is entirely possible for both sets of plants in this new generation to go through exactly the same energy cycles, and to have the same values for dU and dH throughout. However, the work done in the environment—again an imperfect differential—cannot now be the same. The first plant can repeat and hand on all the same values as its predecessor to another generation, for its conditions remain the same. It is therefore likely to be more successful in transferring its masses and energies to that next generation. The second plant, however, will have to contend with more wasted heat, and will produce less work. Thus even though the transitions are identical and their exact differentials match, their results in the inexact differentials will differ in the paths and actions concerned.

The interplay between exact and inexact differentials means that:

1. two systems can start off from the same initial states and conditions and release the same total energy, but end up in different states and conditions because of differences in their partitionings of work and heat;
2. two systems can start off from different initial states and conditions and release the same total energy, yet end up in identical states and conditions again because of differences in their partitionings of work and heat;
3. two systems can start off from the same initial state and condition and release different total energies yet end up in the same end states and conditions because of differences in their partitionings of work and heat;
4. two systems can start off from different initial states and conditions and release the same total quantities of energy, and end up in different states and consitions, but still nevertheless have partitioned their energies identically between work and heat.
5. two systems can start off from different initial states and conditions and release the same total quantities of energy, and end up in different states and consitions while also having partitioned their energies differently between work and heat.

We now recollect our bristlecone pines. The transmission of dU and dH over time is a part of the energy flux arising from the processes of work and heat undertaken by all biological entities. The chemical bond energy of those material components impacts on the environment, through biological entities, in the balance of the capturing and escaping tendencies, and as the work done and heat emitted … both of which we can measure. The succeeding generations for our second plant in Figure 21 will exhibit the differences in work. They will have smaller quantities of mass handed on; and/or will suffer a less efficient distribution of bond energies; and/or will produce a smaller number of successful progeny. If they wish to survive then those successor biological organisms will have to adopt the north-facing bristlecone pine strategies.

This is now a very significant success. We have identified the source of natural selection and of Darwinian competition. It is nothing more than the workings and the consequences of the first law of thermodynamics. Its effects are achieved—or not achieved—by any given changes in state, dU and dH, through the energy flux expressed in the sum of the capturing and escaping tendencies. Work and heat are therefore also field lines. They join our chosen flux commodities. They pass through our Gaussian surface. As in our Brassica rapa experiment, they are also eminently measurable.

By the first law of thermodynamics—which distinguishes between the two energy transfers and the two physical processes of work and heat, both of which are inexact differentials—we know that δδdU. This states that the total energy flux emerging from the population is equal to the heat evolved by that population. And since the energy flux emerges entirely because of the entities’ chemical configurations and the work they do, we have:

1. dH δδdU
1. Maxim 2: Proviso 3

All biological activity begins and ends in the environment. It passes through the chemical bonds of our population of biological entities. Energy is temporarily retained there, as , over the population to do biological work. Our energy field lines of flux enter and leave our biological entities, and we again measure them as work and heat. But since it is a flux, it must also have a flux density or divergence. And since both internal energy and enthalpy are state variables, then even in reproduction the volume or count of biological point entities depends entirely upon the system’s state, which is its current number. That total chemical bond energy, H, is always shared amongst that number as the divergence over that volume and so as the flux density. It is the net of the vector capturing and escaping tendencies over the population, and is therefore:

1.  •

We can now combine our three new provisos. This dependency of biological populations on H and n is the dependency of given biological and thermodynamic systems on the capturing and escaping tendencies. But since n is involved, then it is also a dependency on the divergences of the mass flux, which incorporate reproduction. It duly combines with the net divergence in energy over the population to produce another concise vector mathematical description for that population:

The Second Maxim of Ecology: The Maxim of Number

H = W/n =

The number of progeny produced depends upon the number of progenitors maintained.