30. Natural selection is a vector and a force of nature

It is time to reveal our first vector. With it, we can gradually show that natural selection, Darwinian competition, and Darwinian evolution are not merely aspects of nature that can only be described. We can show that they are forces of nature capable of accurate and precise measurement.

Figure 29: The Simplest Possible Biological Population

In Figure 29 we have a tunnel or line segment of molecules. We therefore conceive of this line segment or tunnel of molecules stoichiometrically. It is entirely filled with, or composed of, whatever molecules any given population of biological entities needs to go through its cycle and create its distribution. Since all terrestrial biological organisms are composed of DNA configured in different ways; and further since they all sustain themselves by absorbing ingredients that are ultimately molecular; then all necessary molecules are laid down upon our line segment, and in suitable proportions. Our biological entities are therefore able to move along our line segment or through our tunnel; take up molecules; and use them for their sundry metabolic purposes.

Our line segment is in fact the environment. It is the surroundings. All elements composing each molecule are allocated their atomic number from the periodic table of elements. They are therefore declared in amount of substance, or moles. So to state the mass of molecules absorbed or emitted by the entities to and from our line segment or tunnel is simply to state the number and the types of molecules utilized. It is also to state that quite independently of how those molecules might be configured or arranged within the entities. We shall deal with the issue of configuration—a different dimension—later.

Our line segment is of length T. It is a given generation length for some given population. Since this is an equilibrium age distribution population, then whatever processes the individuals undertake upon that line segment are sufficient to allow the generations to succeed to each other.

It is important to note that the absorption and/or emission of molecules to and from this line segment in no way changes this system’s volume. We tightly define a system’s size—in biology—so that the volume only changes if and when the entity count, n, changes … and the volume only changes when a given and specified collection of molecules is either incorporated into or else is removed from, a specified entity that is itself being either lost or gained. Volume is therefore linked to natality and/or mortality, and so to δ(t, l, ψ) and β(t, l, ψ). Thus the question of molecules—and energy—either entering or leaving this tunnel or being picked up from or set down on this line segment does not in and of itself render this sytem open or closed. Those terms still await a precise definition. We are only clarifying that the mass and the energy fluxes involved in the take up or set down of molecules have nothing to do with volume, or with inertia, or with mass, or with open, or with closed, all of which we are free to define as we choose, and as we proceed.

Now that we have a suitable line segment, we can gather up the entire count of whatever biological entities happen to constitute an entire generation for a given equilibrium age distribution population. We can then distribute those entities upon our line segment. We distribute them with a mapping and at the specific number density appropriate at each t over T, and so that they act first as progeny, and then as progenitors, as they produce more like themselves, using the molecules provided. We will eventually have a trail of molecules of specified quantities per each entity, and all with the appropriate distribution of both entities and molecules. We can now study this line segment of trails of molecules stretched out over length T, as well as the entities and molecules that compose it.

Our vector arises immediately. It does so by virtue of Maxim 1, Proviso 1, of our maxim of dissipation, ∫dm < 0, which is our biological expression of the second law of thermodynamics. When translated to our line segment of molecules stretched out all along T, then Proviso 1 gives us a terminal point for each entity along our line segment. Each entity’s molecules will at first ray out before it … but will then fall away from the line segment or tunnel as the entity inevitably dissipates.

However … since this is an equilibrium age distribution population, then in spite of the inevitable termination point awaiting every entity somewhere or other on our line segment of length T, the population at large will continue. There are different trails all over T. The entire line segment therefore proceeds apace as the population enjoys successive Ts or generations, and in spite of the loss of individual entities and their constituent molecules. This simply means that whatever may be the chemical components laid out over any given time span for the population, some of these trails of molecules, and therefore their entities, are certain to degrade. But if the population is to continue then at least some trails must also always be recoverable over T, and therefore at at least some ts over T.

If a degradation is certain everywhere … while recovery is only possible in some locations … then there is a definite directional sense upon our line segment of molecules. The direction that tends towards the ultimate dissipation of all possible components is now very different indeed from the direction that tends away from such certainties.

Reproduction still awaits its clear definition, but a directed line segment of molecules that points in the direction of the ultimate dissipation of all entities need not of necessity, and on any given entity trail, lead to a moment of reproduction in that direction. However … a directed line segment or trail that points away from the direction of ultimate dissipation will definitely lead towards such moments of reproduction.

We have discovered a very important difference in the directions upon our line segment or tunnel of molecules. A direction exists, along any random trail of molecules, in which dissipation is certain, while an act of reproduction and continuation is very much less so. But to set against this, we have a contrary direction upon all trails in which reproduction events are certain. Not only that, but it is equally certain that continuing in that same reproductive-direction, from any point on this line segment, and so yet further away from any given acts of dissipation, will lead to still yet further reproductions beyond those … and yet again behind those … and so on and so forth. And since these two directions are completely different in that the one direction heading towards dissipation sees at least some terminations without continuations; whilst the other is guaranteed to demonstrate a constant and ongoing succession of continuations and reproductions; then distances on our line segment—which are periods of time, Δt—are vectors. They have both magnitude and direction in their lengths, and also in the quantities of mass and energy pertaining to those molecules. Therefore: this line segment of molecules is a vector … which is what we have long been looking for.