31. The range of natural selection

Now we have our vector for biology, it has become important to clarify the terms ‘intrinsic’ and ‘extrinsic’. But this unfortunately takes us back to the many confusions regarding basic scientific terms extant in biology and ecology, examples of which could be multiplied endlessly. Since finding and then correcting inappropriate justifications, by biologists, of supposed correspondences between biology and thermodynamics in particular would require a lengthy excursion into thermodynamics as well as more than double the length of this paper to no good avail, we stick to the ones exemplified by Harte et al in their paper Maximum Entropy and the State Variable Approach to Macroecology (Harte et al, 2008). To recollect, they justify asking us to take area, A0, as a “state variable”: (1) “because it is the obvious measure of the physical scale of the system, in analogy with the state variable, volume, in thermodynamics”; and they do this because (2) they find a correspondence of “the individual organism and its energy requirements”, by noting (3) “a close analogy with the number of molecules and the total internal energy in thermodynamic systems”.

The lack of rigour in the above usages of the respective terms is common in the literature. It is of concern because scientific laws form an interconnected web, with the Biot-Savart law, the Liouville theorem and the Helmholtz theorem being particularly important in this context. If we propose a new state variable without taking note of the effect on those foundational laws, and whether they are breached or observed, then there is little hope of producing anything with the coherence of the fields from which the terms are extracted.

It is important to appreciate that volume—to stick for the case of brevity to that example—performs at least three different functions in thermodynamics, each of which it is important to recognize. This also means that once those three are recognized, it is actually quite straightforward to design a system free from volume, simply because all the stated functions are being fulfilled either singly by one replacement variable, or else because they are distributed amongst several. These three functions are: (a) specifying a system’s size; (b) stating the joules per unit volume it imposes across its extent upon all entities or particles contained within it; and (c) specifying the joules per unit mass that all material media within it experience and transport. It is particularly interesting to note that although volume is classed as an extensive variable, two of its three functions are intensive.

Figure 30: The Intrinsic and the Extrinsic
 The Intrinsic and the Extrinsic

We can again observe the Joule experiment as presented in Figure 30 . Since the temperature remains the same once the gas has expanded, then the average velocity of the molecules, along with their distribution, remains the same. Even if the range of velocities explored changes when the volume changes, the distribution will stay the same. This is the Maxwell distribution of molecular velocities (Atkins, 1990; Callen, 1985; Encyclopaedia Britannica, 2002; Ohanian, 1987; Thomsen, 2011):

f(vx)f(vy)f(vz) = N(m/kT)3/2 e-½mv2/kT

We can analyse this situation by saying that in the initial state, the molecules spend relatively more of their time and energy interacting intrinsically with each other, and less interacting extrinsically with the environment. The pressure is high at 2P, while the quantity of Helmholtz energy or the expansion work that the molecules can do is also high. The pent up energy from that relatively larger number of intrinsic mutual interactions is therefore high. The entropy is low.

The pressure after the expansion now decreases to P, while the volume doubles. The molecules now spend more time interacting directly with the environment and less time, relatively, interacting with each other. There has therefore been a qualitative change in the prevailing balance of intrinsic and extrinsic interactions … which is reflected in the increase in the entropy and decrease in the Helmholtz energy, as well as in the fall in the ability to do further expansion work. The pent up energy from intrinsic mutual interactions is now low.

The quantum theory, especially relativistic quantum theory, threw physics into crisis by forcing the re-evaluation of basic terms and concepts. As quantum physicists eventually realized, a particle’s mass cannot be properly debated without also debating its total energy as transported through each volume element it enters. And since mass is merely the quantification of a given particle’s inertia, then it is always possible to determine the mass of any desired object by determining its energy, along with the inertia of that energy, for inertia is simply the resistance to the receipt of further increments of energy at whatever locus the force of the moment is being applied. If we measure that force and measure that energy, we are also measuing the resistance to the energy, and we can therefore allocate a value for mass to whatever appears to be occupying that volume element of space, whether it is biological or non-biological. It should again be stressed that “mass” is the measure of the inertia of energy and has no other attribute in science.

However it is executed, the Joule experiment's change from one condition to another requires a molecular movement. When a force is applied to an ordinary particle, such as a molecule, it changes its velocity, v, and leaves its mass, m, constant. Its kinetic energy of E = mv2/2 is transported through each unit volume of space as a cloud of photons which mediate its impact in the material world. Thus when an ordinary material particle—such as a molecule—is in motion, there are changes through space and time and energy is absorbed and released as photons. The particle therefore has an accompanying energy density.

Things are somewhat different when a force is applied to a particle of constant acceleration, such as a photon. Its kinetic energy is given by (at its simplest) E = pc, where p is its momentum. Although photons both transport and respond to energy, since they always travel at the speed of light, c, and never deviate from that, then their acceleration, a, remains constant even when under the impress of some force, F. Since a photon cannot change its velocity away from c, its only recourse is to change in its mass, m. The energy concerned is again carried through every volume element of space to affect the material world, and there is again an accompanying energy density. Also and commmensurately … if a particle must demonstrate a constant acceleration of a, yet its mass, m, insists on increasing, then by Newton’s F = ma, the size of the force being impressed must itself be increasing commensurately with that increase in mass. We can measure that force through the photon’s inertia as we observe it increasing through its mass, which we can note via the effects it has on the environment. Those effects again allow us to draw conclusions about the force and the photons even if neither is directly observable. The similarity is that mass is the measure of the inertia of energy whether the object is supposedly material or supposedly non-material.

The photons that surround any particle in motion, and that thereby carry its effects, occupy unit volume elements of space around that particle. Those unit volumes therefore have an energy density … meaning their kinetic energy can be expressed as joules per unit volume. But since the energy density concerned is caused entirely by the particle’s mass, that energy density can also be expressed as joules per unit mass. These two are equivalent. They each state an important purpose which is why volume is a critical part of thermodynamic theory.

We now carefully note that in the Joule experiment (a) the system count or size has remained the same; (b) the joules per unit mass or the “specific energy” has changed; and (c) the energy density or the joules per unit volume has also altered. We cannot immediately assume that these latter two changes are of the same magnitude, for that depends entirely upon the heat capacity of the material medium and how the energy is therefore apportioned. Nevertheless, a change in the range—but not in the distribution—of the molecular velocities could easily accompany those two specific changes in joules per unit mass and per unit volume, thus leaving the temperature constant. So when Harte et al say ‘in our theory, the inverses of the Lagrange multipliers are neither intensive nor extensive, and we do not know if they can be associated with a generalized ecological ‘‘temperature’’’, then it may well be that if we take great care with all these kinds of correspondences; and that if we take explicit care to make sure that we constantly abide by the above stated Liouville and Helmholtz theorems as well as the Biot-Savart law; then we may well find that either ecological systems simply do not need a “temperature”, or else that we have carefully fulfilled the functions of temperature with some other variable or variables that stand duty for that thermodynamic function within biology and ecology. But it would of course first be necessary to examine that basic concept of temperature as closely as we are examining the basic concept of volume. For now … we simply and carefully note that the range of molecular velocities is free to change in this Joule experiment, while the distribution of those same velocities remains invariant, so leaving an important variable—in this case temperature—unchanged.


Figure 31: The Volume of a Biological Population
 The Volume of a Biological Population

We now further clarify volume by examining Figure 31, which is a biological system engaged in reproduction. Before reproduction, there is a count of N cells composed of 2Q moles of chemical components whose total mass is 2M. The average individual values over the population are 2 and 2 respectively. The cells then prepare to reproduce. Energy is expended, so we can straight away observe that—at the very least—the joules per unit mass must change; and if the joules per unit mass change, then so also must the joules per unit volume. But since certain other aspects of this system nevertheless remain the same, then we now know that we must search for some kind of accompanying change in range, but one that is not also a change in distribution. We therefore note this requirement most carefully.

After reproduction, and the cells have split, we have 2N cells. However, neither the total number of components over the entire population, nor their mass has changed. But since N has doubled, then both the number-of-components-per-cell and the mass-held-per-cell must have changed. These have each halved. The population average individual values are therefore now and respectively. This does not immediately tell us what has changed in its range or possibly distribution … but we can surely suspect that it is in q, or in m, or else in some ratio such as Q/N or M/N and therefore in q/m. We also note with great interest … we have already met for it is the divergence in the mass flux.

Since the number of cells has doubled then the population count has doubled … and the system’s size has also doubled … which is entirely consistent with the increase in the volume recorded in the Joule experiment. Number count is therefore holding as our proposed determiner of size. Thus the size of a biological system is again established by counting … and a biological count is the required biological analogue for this first aspect of thermodynamic volume.

We shall now set all of this aside. We shall try later to find analogues for the other two aspects of volume because we cannot proceed without devoting some attention to these intrinsic properties of biological populations. They are enshrined in what we have called our laws of biology. We therefore turn to a series of largely “hand-waving” arguments to declare them, for we do not have time for their complete and thoroughly logical derivations. We nevertheless require them to ensure that biological populations do indeed abide by such important physical parameters as the Laplace operator and the already stated Liouville and Helmholtz theorems—as well as the Biot-Savart law—and so that we can then find further maxims of ecology, our primary interest in this paper. And so therefore: