by means of relative selection
Using the Weyl and Ricci tensors to prove Darwin’s theories of competition and evolution
and to refute creationism and intelligent design
Since we have clarified the important distinction between mechanical and nonmechanical chemical energy as expressions of the more general internal energy, we can draw some very important conclusions about biological entities. We can do so even though this whole situation facing us and our population is so unrealistic.
We may now understand energy and the planetary motion that gives rise to it, but no walls in the real world are perfectly adiabatic. Energetic conversions from one type to another do not happen faultlessly … not even for biological entities.
It is in practice impossible to remain within the same isentropic set. Molecules are not like that. Heat is therefore always radiating into the environment. And when heat is lost, the piston in Figure 9.1 on the previous page does not return to its prior position. It cannot then do as much work as it could before, for it is not in the same isentropic set. This is the famous second law of thermodynamics.
Carathéodory defined entropy and the second law along these lines. He used his simple system to point out that only a system that can move adiabatically, in the way he suggested, is free from changes in entropy.
There are many paths a system can follow when it is not perfectly adiabatic … which is all real systems. And all those other paths share a singular characteristic: they all involve the surrender of some very definite quantity of heat to the surroundings. That heat is irrevocably lost. That loss increases the universe's entropy by surrendering that heat to the surrounding universe. It compromises any system as the molecular vibrational capability is compromised. Since it is molecular in scale, the heat loss occurs at different rates. It is lost from each system but gained by the cosmic background surroundings, which is why it is a net increase in entropy.
A mathematical aside
Carathéodory's definition for the second law states: “in every neighborhood of any state S in an adiabatically isolated system there exist other states that are inaccessible from S”. Those other states are inaccessible from S because if the system enters them, then it cannot return to its previous isentropic state, S.
If a system—such as one in the real world—follows any of the paths on which it surrenders molecular vibrations as the energy of heat, then it is following a path on which its entropy is increasing. It is surrendering a quantifiable amount of molecules and energy. It is also surrendering the ability to do the previous quantity of useful work. It is surrendering some of its stock of potential planetary motion, both microscopically and macroscopically, to the cosmos.
The possible work a system can do just keeps on diminishing (unless we can do something to restore it … although even if we completely restore this particular system, it is going to cost us something somewhere else within the universe). Only a perfect system that can move adiabatically—i.e. with no molecular vibrating capability or heat crossing its walls—is free from the steady increase in entropy. But those ongoing changes within the system are equal to the heat it surrenders to the surroundings. That lost heat is the system's decrease in internal energy and is the reduction in the amount retained, so affecting its future states and capabilities. Only an adiabatic system can retain its potential for work indefinitely. But the critical factor is that we can measure the effects the entire and surrounding environment is having upon the system simply by measuring this particular aspect. Entropy is the link between the system and its environment. By measuring entropy—which we can always do externally—we gain information about the system's interior, even on a molecular level. Entropy tells us about the scale and scope of the ongoing transformations and interactions that system is going through. That is precisely why it is so useful in scientific work.
A mathematical aside
The essence of this second law of thermodynamics is that all the entropy-creating changes in state, dS, that a system goes through at that given temperature, T, are quantifiable as TdS.
We can now use the above information in conjunction with Figure 10.1 to draw our first set of conclusions about biological entities:
A mathematical aside
The above four conditions can be expressed mathematically as:
We can now draw these various statements together as:
Energy is universal. It penetrates all parts of the cosmos without let or hindrance. If it penetrates everywhere, then nothing can be adiabatic. Everything must be the opposite of adiabatic, which is “diathermal”. Everything must in fact follow a diathermal, rather than adiabatic, path.
All non-adiabatic paths are similar simply in being diathermal. The zeroth (or fourth) law of thermodynamics can therefore be equivalently restated as: “all diathermal walls are equivalent”; or else as: “all heat is of the same kind”. All heat is of the same kind because it can all penetrate everywhere. Biological organisms are not exempt.
All diathermal walls and boundaries may be equivalent in being porous to molecules and heat, but they are not all equally porous. Some things pass heat more easily than others.
If we think more carefully about what is involved in taking a temperature, then we can see that two bodies will have the same porosity to heat—or temperature—if they could have interacted thermally in a given situation; if they could have transferred heat energy to and from each other; but did not do so, and continue not to do so. If there is an ongoing lack of heat transfer between two bodies, then they are effectively thermally blind or adiabatic with respect to each other. They are of course not adiabatic with respect to all other systems. They are simply identical to each other, and so are being non-responsive to each other.
Bodies have the same temperature if they have the same porosity to heat, and so possess the same states and rates of change. They are then on the same path in terms of their energy, entropy, and capacity to do work, which is measurable as a given amount of planetary motion.
We can again reconsider a thermometer. It has this kind of mutual invisibility and porosity with respect to whatever we are choosing to measure with it. A thermometer, as a Body A in the zeroth law, in essence measures its own state. It does not measure the other body. Heat simply flows to and from it until it achieves a Prévost style equilibrium with respect to the body it is measuring, and so experiences no further transformations relative to that other, but by measuring its own changes in state. It measures their joint entropies or rates of change. When the temperatures are the same, then these rates are the same and they are on the same path. Their molecules are behaving in the same way and there are no more relative changes. The thermometer knows this simply by measuring its own state.
We go back to our opening propositions: “things which equal the same thing also equal one another”, and “if a Body A be in thermal equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another”. This is about transformations and rates of change under energy.
A mathematical aside
The equality of rates under entropy, which is the equality of temperatures, is formally stated as: [(T_{A} = T_{B}) ∧ (T_{B} = T_{C})] ⇒ (T_{A} = T_{C}).
We link this equality in transformations to the understanding that heat energy, in this case as solar radiation, can initiate from one stock of internal energy and be projected across free space at the speed of light to another. It is the only interaction that can change internal energy and that is: (a) nonmechanical, and (b) does not involve transfers of mass.
But even though heat energy does not in itself create transfers of mass, under the first law of thermodynamics it is always equivalent to some such transfer for it can always be converted and achieve that effect, such as with a solar cell or battery that then does some mechanical work. Heat energy ultimately increases internal enery by a specific amount, and by the first law of thermodynamics, there are equivalent ways the same increase could have been effected.
Nonmechanical energy may not be so evidently visible, but it can always do work that is. Since energy is always interconvertible, it always has a measurable potential. So if a first mass moves through a given height in a given gravitational field, then it does a given quantity of work. There is always some associated planetary motion. It is immediately equivalent to any second mass that moves through an equivalent height in an equivalent gravitational field, producing an equivalent amount of work. These are the same quantities of planetary motion … with both still and nevertheless being equivalent to a given amount of available heat energy..
If a first system receives energy by interacting with our first mass; and if it provides a given amount of heat or nonmechanical energy under a first given conversion; then it is equivalent to a second system that does the same quantity of work using our second mass, and under the same mechanical-to-nonmechanical conversion. These two will remain equivalent if they follow paths that involve the same entropy transformations. They can be freely substituted for each other in any and all pertinent interactions involving mass and energy, both mechanical and nonmechanical. They will continue to remain equivalent for as long as these mass, energy, and entropy conversions and conditions continue to hold … and all under the discernible laws of the attending planetary motions both microscopic and macroscopic.
A mathematical aside
Let a first mass now follow a path and do the work δW1; let a second mass follow a path and do δW2; and let a third mass follow a third path and do δW3. We then have: [(δW_{1} = δW_{2}) ∧ (δW_{2} = δW_{3})] ⇒ (δW_{1} = δW_{3})
The above relations now allow us to declare:
Now we have a guarantee that our entity really can reproduce another like itself within its population, we can watch it convert the incoming nonmechanical solar radiation into the equally nonmechanical chemical energy that it needs to survive and to reproduce its population.
An algebraic and geometric topology based proof.
A vector calculus based proof.
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