This is sort of another "note to self".

The entropy of a gas of photons in a box is S=4/3 U/T. A 4/3 occurs in physics because a volume has been divided by an area (if there is not a hidden "pi" next to it like in a constant or the units like U or T, but in this case I do not believe that is the case). Another possibility is that a space-time (4D) has been divided by a space (3D).

The entropy of the universe on an expanding-volume basis is constant. It seems to be get the extra entropy per fixed volume it needs in order to expand from mass (mc^2=U=3/4 T S) in gravitational systems being converted to heat and thereby black-body radiation. For example, the Sun at a hot 5000 K is emitting "high" energy photons, so their are fewer of them. This means lower entropy and this relationship shows lower S for the higher T. Earth, after receiving some of these photons, absorbs them and heats up, until the energy in equals the energy out. But it does not get nearly as hot, so it is more "efficient" at converting converting photons to entropy compared to the Sun. I've read there are 17 photons emitted from the Earth for each photon received from the Sun. Now I see why: 5000 k of the Sun divided by the 300 k of Earth is 17.

The order created by life that is more tightly packing the mass on Earth (as compared to if there was no life) is the result of using the Sun's photons (or the rotational energy from the Earth as delivered to the tides and mantle from the moon) to create higher-energy bonds which means lower entropy in those atoms, at least in the mass we consider important to life and economics if not a net total reduction in entropy for the Earth.

The 4/3 is interesting because it might signify the relation of entropy to an energy of mass (a 4D space-time contortion via M=8 pi T) inside a 3D spatial volume. And/or it might also give insight into how Energy per particle (temperature) is different from Energy.

I could look at the derivation, but I want to understand it from a deeper level. Physics equations are usually derived from long explanations that involve axioms that are no less simple than the result. This indicates we could work backwards from the simple equations to simpler axioms. I'm sure I'm not capable of that, unless the errors I'm aware of in how physics normally applies units can help. One units problem I'm talking about is that temperature is not fundamentally different from energy and therefore entropy fundamentally has no units (J/K is as fictitious as the distinction between J and K). I also would like physics to explicitly use the precise and complete meters=i*c*seconds equivalence. (It might be that I should say c*seconds + i*meters = 0, using the 1/i = -i relation.)

I should mention E = 3/2 kT = 1/2 m v^2 for the kinetic energy of a particle. The "3" comes from entropy (and thereby k) depending on 3 different possible directions for the momentum that holds the energy. S=k ln(states in 3D) = 3 k ln(states in 1D). Maybe the 1/2 has something to do with the average of entropy when raising the v from 0 to v. This is another rare instance in physics of where an integer shows up and it's not exactly clear why. I do not believe anything in fundamental physics should involve integers even as I leave open the possibility that all of physics is based on 1 and 0, 1 and -1 or "i". The "3" of space and 4 of space time apparently are the result of our 6-layered brain enabling us to conceive of mass with 6 degrees of freedom of motion (3 translational and 3 rotational). If we had 10 layers, we would see less movement in the world and the "mass" we would perceive would exist in 4D space (5D space-time) and e=mc^3 would probably apply.

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