To a first approximation, the answer is almost certainly “astronomically larger.”
I think I see a way to hand-wave some dimensional analysis and get an answer in the correct units by applying Boltzmann’s constant and this equation from @PdotdQ’s work above: \frac{4 \sigma T_s^3}{3} \cdot \frac{4R_e^2}{4 \pi AU^2} \cdot 2 \pi R_s^2 and stating the problem more precisely to involve the flux of entropy from the sun for 1 second. But I’m not entirely clear on where \sigma comes from to calculate an actual numerical result.
\frac{4}{3} \frac{\rm{Energy \; Flux \; of \; blackbody \; radiation}}{\rm{Temperature}}
This is because the entropy of blackbody photon gas is (4/3)(Energy/Temperature). Note that the energy flux of blackbody radiation is F=\sigma T^4.
The second term takes into account that the Sun spans a certain angular size in the sky as seen from a point on the Earth. The third term takes into account that an entire half of the Earth receives entropy from the Sun.
Join the club! In another thread there was some calculations related to the size of the human genome in bits. The thread referenced a news story that was ambiguous in how it was using ‘bit’. Josh knows that I’m interested in overloaded terms and brought it to my attention. I made a highly amusing (read: dubious) joke involving yet another meaning of ‘bit’ (half of a quarter, as in ‘shave and a haircut, two bits’). Then I doubled down on the (again, dubious) humor by calling back to the discussion here about adjusting the relative size of the Sun, based solely on the fact that entropy also appears in information theory and its bits.
For my sins, I then immediately fell down my own pit trap of wondering whether it would be actually possible to calculate an “actual” answer to my nonsensical question. To wit: how much bigger would the human genome have to be (in information-theory-bits) to have more entropy than the entropy arriving on Earth in one second from the Sun? I have no delusions that the answer will be meaningful in any sense other than the tomfoolery of making units match up.
Thanks!
Seeing as how I went awry previously by trying to do math late at night, I think I will put off the actual calculation for now. But I do appreciate the added info.
Yes, the Sun sends entropy to the Earth through photons, much like the Sun sends energy to the Earth. Entropy is a thermodynamically extensive quantity, like mass, energy, and number of particles. These things can typically be transported from one system to another.
@AndyWalsh I must say I wanted to know the answer too. I don’t know what the answer might mean physically, but if anything it might be a good question to put in a final exam.
Using the flux of entropy formula \frac{4\sigma T_s^3}{3} \cdot \frac{4R_e^2}{4\pi AU^2} \cdot 2\pi R_s^2 we get \frac{4}{3} \cdot 5.67 \cdot 10^{-8} \frac{kg}{s^3 \cdot K^4} \cdot (5777K)^3 \cdot \frac{4 \cdot (6.38 \cdot 10^6 m)^2}{4\pi \cdot (1.50 \cdot 10^{11} m)^2} \cdot 2\pi \cdot (6.96 \cdot 10^8 m)^2 = 133 \frac{kg \cdot m^2}{s^3 \cdot K}. if we multiply by 1 second to get the entropy of photons arriving over 1 second, we get 133 \frac{kg \cdot m^2}{s^2 \cdot K} or 133 Joules per Kelvin.
Now here’s where we get into territory where I’m less certain if what I’m doing is actually meaningful. If we use the formula for entropyH = \kappa \ln W where W is the number of microstates, then we have a relationship between the thermodynamic entropy and the number of bits needed to specify which of the W microstates the photons are actually in.
So 133 \frac{kg \cdot m^2}{s^2 \cdot K} / 1.38 \cdot 10^{-23} \frac{kg \cdot m^2}{s^2 \cdot K} = 9.6 \cdot 10^{24}. Notice that all the units cancel out, unless we understand \kappa to be in terms of Joules per Kelvin per nat and then the answer is in nats. We can then do a logarithm base conversion from e to 2 to get 1.4 \cdot 10^{25} bits.
And how many bits are in the human genome? If we say it has 3.2 \cdot 10^9 base pairs and use the maximal information content of 2 bits per base pair (an overestimate biologically, but I hope it is obvious that a factor of 2 or 4 isn’t going to matter in the answer we are about to get), then we get 6.4 \cdot 10^9 bits. Which means the human genome would need to be 2.1 \cdot 10^{15} times larger (in terms of bits) to eclipse the entropy of the photons arriving on Earth from the Sun in one second. To try to put a little more perspective on that, if we say there are roughly 10 million eukaryotic species (the upper range of the estimate) and assume that to a first approximation their genomes have the same information content as humans’ (a very crude estimate), we could use the bits in all eukaryotic genomes and still be short by a factor of 100 million.
This is good, because you are using two different senses of the term entropy, as you are aware.
More entropy in terms of information-theory-bits? IOW are you using the same measure for both measurements?
But i think we know how the sun sends energy to the earth through photons, while it is not at all clear to me how the sun sends entropy to the earth. You seem to be saying that entropy is something like energy. But has the question" what is entropy" really been answered yet?
The same way as how the Sun sends energy to Earth. There is a entropy that is locked up in the electromagnetic degrees of freedom of a photon, similar to how there is energy that is locked up in a photon.
As I mentioned in a previous post in this thread, the photon entropy is given by the von Neuman entropy, S=-\rm{tr}[\rho ln \rho], where \rho is the quantum density matrix which in words measure how mixed the quantum states of the photon is.
That is no doubt a simple statement for a physicist but mind-blowing for me. I had never given that a thought before. I like the way you describe it: “entropy that is locked up in the electromagnetic degrees of freedom of a photon.”
In physics the entropy of photons is well defined. You cannot refute this by saying that a lot of people are confused by it. I am sure I can find a lot of people who are confused about quantum mechanics through a Google search - this does not mean that quantum mechanics is not well understood amongst professional physicists.
Again, as I kept saying in this thread, entropy plays a part in determining the amount of useful energy that can be harvested from an energy source. The simplest way this is done is through a heat pump, and then the change in entropy is given by \Delta S = \Delta E/T
No, the entropy for a photon gas streaming in vacuum does not change. Note that energy is not lost either.
I don’t know what you mean by “frozen” for the cases of photons. What is the context of this question?
Physics Essays is a fringe/crackpot journal with an impact factor of 0.245. They have a low standard for the quality of papers that they accept. Typical physics journals have impact factors of ~3.
The usual cited figure for the length of a fully unraveled set of human chromosomes is 2 meters. Multiplying that by 2.1 \cdot 10^{15} gives us roughly 4 petameters, or about 0.4 light years.
The typical human nucleus has a diameter of 6\mu m. If we treat it as a sphere, we get a volume of 1.1 \cdot 10^{-16} m^3. Assuming the same density of base pairs per cubic meter, our expanded genome would need a nucleus with a volume of 0.24 m^3, or a sphere with a diameter of 0.77m.
Bonus calculation(!): We said our expanded genome has 1.4 \cdot 10^{25} bits. The Bekenstein bound for the maximum entropy of a sphere with radius 0.77m and mass of 12.6 kg (6 picograms of DNA in the actual human genome expanded up by the same factor as everything else) is 1.3 \cdot 10^{44} bits.
