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I once listened to a lecture by Sir Roger Penrose on life and also read the second answer to this question. If I understood correctly, then the reason why life still exists on earth inspite of the second law of thermodynamics is due to the low entropy of sunlight. But what does it mean for light to be of low entropy? I.e. how would one model mathematically the low entropy of sunlight?

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One way to think about it is to consider that sunlight comes only from a small part of the sky and that it is of a high frequency, way above what warm life emmits as a byproduct, in a form of infrared.

If the whole sky would shine like a sun, life would not be possible. Not just because of overheating, but fundanentally because there would be no difference in entropy available.

If sky would shine like a sun, but far away from thermal equilibrium, so that there is no way a one temperature sky could make such a light, like sun light diffused in clouds, life still could exist by absorbing light and emitting IR.

You need at least one non-thermal equilibrium effect, either one side of the sky is hot and another is cold, or sky emmits much more higher frequencies than lower frequencies. Both option allow to do some work from it.

There are a few more properties of light, like coherency, that would give another option to use it for work, but sunlight doesnt have these properties.

TLDR: it is not just the sunlight, but also lack of it in some places

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  • $\begingroup$ That's interesting. I have a question though: a single plant relies on the sunlight it absorbs. The spatial variation of sunlight is much larger than the plant's size. So maybe do temporal variations also contribute to the low entropy? Or does light really have properties such as Gibbs or Helmholtz free energy? $\endgroup$
    – eeqesri
    Feb 6, 2022 at 9:51
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    $\begingroup$ Free energy that can be extracted from sunlight on earth is about 800w/m2 when sun is in zenith, momentarily. Thats from difference of the sun and sky around it, the light itself. Day-night cycle can add some 100w/m2 or so. There is no simple way to calculate it like gibbs energy, especially if you add day-night cycle - it is really far from optics at this point. You will need to go down to 'how much work can be extracted from this by the best machine possible?' $\endgroup$ Feb 6, 2022 at 10:04
  • $\begingroup$ Nice answer! I think it may be worth expanding on the idea that the entropy of the sunlight is low because the frequency is higher than waste radiation from Earth. There's nothing intrinsically low entropy about high frequencies: if we put the sun in an insulated box, the sun would be in thermal equilibrium and so maximize its entropy, with the same Planck spectrum we observe. I think what you're getting at is that the surface of the Sun is hotter than the Earth, and therefore radiation from the Sun heats up the Earth. The absorption/reemission leads to an increase in entropy, like you said. $\endgroup$
    – Andrew
    Feb 7, 2022 at 2:55
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The surface of the sun is very hot. Not as hot as the interior, but still hot, about $T_{sun}$ = 6000°K

The photons each have a large individual energy, and their spectrum almost fits the "black body" spectrum. The entropy associated at a given quantity of energy $E$ received from the sun is $$S_{in}= E/T_{sun}$$

Most of this energy is sent back to outer space by infrared radiation form the Earth. During the day, one receives more, during the night more is sent back. But day by day, or at least year after year, almost all the energy received from the sun is reemitted in the form of infrared radiation, to outer space. The Earth is not a black body, and not even at the same temperature all around. However, very roughly, one can estimate the mean temperature of Earth at about $T_{Earth}=300$°K (that would be 27°C, but as I say, I am aiming for a very rough estimate).

The entropy carried away, to outer space, by the infrared radiation is, as I wrote above, essentially the energy received from the sun, divided by this rough estimate $$S_{out}= E/T_{Earth}$$ this is about 20 times larger than the entropy received from the sun, because Earth temperature is about 1/20th of the temperature of the sun. This "entropy sink" is whats allow life on Earth.

All phenomena, weather, life, everything, produce entropy. If it was not for the evacuation of entropy to outer space, we would reach the maximum of entropy compatible with the energy present on Earth and everything would stop.

The arrival of "low-entropy" energy form the sun, and departure of the same amount of "high-entropy" infrared radiation is what allows life.

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  • $\begingroup$ "The entropy associated at a given quantity of energy E received from the sun is Sin=E/Tsun" No, that's not how entropy works. That is the entropy that the Sun loses by radiating that energy. You can't equate the entropy that the Earth receives with the entropy that the Sun loses. That the Earth gains more entropy than the Sun loses is exactly what gives rise to the Second Law of Thermodynamics, in which the total entropy always increases. $\endgroup$ Feb 6, 2022 at 21:18
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    $\begingroup$ "almost all the energy received from the sun is reemitted in the form of infrared radiation, to outer space. " Actually, the amount of heat that the Earth emits is larger than the heat that it receives from the sun, because of the radioactive elements in its core. $\endgroup$ Feb 6, 2022 at 21:18
  • $\begingroup$ @Acccumulation About energy coming from the core you are right. But it is paltry compared to the energy of the sun. I said the factor of 20 is just a rough estimate, not that it is exact. The main sink of entropy is the heat coming from the sun and radiated in the infrared. $\endgroup$
    – Alfred
    Feb 6, 2022 at 22:54
  • $\begingroup$ @Acccumulation Re: the Second Law of Thermodynamics. You are wrong It does not say that entropy always increases. What it says is that entropy never decreases. But entropy can stay constant. Or, if not strictly constant, at least almost constant. Once more, the increase in entropy of sunlight between leaving the sun and reaching Earth is not large. The factor is still of the order of 20. $\endgroup$
    – Alfred
    Feb 6, 2022 at 22:57
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I don't think it's quite right to say that sunlight has low entropy. Sunlight has very high entropy. Or, put another way, it has very low negentropy, where negentropy is how much entropy in a system can be increased. But the weird thing about negentropy is that it's superadditive. That is, the negentropy that two systems have together is larger than the sum of their individual negentropies. So while neither sunlight nor the Earth have large negentropy individually, together they do. So sunlight adds negentropy to the Earth, even though its intrinsic negentropy is very low.

The Earth is around 300 degrees Celsius. Sunlight comes from the surface of the sun, which around 6000 degrees Kelvin. So, sunlight allows the sun to thermodynamically interact with the Earth through radiation, and the large difference in temperature means large negentropy; heat flowing from hot bodies to colder ones creates opportunity for negentropy. If the surface of the Earth were also 6000 degrees Kelvin, sunlight wouldn't add negentropy.

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