# Entropy of photons in the Sun / Earth system - Do infrared photons have higher entropy than ultraviolet/visible photons?

The entropy of a photon gas in equilibrium (emitting e.g., black-body radiation; BB) is

$$S \propto V \cdot T^3$$

where $$V$$ is the volume and $$T$$ is the temperature of the gas [see https://en.wikipedia.org/wiki/Photon_gas].

Now, in case of a BB, $$T$$ is linked to the peak frequency of the BB, $$\nu_{\rm peak}$$, according to the Wien's law:

$$\nu_{\rm peak} \propto T$$

[see https://en.wikipedia.org/wiki/Wien%27s_displacement_law].

So, the entropy of a BB in a unit volume is proportional to the to the third power of peak frequency:

$$S \propto \nu_{\rm peak}^3$$.

Thus, from this I understand that, for instance, a BB radiation peaking at visible wavelengths (like the Sun) would have higher entropy than a BB radiation peaking at infrared wavelengths (like the Earth).

However, this looks in contradiction with many arguments saying that the Earth is "powered" by low-entropy photons coming from the Sun, which are absorbed and then irradiated as high-entropy infrared photons [see e.g., https://www.preposterousuniverse.com/blog/2016/11/03/entropy-and-complexity-cause-and-effect-life-and-time/].

Where am I getting wrong?

• I don't think it makes much sense to refer to a single wavelength as having entropy. It's only when you look at the ensemble and see how "packed" the energy is vs. the number of photons that it matters. Jul 6 '20 at 18:16
• @CarlWitthoft So why infrared photons have higher entropy than ultraviolet / visible? Jul 6 '20 at 19:10
• Infrared photons don't have "more entropy" intrinsically, that comparison makes no sense, as Carl says. You do however end up with more photons for the same energy in the infrared rather than higher frequencies, which if you keep the same bandwidth etc. (or have it modify in a way that doesn't completely offset/overwhelm this effect) means you have increased entropy. Jul 7 '20 at 1:49
• Here, for the special case of the entropy of the frequency distribution corresponding to a photon gas in equilibrium, the frequency distribution modification itself with different temperature is causing the effect. Jul 7 '20 at 1:56

Let us demonstrate by showing that re-radiating the energy as BB radiation at $$R-1$$ radii away from the sun's surface results in higher entropy.
At $$R$$ radii away, the entropy density due to the sun's radiation is now: $$S/V=\frac{k_1T^3}{R^2}$$ The energy density is now: $$U=\frac{k_2T^4}{R^2}$$ If we reradiate BB radiation of this energy, the new temperature $$T'$$ follows: $$U=k_2T'^4$$ Which gives $$T'=\frac{T}{\sqrt{R}}$$ Which has an entropy density $$S'=k_1T^3=\frac{T'^3}{{R}^{3/2}}$$ Which is greater than our original entropy density from the sun's radiation, since we are $$R-1>0$$ radii from the sun's surface