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Consider two different temperature black bodies connected by an ideal circulator (or similar passive and directional device) such that thermal radiation emitted from the cooler black body is able to reach the hotter black body, but thermal radiation from the hotter body is redirected in some other direction and not able to reach the cooler black body (image provided to illustrate system).

Black Body Diagram

If the black bodies were replaced with (ideal) electronic transmitters having identical emission characteristics to the black bodies, there would be no issue with an ideal circulator directionally passing the radiated signals as described. However, the second law of thermodynamics indicates that the system can not function as described when the thermal radiation sources are black bodies, since it implies that the passive circulator is capable of decreasing entropy. Since a circulator is not capable of distinguishing between electromagnetic radiation that originates in a black body or a transmitter, it's not clear how the circulator's behaviour could change to ensure that the second law of thermodynamics isn't violated.

There must be some aspect of the system description, the behaviour of circulators or the second law of thermodynamics that avoids the apparent contradiction. What is it?

This question is an extension of questions 661991 and 136083, but substantively different.

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    $\begingroup$ How exactly does your "passive circulator" work without requiring an external energy source to power it ? $\endgroup$
    – gandalf61
    Aug 31 '21 at 14:51
  • $\begingroup$ To add to @gandalf61 comment, you would also need to show that such a device does, in fact, perfectly transmit energy to the appropriate output and is not outputting some amount of heat into some other heat sink, increasing its entropy $\endgroup$ Aug 31 '21 at 15:46
  • $\begingroup$ @gandalf61 - Passive circulators are an existing real world device, commonly found in many radio installations, so I think it's fair to assume that they exist. Real world circulators are not lossless of course, but I don't think that adding a small amount of loss in the circulator substantively changes the problem. $\endgroup$ Sep 1 '21 at 0:10
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Let us assume the perfect passive circulator.

We do not only have 2 heat baths (blue and red). We have a third, at the bottom of the circulator. It surely has some temperature.

Over time, all three heat baths will reach the same temperature.

Suppose T < B < R. Then B is cooling as it sends radiation to R, and R is cooling as it sends radiation to T, and T is sending less radiation to B which allows B and R to cool because they ultimately contact T.

Suppose B < T < R. Then although B cools towards R, it receives more from T, so B, the coolest blackbody, is warming up in net, as you expect.

Suppose B < R < T. Then B is warming up in net, as you expect, due to T.

The net result of the three bodies coming to equilibrium is an increase in entropy.

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    $\begingroup$ this answers the case of a circulator, as was posed, but not that of a lossless ideal nonreciprocal "isolator". The resolution of the paradox going back to Rayleigh was given by Ishimura here apps.dtic.mil/sti/citations/AD0297019 $\endgroup$
    – hyportnex
    Aug 31 '21 at 15:52
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    $\begingroup$ Stating this a slightly different way, does the system avoid violating the second law of thermodynamics because B is not really 'heating' R, it's just slowing down the rate at which R cools down? $\endgroup$ Sep 1 '21 at 0:18
  • $\begingroup$ That's a fine way to state it. Imagine a simple scenario with a cold block next to a hot block. Atoms from the cold block must occasionally strike slow atoms in the hot block, moving energy from cold to hot. But overall, on average, it is more frequent for atoms from the hot block to bring energy to the cold block. The circulator just makes the net exchange more roundabout. $\endgroup$
    – Alwin
    Sep 1 '21 at 2:26

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