Equilibrium between objects at different temperatures?

[Philip Koeck]

I've come across a puzzling thought experiment.

Consider two black bodies surrounded by vacuum.
The surrounding temperature is exactly 0 K.

By some ingenious optical device all the radiation from body 1 is focused onto body 2 and vice versa.

If left alone sufficiently long the two bodies will reach equilibrium.
This means that the heat current emitted by body 1 is equal to the heat current absorbed by body 1. The same is also true for body 2.

According to the assumption made the heat current emitted by body 1 is absorbed by body 2 and vice versa.
This also means that the heat currents emitted by the two bodies are equal.

If we also assume that the surfaces of the two bodies are not the same I would conclude that the temperatures of the two bodies have to be different since the emitted heat is proportional to the surface area and to T⁴.

 

[.Scott]

Small black holes radiate much faster than large black holes. So the equilibrium state will be reached when the smaller BH evaporates.

 

[Philip Koeck]

Small black holes radiate much faster than large black holes. So the equilibrium state will be reached when the smaller BH evaporates.

Black bodies, not holes. You can imagine a cavity that's perfectly black inside and radiation is allowed to exit and enter through an opening.

 

[Steve4Physics]

By some ingenious optical device all the radiation from body 1 is focused onto body 2 and vice versa.

I’m on shaky ground but I’m sure that someone else will be able to correct (or expand upon) this if needed...

I think the system you describe is infeasible because it is not possible to have an (optical) system such that “all the radiation from body 1 is focused onto body 2 and vice versa”.

This would violate conservation of etendue. ‘Etendue’ is a measure of how much light (or radiation in general) ‘spreads-out’. Conservation of etendue can be derived from the second law of thermodynamics.

For example, it tells you that you can’t focus sunlight to such an extent that the energy-density corresponds to a temperature higher than that of the sun’s surface.

Take a look at https://en.wikipedia.org/wiki/Etendue and/or do a search.

 

[Philip Koeck]

I’m on shaky ground but I’m sure that someone else will be able to correct (or expand upon) this if needed...

I think the system you describe is infeasible because it is not possible to have an (optical) system such that “all the radiation from body 1 is focused onto body 2 and vice versa”.

This would violate conservation of etendue. ‘Etendue’ is a measure of how much light (or radiation in general) ‘spreads-out’. Conservation of etendue can be derived from the second law of thermodynamics.

For example, it tells you that you can’t focus sunlight to such an extent that the energy-density corresponds to a temperature higher than that of the sun’s surface.

Take a look at https://en.wikipedia.org/wiki/Etendue and/or do a search.

I was thinking the problem must lie in the assumption about the optical device.

I don't think that I run into problems with the second law however, at least not directly.
When the two objects are in equilibrium there is no net heat transfer between them.

It does seem to violate the zeroth law, though, or at least the usual conclusion that objects in equilibrium have to have the same temperature.

 

[Steve4Physics]

I don't think that I run into problems with the second law however, at least not directly.
When the two objects are in equilibrium there is no net heat transfer between them.

One way of stating the second law of thermodynamics is that the (natural) direction of heat flow is from a hotter object to a colder one. (A consequence of this is that when two objects are at the same temperature, the net heat flow between them is zero.)

I guess the link between the second law and etendue is that they both relate to the flow of radiation (in appropriate contexts).

It does seem to violate the zeroth law, though, or at least the usual conclusion that objects in equilibrium have to have the same temperature.

The zero-th law (as far as I understand it) is about something slightly different. It says if two bodies are both in equilibrium with a third body, the two bodies are in equilibrium with each other. (This allows us to establish the concept of temperature.)

Referring back to original problem, for a system of two black bodies, there should be no doubt that their final temperatures will be equal. Any argument that says otherwise must have one or more mistakes.

 

[Philip Koeck]

One way of stating the second law of thermodynamics is that the (natural) direction of heat flow is from a hotter object to a colder one. (A consequence of this is that when two objects are at the same temperature, the net heat flow between them is zero.)

The usual formulation is that no heat can flow from cold to hot without any other change somewhere.
Some other change could be potential energy being converted into work, for example.
So if no heat flows in effect, I'm not sure if the 2nd law is actually violated.

Another question is of course the process leading to this presumed equilibrium at different temperatures.
That does involve heat flow from cold to hot.

The zero-th law (as far as I understand it) is about something slightly different. It says if two bodies are both in equilibrium with a third body, the two bodies are in equilibrium with each other. (This allows us to establish the concept of temperature.)

I agree. Often textbooks add that one of the bodies could be a thermometer and therefore the 3 bodies have to have the same temperature.
That's what I meant by "conclusion from the 0th law".

Referring back to original problem, for a system of two black bodies, there should be no doubt that their final temperatures will be equal. Any argument that says otherwise must have one or more mistakes.

Yes, probably. The mistake ought to be in the assumption about the optical system.

So, in a way, thermodynamics determines what is technically possible in optics.

 

[Philip Koeck]

I’m on shaky ground but I’m sure that someone else will be able to correct (or expand upon) this if needed...

I think the system you describe is infeasible because it is not possible to have an (optical) system such that “all the radiation from body 1 is focused onto body 2 and vice versa”.

This would violate conservation of etendue. ‘Etendue’ is a measure of how much light (or radiation in general) ‘spreads-out’. Conservation of etendue can be derived from the second law of thermodynamics.

For example, it tells you that you can’t focus sunlight to such an extent that the energy-density corresponds to a temperature higher than that of the sun’s surface.

Take a look at https://en.wikipedia.org/wiki/Etendue and/or do a search.

Not sure if conservation of etendue is the solution to the puzzle.

In the wikipedia page shared above they quote the Brightness theorem, which states that power per unit solid angle and per unit area cannot increase, but that doesn't mean that power per unit area cannot increase.

 

[DrClaude]

By some ingenious optical device all the radiation from body 1 is focused onto body 2 and vice versa.

This is impossible if the bodies have a different finite extent.

Theodore J. Yoder and Gregory S. Adkins, "Resolution of the ellipsoid paradox in thermodynamics", American Journal of Physics 79, 811-818 (2011) https://doi.org/10.1119/1.3596430

Free full text: https://www.researchgate.net/public...on_of_the_ellipsoid_paradox_in_thermodynamics

 

[hutchphd]

Does this help: Perfect thermal conductors. Make one body a sphere and the second body a larger concentric spherical shell. The central sphere is kept at temperature T₀. In steady state the T_{outer shell}<T₀
But if you instead hold outer shell temperatere to T_{outer shell} the inner T₀ will never be larger than T_{outer shell}
I think the generalization is obvious.
Does that help?

/

 

[Lnewqban]

I've come across a puzzling thought experiment.

Consider two black bodies surrounded by vacuum.
The surrounding temperature is exactly 0 K.

By some ingenious optical device all the radiation from body 1 is focused onto body 2 and vice versa.

Could we eliminate the optical device by assuming one body to be a hollow sphere that contains the other one suspended inside?

We could then assume emisivity or out-radiation of the exterior side of the wall of the hollow sphere to be zero, so all the thermal energy remains inside the two-body system, flowing or not.

 

[hutchphd]

It depends upon what you are trying to show. I

 

[Philip Koeck]

Could we eliminate the optical device by assuming one body to be a hollow sphere that contains the other one suspended inside?

We could then assume emisivity or out-radiation of the exterior side of the wall of the hollow sphere to be zero, so all the thermal energy remains inside the two-body system, flowing or not.

I considered that too, but the problem is that a lot of the radiation from the surface of the hollow sphere will go straight to the hollow sphere again and not to the solid sphere in the center.
So it's actually more complicated.
I liked the situation where all the radiation from one body is absorbed by the other and vice versa.

The ellipsoid paradox quoted in post 9 is about exactly such a situation.

 

[Bystander]

It does seem to violate the zeroth law, though, or at least the usual conclusion that objects in equilibrium have to have the same temperature.

Are you trying to evade/avoid the forum policy regarding PPMs?

 

[Philip Koeck]

Are you trying to evade/avoid the forum policy regarding PPMs?

Can't really say. What's a PPM?

 

[DrClaude]

Can't really say. What's a PPM?

A misspelled PMM

(perpetual motion machine)

 

[Philip Koeck]

A misspelled PMM

(perpetual motion machine)

Okay, I see.

I did say it's a puzzling thought experiment and I was trying to locate the wrong assumption.

The paper on the ellipsoid paradox you quoted is really to the point and it shows that this paradox has a long history and isn't trivial at all.

 

[Demystifier]

By some ingenious optical device all the radiation from body 1 is focused onto body 2 and vice versa.

It's not much different from the idea of perpetuum mobile; by some ingenious mechanical device all the heat is transferred to work ... The point is that there is no such device, due to the 2nd law. Light naturally tends to spread (rather than focus), which is a version of the law that entropy naturally tends to increase (rather than decrease). A spread light occupies a larger portion of phase space and hence has larger entropy.