I came up to an interesting physics problem:
The way, how vacuum tube diodes work:
If their anode is placed in the area of the space charge region (the area, where the cloud of free electrons forms around the cathode), some electrons land on the anode and so form an electric current, even when there is no voltage between the anode and cathode.
Or you need to build up certain negative voltage on the anode, in order to repell the electrons so they wont land there anymore, so the current stops flowing (so anode being negative).
So if the negative voltage on the anode is present, but a bit lower than what cuts the current off, you have a positive anode current (current flowing into anode and out of the cathode, so electrons flowing in the opposite direction), so the diode is in fact an electrical generator. So far nothing wrong, there is just huge (compare to the power you get in that way) heating power required to be supplied externally and the electron emission current is known to take quite some heat away from the cathode (well, at the end the electrons need some energy to become liberated from the material).

The way, how an electron emission is usually explained in books:
When a surface is hot enough, the electrons are moving so fast, so some may just happen to escape the field of the atoms in that solid and so turn into an electron cloud around. The electron speed required, so the temperature, depends on how strongly the electrons are held in the material, in other words what is the energy required to liberate them.
Some materials have low energy barrier, so require lower temperatures, some have higher energy barrier, so require higher temperature to start emitting.
On the other hand a conductor is "always ready" to accept an electron if it happens to land on its surface.
So you may find some combination of materials and temperature, where all parts would be at exactly the same temperature, while one will happily release electrons because of its low energy barrier, while the other wont like to release any because having its energy barrier high.
So in theory the first material should form the cathode, the second an anode of a vacuum diode, although all at the same temperature.
Now if we place the anode into the space charge of the cathode and connect a resistor to them, because being a diode, a current should flow, so some power would have to be delivered to the resistor.

Now if all happens to be a system perfectly thermally insulated from anything around (and also from the infinity space - e.g. by reflective coatings to the outside world and in a perfect vacuum), it should nor gain, nor loose any heat, nor any other form of energy, while all inside starts at exactly the same temperature.
But inside we would have something delivering power to the resistor, so the resistor would have to become hotter than the rest of that system. And the part serving as the cathode would have to be cooled down below the initial temperature of the whole system.
But there is a problem: That would become a device transferring heat from the colder point (the cathode) to the hotter point (the resistor) without any external energy delivered.
And that is what the 2nd law of thermodynamics do not allow (hence the "perpetual motion of a 2nd kind" name for a mythical apparatus claiming to do just that).
So it just does not add up, there should be something wrong, in all the explanations how the electron emission or electron energy balance on a solid surface works.

So could someone knowledgeable in the physics of electrons on solid surfaces in a vacuum cast some clarification?


And no, I do not believe in anything being wrong with the 2nd law of thermodynamics, the title is just meant as a form of clickbait to draw some attention...

Interesting and stimulating topic here! That changes from discussions like "do you like kitchen equipments" or "turn signal sounds". I bet there'll be much less comments here .

The reasoning here is good, you indeed get a continuous flow of heat from the resistor to the cathode - I have nothing to add to your excellent post. This also does not violate the second law of thermodynamics since the entropy of the overall system doesn't change, and at equilibrium the entropy of its components remains also constant since the temperature doesn't change as well. I don't think the second law prohibits a perpetual flow of heat or matter, it only forbids a perpetual extraction of work (or energy) from this system while maintaining a constant entropy.

Perpetual motions and flows are possible and there are actual examples of those, i.e., current flow in superconductor loops, circular flows in superfluid helium, etc. Only, you cannot extract energy/work from those processes without destroying them, and that's why power sources with infinite capacity do not exist.

Well, if you replace the resistor with some external load (on whatever temperature, so e.g. the same or even hotter than the rest) and thermally tightly couple the rest together, what you get is work, so clear violation of the second law of thermodynamics - the entropy of the thing without the resistor does gets reduced.
The statements "heat can not flow from colder to hotter without external source of work" or "can not extract work by reducing entropy" or so  are just variants of the same - any heat engine can transform one to another.

The things about superconductors or superfluids: These are just things of zero losses (current flow withot voltage drop means zero power, the same the fluid flow without any resistance, again no power gain, nor loss), so nothing against anything.
But what I've described would make an apparatus extracting electrical work just from the heat, without transferring the heat onto something colder (so increasing the total entropy there, so as all heat engines do).

And yes, I do expect less comments here, don't worry...