Typically, in the derivation used for resistor noise, I see a hypothetical voltage source (the equivalent noise source) with a source resistance of R, connected to a transmission line of characteristic impedance R, terminated in a resistor of value R. So in the below figure, Vg is the hypothetical equivalent voltage noise, Zg is R, and Zf is also R.

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We can then argue that this transmission line, being a transmission line, has electromagnetic modes that can be occupied by a number of photons. The occupancy of a mode of frequency $\omega$ is just given by the Bose-Einstein occupancy factor (photons are bosons) $$\frac{1}{e^{\hbar \omega / \tau}-1}$$ where $\tau=k_B T$, and $k_B$ is Boltzmann's constant. At high temperatures, this occupancy factor tends to $\hbar \omega / \tau$, and the average energy per mode is just equal to $\tau$. This all makes sense to me. Where things start to get confusing is usually people then start talking about the modes of the transmission line. They usually make a boundary conditions argument, that the field must go to zero at the boundaries, in which case you'll get the modes have frequencies $f=\frac{v}{2L}*m$, where v is the velocity of the mode (nearly the speed of light), L is the length of the transmission line, and m is a positive integer. You then interpret the frequency spacing of the modes as the bandwidth, remember to account for the initial voltage division on the line, and viola, you have that the noise voltage power is $4k_B T R \Delta f$.

There's a couple of things that bug me about this. First of all, for a matched line, modes wouldn't exist on the line. We can't apply boundary conditions, as the line isn't shorted or open at either end - it's perfectly matched. Applying these boundary conditions are inconsistent with the model used. Traveling waves going either direction are certainly allowed, but these don't have a particular mode spacing, as any value of $f$ would be allowed. Second, even if this was a valid thing to do, most physical resistors have dimensions of ~mm, which makes the first mode frequency in the ~GHz, but Nyquists' formula works well down to <Hz. This tells me that the transmission line argument is somewhat hacky and doesn't make sense. In order for it to work we need to take $L \rightarrow \infty$, but this isn't justified because resistors can be as physically small as we please and this should still work. $L$ is clearly not infinity. I get that the transmission line is meant to be an intermediate step, but it's an intermediate step I can't see having any physical justification.

A similar argument is made in the context of blackbody radiation in 3D (an artificial cavity is introduced which is then made to be the size of the universe), and for semiconductors, but this argument does seem to hold water because usually we can take the cavity to be the actual universe. What am I missing here?


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This transmission line argument appears to originate with this paper: https://journals.aps.org/pr/pdf/10.1103/PhysRev.32.110

Some important points. First, to quote the paper: "At any instant after equilibrium has been established, let the line be isolated from the conductors, say, by the application of short circuits at the two ends."

So the boundary condition issues you raise are dealt with. The point is that if sudden cuts are made then the thermal equilibrium that already existed should not be ruined (at least not by much). There is possibly something fishy here as sudden boundary condition changes can cause excitations (heat), but I think we can expect these effects to be small.

The next point is that the physical constraints of the transmission lines we are used to (bandwidth, size) are not really relevant to the thought experiment. We can just assume the transmission line is as small as we like and has as much bandwidth as we like. If you wanted to attack the validity of the argument on these grounds you would need to prove that certain transmission line properties are in-principle impossible. It would seem very weird if thermodynamics was held hostage by the limits of transmission line engineering.

Finally, the paper linked hedges its bets slightly. It appears to acknowledge near the end that the transmission line argument, while simple and direct, might not be fully convincing. It suggests that a more convincing argument could be developed. This proposal involves a piston of hot gas connected to a diaphragm telephone-receiver linked to the resistor. Here the electrical noise in the resistor moves the diaphragm (heating up the gas in the piston), while the gas in the piston hits against the diaphragm, which creates electric current that heats the resistor. At some point an equilibrium is found where the resistor temperature matches the gas temperature.

Given all this I think that the transmission line argument should be taken as one of those arguments that is not completely airtight (somewhat hacky as you say). But it is simple, and it turns out to get the right answer.


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