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In Nyquist's 1928 paper, Thermal agitation of electric charge in conductors, the voltage noise for a resistor is derived assuming a circuit in thermal equilibrium and one temperature.

How does the thermal noise of a resistor change when there is a temperature gradient? For example, a circuit that has different resistors at extremely different temperatures, or one long resistive element that has a temperature gradient across the whole thing?

Nyquist looks at a circuit like this one

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where both $R1$ and $R2$ are at the same temperature $T$. Each noisy resistor is modeled as a noisless resistor in series with a noise source so that they can be considered one at a time.

Nyquist uses the fact that the circuit is in thermal equilibrium twice in his derivation: once to argue that the power from $R1$ to $R2$ is equal to the power from $R2$ to $R1$, and once to use the equipartition theorem and assign an energy $k_BT/2$ to each degree of freedom of each mode in a transmission line between the two resistors.

If the temperature of the resistors are different, can you really say that the power from each is the same? If $R1$ and $R2$ are at different temperatures and not at thermal equilibrium, then it seems like you can't use the equipartition theorem and the whole derivation falls apart. Is there a different way to treat the case of resistors at different temperatures?

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  • $\begingroup$ Would it be more proper to show the whole derivation and then ask the question about it? I'm just worried it would make for a very long question. $\endgroup$ – C Earnest Oct 28 '14 at 14:34
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"a circuit that has different resistors at extremely different temperatures" -- Each resistor independently puts out its own noise related to its own temperature.

"one long resistive element that has a temperature gradient across the whole thing" -- That's actually the same thing again. Treat it as a large number N of resistors in series, each with resistance of R/N, and each with a slightly different temperature.

UPDATE: I think you misunderstand the role that thermal equilibrium plays in the derivation. Nyquist is trying to answer the question "What is the noise created by a resistor at temperature T?" Then he proposes a thought experiment, a particular circumstance in which the answer to the question is especially obvious. But nevertheless the answer is general. This kind of thing, where you derive a general principle using a specific thought experiment, is common in physics derivations.

The noise voltage created by a resistor is obviously independent of what else is in the circuit, or what temperature that other stuff has. (The consequences of the voltage of course does depend on the whole circuit via Kirchhoff's laws.) Therefore the conclusions of Nyquist's thought experiment can be applied in any other circumstance.

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  • $\begingroup$ Steve, Thanks for the answer. I'm concerned because the derivation requires thermal equilibrium at a fixed temperature, and I'm not sure that it's clear that you can apply it to a different case. Can you maybe provide an argument for why it is still applicable? $\endgroup$ – C Earnest Oct 28 '14 at 14:20

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