# Johnson-Nyquist noise for resistive element with temperature gradient

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

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?

• 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. – C Earnest Oct 28 '14 at 14:34