What is Johnson noise? I'm trying to understand Johnson noise and I already read Nyquists paper and did some other reading, but I'm pretty stuck. As I guess there is this missing piece that relates everything, I'll just write down my many points in the hope for an answer that relates everything more or less:


*

*What do we measure if we take a resistor $R$ and measure the thermal voltage with a perfect voltmeter?

*If we take two resistors $R_1$ and $R_2$ at random temperatures and random resistances, connect them and put them into vaccuum. Will their temperatures converge to some temperature $T$?

*If we take again two resistors $R_1,R_2$, but with different resistances. Now put two voltmeter directly besides of each of them and then connect them again. Will both of the voltmeters always show the same voltage? What if the temperature differs at the beginning?

*Is there a general equivalent circuit for a resistor to account for the thermal noise? What would that look like?

*Assuming that in #2 their temperatures converge. Eventually they are pretty close to being in thermal equilibrium and the power transfer in both direction would need to be the same. How would I calculate/show that for $R_1\ne R_2$?


Unfortunately my knowledge about transmission lines is nonexistent, and I didn't understand the part in Nyquist's paper, where he requires the impedance of the transmission line to be $(LC)^{\frac{1}{2}}=R$. How important is this detail?
 A: *

*assuming that your resistor is connected to a lossless ideal flat bandpass filter of bandwidth $W$ then a perfect voltmeter will measure at the output of the filter an average 0, but a perfect rms meter will measure a voltage $v_{rms} = \sqrt{Rk_BTW}$

*if you connect to resistors at different temperatures then they equilibrate, i.e., then their temperatures will be equal after a while.

*see the formula above in 1

*The Thevenin equivalent would be a resistor connected in series with a random voltage source of $v_{rms}$ in 1. There is an obvious analog of this for the Norton equivalent current source.

*Use the Thevenin equivalents and voltage division.

*Nyquist assumes that the transmission line connecting the resistors at either end is such that it is terminated without reflection, thus whatever noise wave one resistor emits at one end is fully absorbed by the other.

A: Johnson noise, like Brownian motion, is an observable effect of temperature in a population of identical particles.
What you measure, with an ideal voltmeter across a resistor,
is the small fluctuations in local charge density at the  terminals
of that resistor, basically the charge on a (very small) parasitic capacitor.

If we take two resistors R1
  R
  1
   and R2
  R
  2
   at random temperatures and random resistances, connect them and put them into vaccuum. Will their temperatures converge...
If we take again two resistors R1,R2,
  but with different resistances. Now put two voltmeter directly besides of each of them and then connect them again....

If you connect two resistors, they DO come to the same temperature,
because electrons flow freely through an electrical connection,
and electrons have temperature (and carry heat).   This is why metals
are usually good conductors of heat, the freely moving electrons
carry heat.   The sum of Johnson voltages (which are random fluctuations) is done in quadrature (i.e. the squared voltages 
add) because there is no phase synchronization possible.   In-phase (zero degrees)
voltages add, out-of-phase (180 degrees) voltages subtract, and
random (orthogonal, 90 degrees average) voltages add in quadrature.
So the mean-square Johnson voltage is proportional to resistance,
and the mean-square Johnson voltage of two resistors in series is
proportional to the sum of the two resistors.   There's no significance to temperature distributions in the resistor(s), it just
means a messier calculation.  EVERY resistor is a series-connected
multiplicity of smaller resistors, each at a temperature.
