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# ThermalNoise

In 1928 Johnson measured and Nyquist (Phys. Rev. 32, 110) calculated the thermal noise in a circuit. It is the noise due to thermal fluctuation and it produces a non null mean square voltage across any element of a circuit.

Nyquist considered an ideal coaxial cable of characteristic impedance R, length l, closed at both ends on two equal resistors R. Thermal fluctuations generate a random voltage in one resistor, the corresponding signal travels to the other, where it is totally dissipated and vice-versa.

Alternatively one may consider the modes of the line, $\nu_m=\frac c {2l} m$. Here we assume for simplicity that the speed of light in the coaxial is $c=1/\sqrt{\varepsilon_0\mu_0}$ (this quantity drops out, so any lower velocity would do). Consider a broad enough frequency range $\Delta\nu=\frac c {2l} \Delta m$ to contain several modes ($\Delta m\gg 1$). They may be considered as averagely populated double degrees of freedom (one for the magnetic and one for the electric component), each with mean thermal energy $k_B T$. This energy takes $\Delta t = l/c$ to travel to both ends, where it is dissipated on a total resistance 2R, hence the dissipated power is

$P = \frac{\langle V^2\rangle} {2R} = \Delta m \frac {k_B T} {\Delta t} = 2\Delta \nu k_B T$

It follows that $\langle V^2\rangle=4Rk_BT\Delta\nu$. This is the minimum mean square voltage that one measures across any resistor in equilibrium at temperature T, if one carefully avoids any other signal.