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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, . Here we assume for simplicity that the speed of light in the coaxial is (this quantity drops out, so any lower velocity would do). Consider a broad enough frequency range to contain several modes (). 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 . This energy takes to travel to both ends, where it is dissipated on a total resistance 2R, hence the dissipated power is
It follows that . 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.