Microscopic Derivation of Johnson Noise
We consider a conductor and perceive it as a one-dimensional case of black-body radiation. We consider a transmission line terminated by conductors with resistance R at each end. We also choose the transmission line such that its characteristic impedance is also R.
Voltage waves of the form  propagate down the transmission line. The xcomponent of the wave-vector is kx and the oscillatory frequency ω; the velocity of the waves is given by the expression  .
Imposing periodic boundary conditions upon these voltage waves gives us V(0) = V(L). This constrains the wave-vector to satisfy  for some integer n.
We then consider the density of modes of oscillation
Statistical Mechanics (and the Equipartition Theorem) give us the result that the average energy per mode of oscillation is given by
The energy density per unit frequency U and the power per unit frequency P are then given by the following expressions:
This expression describes the power absorbed by the resistor. This must therefore be equal to the power emitted by the resistor. The force generated by the resistor sets up a current in the transmission line equal to  . Thus, the power absorbed by the other resistor is
Setting this equal to the power emitted, we have an expression which we then integrate over the whole range of frequencies to obtain the Johnson Noise
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