Macroscopic Derivation of Johnson Noise
We consider a conductor with resistance R, length l, cross sectional area A, carrier density n and relaxation time τc. The voltage across the conductor related to the current I, the current density j, the charge on an electron e and the drift velocity  by the following expression:
We consider this average drift velocity:
This gives us a random variable that sums to be the voltage:
Now, we know that the spectral density J(?) satisfies
The correlation function (a measure of the strength of the liner relationship between random variables at different points in time) for this random variable Vi is dependent upon our assumption of the correlation time τc:
Now we substitute our results into the Wiener-Khintchine theorem (which states that for a well behaved stationary random process the power spectrum is equal to the inverse (cosine) Fourier transform of the (auto)correlation function):
This sequence of equalities involves the Fourier transform of the function  which we look up in a table, and the equipartition theorem from statistical mechanics, which gives  . We also make the one approximation for the result of that Fourier transform; this is valid because τ < 10−13 for metals at room temperature; this is valid for frequencies where 2πντc << 1.
We determine the Johnson noise:
Now we use two results from basic concepts in electrical transmission. The term enclosed in parentheses is precisely the conductivity s, the ease with which current density (current per cross sectional area) flows in a material. And, this σ is the inverse of the cross-section resistivity per unit length ρ. Thus, the whole term in brackets is  where R is the macroscopic resistance of the wire. Putting this all together, we have
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