Editing Fluctuation–dissipation theorem (section)

===Thermal noise in a resistor===
In 1928, [[John Bertrand Johnson|John B. Johnson]] discovered and [[Harry Nyquist]] explained [[Johnson–Nyquist noise]]. With no applied current, the mean-square voltage depends on the resistance <math>R</math>, <math>k_\text{B} T</math>, and the bandwidth <math>\Delta\nu</math> over which the voltage is measured:<ref name="Blundell2009">{{cite book |last1=Blundell |first1=Stephen J. |last2=Blundell |first2=Katherine M. |title=Concepts in thermal physics |date=2009 |publisher=OUP Oxford}}</ref>
<math display="block">
 \langle V^2 \rangle \approx 4Rk_\text{B}T\,\Delta\nu.
</math>

[[File:JohnsonThermalNoise.png|thumb|A simple circuit for illustrating Johnson–Nyquist thermal noise in a resistor]]

This observation can be understood through the lens of the fluctuation-dissipation theorem. Take, for example, a simple circuit consisting of a [[resistor]] with a resistance <math>R</math> and a [[capacitor]] with a small capacitance <math>C</math>. [[Kirchhoff's circuit laws|Kirchhoff's]] voltage law yields
<math display="block">
 V = -R\frac{dQ}{dt}+\frac{Q}{C},
</math>
and so the [[response function]] for this circuit is
<math display="block">
 \chi(\omega) \equiv \frac{Q(\omega)}{V(\omega)} = \frac{1}{\frac{1}{C} - i\omega R}.
</math>

In the low-frequency limit <math>\omega \ll (RC)^{-1}</math>, its imaginary part is simply
<math display="block">
 \operatorname{Im}\left[\chi(\omega)\right] \approx \omega RC^2,
</math>
which then can be linked to the power spectral density function <math>S_V(\omega)</math> of the voltage via the fluctuation-dissipation theorem:
<math display="block">
 S_V(\omega) = \frac{S_Q(\omega)}{C^2} \approx \frac{2k_\text{B}T}{C^2\omega} \operatorname{Im}\left[\chi(\omega)\right] = 2Rk_\text{B}T.
</math>

The Johnson–Nyquist voltage noise <math>\langle V^2 \rangle</math> was observed within a small frequency [[bandwidth (signal processing)|bandwidth]] <math>\Delta \nu = \Delta\omega/(2\pi)</math> centered around <math>\omega=\pm \omega_0</math>. Hence
<math display="block">
 \langle V^2 \rangle \approx S_V(\omega) \times 2\Delta \nu \approx 4Rk_\text{B}T\Delta \nu.
</math>