Editing Fluctuation–dissipation theorem (section)

==Examples in detail==
The fluctuation–dissipation theorem is a general result of [[statistical thermodynamics]] that quantifies the relation between the fluctuations in a system that obeys [[detailed balance]] and the response of the system to applied perturbations.

===Brownian motion===
For example, [[Albert Einstein]] noted in his 1905 paper on [[Brownian motion]] that the same random forces that cause the erratic motion of a particle in Brownian motion would also cause drag if the particle were pulled through the fluid. In other words, the fluctuation of the particle at rest has the same origin as the dissipative frictional force one must do work against, if one tries to perturb the system in a particular direction.

From this observation Einstein was able to use [[statistical mechanics]] to derive the [[Einstein–Smoluchowski relation]]
<math display="block">
 D = \mu \, k_\text{B} T,
</math>
which connects the [[Fick's law of diffusion|diffusion constant]] ''D'' and the particle mobility {{mvar|μ}}, the ratio of the particle's [[Terminal velocity|terminal]] [[drift velocity]] to an applied force; {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]], and {{mvar|T}} is the [[absolute temperature]].

===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>