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

${\displaystyle D={\mu \,k_{\rm {B}}T}}$

which connects the diffusion constant D and the particle mobility μ, the ratio of the particle's terminal drift velocity to an applied force. k_B is the Boltzmann constant, and T is the absolute temperature.

Thermal noise in a resistor

In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance ${\displaystyle R}$, ${\displaystyle k_{\rm {B}}T}$, and the bandwidth ${\displaystyle \Delta \nu }$ over which the voltage is measured:^[4]

${\displaystyle \langle V^{2}\rangle \approx 4Rk_{\rm {B}}T\,\Delta \nu .}$

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 ${\displaystyle R}$ and a capacitor with a small capacitance ${\displaystyle C}$. Kirchhoff's voltage law yields

${\displaystyle V=-R{\frac {dQ}{dt}}+{\frac {Q}{C}}}$

and so the response function for this circuit is

${\displaystyle \chi (\omega )\equiv {\frac {Q(\omega )}{V(\omega )}}={\frac {1}{{\frac {1}{C}}-i\omega R}}}$

In the low-frequency limit ${\displaystyle \omega \ll (RC)^{-1}}$, its imaginary part is simply

${\displaystyle {\text{Im}}\left[\chi (\omega )\right]\approx \omega RC^{2}}$

which then can be linked to the power spectral density function ${\displaystyle S_{V}(\omega )}$ of the voltage via the fluctuation-dissipation theorem

${\displaystyle S_{V}(\omega )={\frac {S_{Q}(\omega )}{C^{2}}}\approx {\frac {2k_{\rm {B}}T}{C^{2}\omega }}{\text{Im}}\left[\chi (\omega )\right]=2Rk_{\rm {B}}T}$

The Johnson–Nyquist voltage noise ${\displaystyle \langle V^{2}\rangle }$ was observed within a small frequency bandwidth ${\displaystyle \Delta \nu =\Delta \omega /(2\pi )}$ centered around ${\displaystyle \omega =\pm \omega _{0}}$. Hence

${\displaystyle \langle V^{2}\rangle \approx S_{V}(\omega )\times 2\Delta \nu \approx 4Rk_{\rm {B}}T\Delta \nu }$

Classical version

We derive the fluctuation–dissipation theorem in the form given above, using the same notation. Consider the following test case: the field f has been on for infinite time and is switched off at t=0

${\displaystyle f(t)=f_{0}\theta (-t),}$

where ${\displaystyle \theta (t)}$ is the Heaviside function. We can express the expectation value of ${\displaystyle x}$ by the probability distribution W(x,0) and the transition probability ${\displaystyle P(x',t|x,0)}$

${\displaystyle \langle x(t)\rangle =\int dx'\int dx\,x'P(x',t|x,0)W(x,0).}$

The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution for the Hamiltonian ${\displaystyle H(x)=H_{0}(x)-xf_{0}}$

${\displaystyle W(x,0)={\frac {\exp(-\beta H(x))}{\int dx'\,\exp(-\beta H(x'))}}\,,}$

where ${\displaystyle \beta ^{-1}=k_{\rm {B}}T}$. For a weak field ${\displaystyle \beta xf_{0}\ll 1}$, we can expand the right-hand side

${\displaystyle W(x,0)\approx W_{0}(x)[1+\beta f_{0}(x(0)-\langle x\rangle _{0})],}$

here ${\displaystyle W_{0}(x)}$ is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for ${\displaystyle \langle x(t)\rangle }$ yields

${\displaystyle \langle x(t)\rangle =\langle x\rangle _{0}+\beta f_{0}A(t),}$

(*)

where A(t) is the auto-correlation function of x in the absence of a field:

${\displaystyle A(t)=\langle [x(t)-\langle x\rangle _{0}][x(0)-\langle x\rangle _{0}]\rangle _{0}.}$

Note that in the absence of a field the system is invariant under time-shifts. We can rewrite ${\displaystyle \langle x(t)\rangle -\langle x\rangle _{0}}$ using the susceptibility of the system and hence find with the above equation (*)

${\displaystyle f_{0}\int _{0}^{\infty }d\tau \,\chi (\tau )\theta (\tau -t)=\beta f_{0}A(t)}$

Consequently,

${\displaystyle -\chi (t)=\beta {dA(t) \over dt}\theta (t).}$

(**)

To make a statement about frequency dependence, it is necessary to take the Fourier transform of equation (**). By integrating by parts, it is possible to show that

${\displaystyle -{\hat {\chi }}(\omega )=i\omega \beta \int _{0}^{\infty }e^{-i\omega t}A(t)\,dt-\beta A(0).}$

Since ${\displaystyle A(t)}$ is real and symmetric, it follows that

${\displaystyle 2\operatorname {Im} [{\hat {\chi }}(\omega )]=-\omega \beta {\hat {A}}(\omega ).}$

Finally, for stationary processes, the Wiener–Khinchin theorem states that the two-sided spectral density is equal to the Fourier transform of the auto-correlation function:

${\displaystyle S_{x}(\omega )={\hat {A}}(\omega ).}$

Therefore, it follows that

${\displaystyle S_{x}(\omega )=-{\frac {2k_{\text{B}}T}{\omega }}\operatorname {Im} [{\hat {\chi }}(\omega )].}$

Quantum version

The fluctuation-dissipation theorem relates the correlation function of the observable of interest ${\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }$ (a measure of fluctuation) to the imaginary part of the response function ${\displaystyle {\text{Im}}\left[\chi (\omega )\right]=\left[\chi (\omega )-\chi ^{*}(\omega )\right]/2i}$ in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called Kubo formula^[6]

${\displaystyle \chi (t-t')={\frac {i}{\hbar }}\theta (t-t')\langle [{\hat {x}}(t),{\hat {x}}(t')]\rangle }$

which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable ${\displaystyle \langle {\hat {x}}(t)\rangle }$ in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as

${\displaystyle {\text{Im}}\left[\chi (\omega )\right]={\frac {1}{2\hbar }}\int _{-\infty }^{+\infty }\langle {\hat {x}}(t){\hat {x}}(0)-{\hat {x}}(0){\hat {x}}(t)\rangle e^{i\omega t}dt.}$

In the canonical ensemble, the second term can be re-expressed as

${\displaystyle \langle {\hat {x}}(0){\hat {x}}(t)\rangle ={\text{Tr }}e^{-\beta {\hat {H}}}{\hat {x}}(0){\hat {x}}(t)={\text{Tr }}{\hat {x}}(t)e^{-\beta {\hat {H}}}{\hat {x}}(0)={\text{Tr }}e^{-\beta {\hat {H}}}\underbrace {e^{\beta {\hat {H}}}{\hat {x}}(t)e^{-\beta {\hat {H}}}} _{{\hat {x}}(t-i\hbar \beta )}{\hat {x}}(0)=\langle {\hat {x}}(t-i\hbar \beta ){\hat {x}}(0)\rangle }$

where in the second equality we re-positioned ${\displaystyle {\hat {x}}(t)}$ using the cyclic property of trace. Next, in the third equality, we inserted ${\displaystyle e^{-\beta {\hat {H}}}e^{\beta {\hat {H}}}}$ next to the trace and interpreted ${\displaystyle e^{-\beta {\hat {H}}}}$ as a time evolution operator ${\displaystyle e^{-{\frac {i}{\hbar }}{\hat {H}}\Delta t}}$ with imaginary time interval ${\displaystyle \Delta t=-i\hbar \beta }$. The imaginary time shift turns into a ${\displaystyle e^{-\beta \hbar \omega }}$ factor after Fourier transform

${\displaystyle \int _{-\infty }^{+\infty }\langle {\hat {x}}(t-i\hbar \beta ){\hat {x}}(0)\rangle e^{i\omega t}dt=e^{-\beta \hbar \omega }\int _{-\infty }^{+\infty }\langle {\hat {x}}(t){\hat {x}}(0)\rangle e^{i\omega t}dt}$

and thus the expression for ${\displaystyle {\text{Im}}\left[\chi (\omega )\right]}$ can be easily rewritten as the quantum fluctuation-dissipation relation ^[7]

${\displaystyle S_{x}(\omega )=2\hbar \left[n_{\rm {BE}}(\omega )+1\right]{\text{Im}}\left[\chi (\omega )\right]}$

where the power spectral density ${\displaystyle S_{x}(\omega )}$ is the Fourier transform of the auto-correlation ${\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }$ and ${\displaystyle n_{\rm {BE}}(\omega )=\left(e^{\beta \hbar \omega }-1\right)^{-1}}$ is the Bose-Einstein distribution function. The same calculation also yields

${\displaystyle S_{x}(-\omega )=e^{-\beta \hbar \omega }S_{x}(\omega )=2\hbar \left[n_{\rm {BE}}(\omega )\right]{\text{Im}}\left[\chi (\omega )\right]\neq S_{x}(+\omega )}$

thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, ${\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }$ has an imaginary part originating from the commutation rules of operators.^[8] The additional "${\displaystyle +1}$" term in the expression of ${\displaystyle S_{x}(\omega )}$ at positive frequencies can also be thought of as linked to spontaneous emission. An often cited result is also the symmetrized power spectral density

${\displaystyle {\frac {S_{x}(\omega )+S_{x}(-\omega )}{2}}=2\hbar \left[n_{\rm {BE}}(\omega )+{\frac {1}{2}}\right]{\text{Im}}\left[\chi (\omega )\right]=\hbar \coth \left({\frac {\hbar \omega }{2k_{B}T}}\right){\text{Im}}\left[\chi (\omega )\right].}$

The "${\displaystyle +1/2}$" can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable ${\displaystyle {\hat {x}}}$. At high enough temperatures, ${\displaystyle n_{\rm {BE}}\approx (\beta \hbar \omega )^{-1}\gg 1}$, i.e. the quantum contribution is negligible, and we recover the classical version.