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Derivation

Quantum version

The fluctuation-dissipation theorem relates the correlation function of the observable of interest $\langle {\hat {x}}(t){\hat {x}}(0)\rangle$ (a measure of fluctuation) to the imaginary part of the response function ${\text{Im}}\left[\chi (t)\right]={\frac {1}{2i}}\left[\chi (t)-\chi (-t)\right]$ (a measure of dissipation), in the frequency domain. A link between these quantities can be found through the so-called Kubo formula

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

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

${\text{Im}}\left[\chi (t)\right]=-{\frac {1}{2}}\left[\langle {\hat {x}}(t){\hat {x}}(0)\rangle +\langle {\hat {x}}(0){\hat {x}}(t)\rangle \right]$

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

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

where in the second equality we re-positioned ${\hat {x}}(t)$ using the cylic property of trace (in this step we have also assumed that the operator ${\hat {x}}$ is bosonic, i.e. does not introduce a sign change under permutation). Next, in the third equality, we inserted $e^{-\beta {\hat {H}}}e^{\beta {\hat {H}}}$ next to the trace and interpreted $e^{-\beta {\hat {H}}}$ as a time evolution operator $e^{-{\frac {i}{\hbar }}{\hat {H}}\Delta t}$ with imaginary time interval $\Delta t=-i\hbar \beta$. We can then Fourier transform the imaginary part of the response function above to arrive at the quantum fluctuation-dissipation relation

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

where $S_{x}(\omega )$ is the Fourier transform of $\langle {\hat {x}}(t){\hat {x}}(0)\rangle$ and $n_{\rm {BE}}(\omega )=\left(e^{\beta \hbar \omega }-1\right)^{-1}$ is the Bose-Einstein distribution function. The "$+1/2$" term can be thought of as due to quantum fluctuations. At high enough temperatures, $n_{\rm {BE}}\approx (\beta \hbar \omega )^{-1}\gg 1$, i.e. the quantum contribution is negligible, and we recover the classical version.

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

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

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.

Mathematically, the motion of a particle of mass $m$ undergoing a classical Brownian motion in one dimension is governed by the Langevin equation

$m{\frac {dv}{dt}}+\gamma v=F(t)$

where $v$ is the velocity of the particle, $\gamma v$ is a frictional force and $F(t)$ is a random force with the property $\langle F(t)\rangle =0$, both of these forces are due to the particle's interaction with a thermal bath. In this example, the observable of interest is the velocity $v$ of the particle and to demonstrate the fluctuation-dissipation theorem, we determine both sides of the relation separately. The response function can be found immediately by Fourier transforming the Langevin equation

$\chi (\omega )={\frac {i\omega }{\gamma -im\omega }}$

from which we can extract its imaginary part

${\frac {2k_{\rm {B}}T}{\omega }}{\text{Im}}\left[\chi (\omega )\right]={\frac {2\gamma k_{\rm {B}}T}{\gamma ^{2}+m^{2}\omega ^{2}}}$

This will be the dissipation side of the fluctuation-dissipation relation. Multiplying the Langevin equation by $v(t')$ and taking the ensemble average, we find

${\frac {d}{dt}}\langle v(t)v(t')\rangle =-{\frac {\gamma }{m}}\langle v(t)v(t')\rangle +\underbrace {\langle F(t)v(t')\rangle } _{=0}$

The last term drops because the random force $F(t)$ is independent of $v(t')$. Solving the remaining equation and using time-translational invariance to shift $\langle v(t)v(t')\rangle \rightarrow \langle v(t)v(0)\rangle$, we obtain

$\langle v(t)v(0)\rangle =\langle v(0)^{2}\rangle e^{-{\frac {\gamma }{m}}t}$

Fourier transforming this correlation function and applying the equipartition theorem for a single particle in one dimension

${\frac {1}{2}}m\langle v(0)^{2}\rangle ={\frac {1}{2}}k_{\rm {B}}T$

we obtain the fluctuation side of the fluctuation-dissipation relation

$S_{v}(\omega )={\frac {2\gamma k_{\rm {B}}T}{\gamma ^{2}+m^{2}\omega ^{2}}}$

demonstrating the validity of the fluctuation-dissipation theorem.

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 $R$, $k_{B}T$, and the bandwidth $\Delta \nu$ over which the voltage is measured :

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

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 $R$ and a capacitor with a small capacitance $C$. Kirchhoff's law yields

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

and so the response function for this circuit is

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

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

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

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

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

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

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

Violations in glassy systems

While the fluctuation–dissipation theorem provides a general relation between the response of systems obeying detailed balance, when detailed balance is violated comparison of fluctuations to dissipation is more complex. Below the so called glass temperature $T_{\rm {g}}$, glassy systems are not equilibrated, and slowly approach their equilibrium state. This slow approach to equilibrium is synonymous with the violation of detailed balance. Thus these systems require large time-scales to be studied while they slowly move toward equilibrium.

File:Violation of FDT1.png

Violation of the fluctuation-dissipation theorem (FDT) in the $D=3$ Edwards-Anderson system with temperature $T$, magnetic susceptibility $\chi$, and spin-temporal correlation function $C$. Figure replotted from .

To study the violation of the fluctuation-dissipation relation in glassy systems, particularly spin glasses, Ref. performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional Edwards-Anderson model using supercomputers. In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature $T=0.64T_{\rm {g}}$ below the glass temperature $T_{g}$, and left to equilibrate for a very long time $t_{\rm {w}}$ under a magnetic field $H$. Then, at a later time $t+t_{\rm {w}}$, two dynamical observables are probed, namely the response function

$\chi (t+t_{\rm {w}},t_{\rm {w}})\equiv \left.{\frac {\partial m(t+t_{\rm {w}})}{\partial H}}\right|_{H=0}$

and the spin-temporal correlation function

$C(t+t_{\rm {w}},t_{\rm {w}})\equiv {\frac {1}{V}}\left.\sum _{x}\langle S_{x}(t_{\rm {w}})S_{x}(t+t_{\rm {w}})\rangle \right|_{H=0}$

where $S_{x}=\pm 1$ is the spin living on the node $x$ of the cubic lattice of volume $V$, and $m(t)\equiv {\frac {1}{V}}\sum _{x}\langle S_{x}(t)\rangle$ is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as

$T\chi (t+t_{\rm {w}},t_{\rm {w}})=1-C(t+t_{\rm {w}},t_{\rm {w}})$

Their results confirm the expectation that as the system is left to equilibrate for longer times, the fluctuation-dissipation relation is closer to be satisfied.

In the mid-1990s, in the study of dynamics of spin glass models, a generalization of the fluctuation–dissipation theorem was discovered that holds for asymptotic non-stationary states, where the temperature appearing in the equilibrium relation is substituted by an effective temperature with a non-trivial dependence on the time scales. This relation is proposed to hold in glassy systems beyond the models for which it was initially found.

Applications

Early Universe Cosmology

In the paradigm of the widely successful Big Bang theory, the Universe started out at an extremely high temperature and gradually cooled down as it expanded. In the process, the temperature of the Universe occasionally went pass various important energy scales such as the Quantum Chromodynamics scale, triggering various phase transitions driving (part of) the Universe out of thermal equilibrium. The evolution of the Universe as it relaxed back to thermal equilibrium following these cosmological phase transitions is governed by some dissipation which inevitably comes with the accompanying fluctuations .