Editing Fluctuation–dissipation theorem

{{short description|Statistical physics theorem}}
The '''fluctuation–dissipation theorem''' ('''FDT''') or '''fluctuation–dissipation relation''' ('''FDR''') is a powerful tool in [[statistical physics]] for predicting the behavior of systems that obey [[detailed balance]]. Given that a system obeys detailed balance, the theorem is a proof that [[thermal fluctuations|thermodynamic fluctuations]] in a physical variable  predict the response quantified by the [[admittance]] or [[Electrical impedance|impedance]] (in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, etc.), and vice versa. The fluctuation–dissipation theorem applies both to [[classical physics|classical]] and [[quantum mechanics|quantum mechanical]] systems.

The fluctuation–dissipation theorem was proven by [[Herbert Callen]] and [[Theodore A. Welton|Theodore Welton]] in 1951<ref name=Callen-Welton>{{cite journal |author1 = H.B. Callen |author2 = T.A. Welton |authorlink1 = Herbert Callen |authorlink2 = Theodore A. Welton | year = 1951 | title = Irreversibility and Generalized Noise | journal = [[Physical Review]] | volume = 83 |issue = 1 | pages = 34&ndash;40 | bibcode = 1951PhRv...83...34C | doi = 10.1103/PhysRev.83.34 }}</ref>
and expanded by [[Ryogo Kubo]]. There are antecedents to the general theorem, including [[Albert Einstein|Einstein]]'s explanation of [[Brownian motion]]<ref>
{{cite journal 
 | last = Einstein 
 | first = Albert 
 | author-link = Albert Einstein 
 | title = Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen 
 | journal = [[Annalen der Physik]] 
 | volume = 322 
 | issue = 8 
 | pages = 549–560 
 | date=May 1905
 | doi = 10.1002/andp.19053220806
 | bibcode = 1905AnP...322..549E
 | url = http://sedici.unlp.edu.ar/handle/10915/2785 
 | doi-access = free 
 }}
</ref>
during his ''[[annus mirabilis]]'' and [[Harry Nyquist]]'s explanation in 1928 of [[Johnson–Nyquist noise|Johnson noise]] in electrical resistors.<ref>{{cite journal | author = Nyquist H |authorlink = Harry Nyquist | year = 1928 | title = Thermal Agitation of Electric Charge in Conductors | journal = [[Physical Review]] | volume = 32 |issue = 1 | pages = 110&ndash;113 | bibcode = 1928PhRv...32..110N | doi = 10.1103/PhysRev.32.110 }}</ref>

==Qualitative overview and examples==

The fluctuation–dissipation theorem says that when there is a process that dissipates energy, turning it into heat (e.g., friction), there is a reverse process related to [[thermal fluctuation]]s. This is best understood by considering some examples:
; [[Drag (physics)|Drag]] and [[Brownian motion]]: If an object is moving through a fluid, it experiences [[Drag (physics)|drag]] (air resistance or fluid resistance). Drag dissipates kinetic energy, turning it into heat. The corresponding fluctuation is [[Brownian motion]]. An object in a fluid does not sit still, but rather moves around with a small and rapidly changing velocity, as molecules in the fluid bump into it. Brownian motion converts heat energy into kinetic energy—the reverse of drag.
; [[Electrical resistance and conductance|Resistance]] and [[Johnson noise]]: If electric current is running through a wire loop with a [[resistor]] in it, the current will rapidly go to zero because of the resistance. Resistance dissipates electrical energy, turning it into heat ([[Joule heating]]). The corresponding fluctuation is [[Johnson noise]]. A wire loop with a resistor in it does not actually have zero current, it has a small and rapidly fluctuating current caused by the thermal fluctuations of the electrons and atoms in the resistor. Johnson noise converts heat energy into electrical energy—the reverse of resistance.
; [[Absorption (electromagnetic radiation)|Light absorption]] and [[thermal radiation]]: When light impinges on an object, some fraction of the light is absorbed, making the object hotter. In this way, light absorption turns light energy into heat. The corresponding fluctuation is [[thermal radiation]] (e.g., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. Indeed, [[Kirchhoff's law of thermal radiation]] confirms that the more effectively an object absorbs light, the more thermal radiation it emits.

==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>

==General formulation==
The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:{{Citation needed|date=October 2010}}.

Let <math>x(t)</math> be an [[observable]] of a [[dynamical system]] with [[Hamiltonian mechanics|Hamiltonian]] <math>H_0(x)</math> subject to thermal fluctuations.
The observable <math>x(t)</math> will fluctuate around its mean value <math>\langle x \rangle_0</math> with fluctuations characterized by a [[power spectrum]] <math>S_x(\omega) = \langle \hat{x}(\omega)\hat{x}^*(\omega) \rangle</math>.
Suppose that we can switch on a time-varying, spatially constant field <math>f(t)</math> which alters the Hamiltonian to <math>H(x) = H_0(x) - f(t)x</math>.
The response of the observable <math>x(t)</math> to a time-dependent field <math>f(t)</math> is 
characterized to first order by the [[Linear response function|susceptibility]] or [[linear response function]]
<math>\chi(t)</math> of the system
<math display="block">
 \langle x(t) \rangle = \langle x \rangle_0 + \int_{-\infty}^t f(\tau) \chi(t - \tau)\,d\tau,
</math>
where the perturbation is adiabatically (very slowly) switched on at <math>\tau = -\infty.</math>

The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of <math>x</math> to the imaginary part of the [[Fourier transform]] <math>\hat{\chi}(\omega)</math> of the susceptibility <math>\chi(t)</math>:
<math display="block">
 S_x(\omega) = -\frac{2 k_\text{B} T}{\omega} \operatorname{Im}\hat{\chi}(\omega),
</math>
which holds under the Fourier transform convention <math>f(\omega)  =\int_{-\infty}^\infty f(t) e^{-i\omega t}\, dt</math>. The left-hand side describes ''fluctuations'' in <math>x</math>, the right-hand side is closely related to the energy ''dissipated'' by the system when pumped by an oscillatory field <math>f(t) = F \sin(\omega t + \phi)</math>. The spectrum of fluctuations reveal the linear response, because past fluctuations cause future fluctuations via a linear response upon itself.

This is the classical form of the theorem; quantum fluctuations are taken into account by replacing <math>2 k_\text{B} T / \omega</math> with <math>\hbar \, \coth(\hbar\omega / 2k_\text{B}T)</math> (whose limit for <math>\hbar \to 0</math> is <math>2 k_\text{B} T/\omega</math>). A proof can be found by means of the [[LSZ reduction]], an identity from quantum field theory.{{Citation needed|date=August 2013}}

The fluctuation–dissipation theorem can be generalized in a straightforward way to the case of space-dependent fields, to the case of several variables or to a quantum-mechanics setting.<ref name=Callen-Welton/>

==Derivation==
===Classical version===
We derive the fluctuation–dissipation theorem in the form given above, using the same notation.
Consider the following test case: the field {{math|''f''}} has been on for infinite time and is switched off at {{math|1=''t'' = 0}}

<math display="block"> f(t) = f_0 \theta(-t) , </math>
where <math> \theta(t)</math> is the [[Heaviside function]].
We can express the expectation value of {{mvar|x}} by the probability distribution {{math|''W''(''x'',0)}} and the transition probability <math> P(x',t | x,0) </math>

<math display="block"> \langle x(t) \rangle = \int dx' \int dx \, x' P(x',t|x,0) W(x,0) . </math>

The probability distribution function ''W''(''x'',0) is an equilibrium distribution and hence given by the [[Boltzmann distribution]] for the Hamiltonian <math> H(x) = H_0(x) - x f_0 </math>

<math display="block"> W(x,0)= \frac{\exp(-\beta H(x))}{\int dx' \, \exp(-\beta H(x'))} \,, </math>
where <math>\beta^{-1} = k_\text{B}T</math>.
For a weak field <math> \beta x f_0 \ll 1 </math>, we can expand the right-hand side

<math display="block"> W(x,0) \approx W_0(x) [1+\beta f_0 (x-\langle x \rangle_0)], </math>

here <math> W_0(x) </math> is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for <math> \langle x(t) \rangle </math> yields
{{NumBlk||<math display="block">\langle x(t) \rangle = \langle x \rangle_0 + \beta f_0 A(t),</math>|{{EquationRef|1}}}}
where ''A''(''t'') is the auto-correlation function of ''x'' in the absence of a field:

<math display="block"> A(t)=\langle [x(t)-\langle x \rangle_0][ x(0)-\langle x \rangle_0] \rangle_0. </math>

Note that in the absence of a field the system is invariant under time-shifts. We can rewrite <math> \langle x(t) \rangle - \langle x \rangle_0 </math> using the susceptibility of the system and hence find with the above {{EqNote|1|equation '''(1)'''}}

<math display="block"> f_0 \int_0^{\infty} d\tau \, \chi(\tau) \theta(\tau-t) = \beta f_0 A(t) </math>

Consequently,
{{NumBlk||<math display="block">-\chi(t) = \beta {dA(t) \over dt} \theta(t) . </math>|{{EquationRef|2}}}}

To make a statement about frequency dependence, it is necessary to take the Fourier transform of {{EqNote|1|equation '''(2)'''}}. By integrating by parts, it is possible to show that
<math display="block"> -\hat\chi(\omega) = i\omega\beta \int_0^\infty e^{-i\omega t} A(t)\, dt -\beta A(0).</math>

Since <math>A(t)</math> is real and symmetric, it follows that
<math display="block"> 2 \operatorname{Im}[\hat\chi(\omega)] = -\omega\beta \hat A(\omega).</math>

Finally, for [[stationary process]]es, the [[Wiener–Khinchin theorem]] states that the two-sided [[power spectrum|spectral density]] is equal to the [[Fourier transform]] of the auto-correlation function:
<math display="block"> S_x(\omega) = \hat{A}(\omega).</math>

Therefore, it follows that
<math display="block"> S_x(\omega) = -\frac{2k_\text{B} T}{\omega} \operatorname{Im}[\hat\chi(\omega)].</math>

=== Quantum version===

The fluctuation-dissipation theorem relates  the [[correlation function]] of the observable of interest <math>\langle \hat{x}(t)\hat{x}(0)\rangle</math> (a measure of fluctuation) to the imaginary part of the [[response function]] <math>\text{Im}\left[\chi(\omega)\right]=\left[\chi(\omega)-\chi^*(\omega)\right]/2i</math> in the frequency domain (a measure of dissipation).  A link between these quantities can be found through the so-called [[Kubo formula]]<ref>{{cite journal |title=The fluctuation-dissipation theorem |doi=10.1088/0034-4885/29/1/306 | volume=29 |year=1966 |journal=Reports on Progress in Physics |pages=255–284 |author=Kubo R |issue=1 |bibcode=1966RPPh...29..255K|s2cid=250892844 }}</ref>

<math display="block">\chi(t-t')=\frac{i}{\hbar}\theta(t-t')\langle [\hat{x}(t),\hat{x}(t')] \rangle</math>

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

<math display="block">\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.</math>

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

<math display="block">\langle \hat{x}(0) \hat{x}(t)\rangle = \operatorname{Tr} e^{-\beta \hat{H}}\hat{x}(0)\hat{x}(t) = \operatorname{Tr} \hat{x}(t) e^{-\beta \hat{H}}\hat{x}(0) = \operatorname{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</math>

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

<math display="block">\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</math>

and thus the expression for <math>\text{Im}\left[\chi(\omega)\right]</math> can be easily rewritten as the quantum fluctuation-dissipation relation <ref>{{cite journal |title=Fundamental aspects of quantum Brownian motion |doi=10.1063/1.1853631 | volume=15 |year=2005 |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |page=026105 |author=Hänggi Peter, Ingold Gert-Ludwig|issue=2 |pmid=16035907 |arxiv=quant-ph/0412052 |bibcode=2005Chaos..15b6105H |s2cid=9787833 |url=https://nbn-resolving.org/urn:nbn:de:bvb:384-opus4-301764 }}</ref>

<math display="block">S_{x}(\omega)=2\hbar\left[n_\text{BE}(\omega)+1\right]\text{Im}\left[\chi(\omega)\right]</math>

where the power spectral density <math>S_{x}(\omega)</math> is the Fourier transform of the auto-correlation <math>\langle \hat{x}(t) \hat{x}(0)\rangle</math> and <math>n_\text{BE}(\omega)=\left(e^{\beta\hbar\omega}-1\right)^{-1}</math> is the [[Bose-Einstein Statistic|Bose-Einstein]] distribution function. The same calculation also yields

<math display="block">S_{x}(-\omega) = e^{-\beta\hbar\omega}S_{x}(\omega) = 2\hbar\left[n_\text{BE}(\omega)\right] \text{Im}\left[\chi(\omega)\right]\neq S_{x}(+\omega)</math>

thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, <math>\langle \hat{x}(t)\hat{x}(0)\rangle</math> has an imaginary part originating from the commutation rules of operators.<ref>{{cite journal |title=Introduction to Quantum Noise, Measurement and Amplification |doi=10.1103/RevModPhys.82.1155 |volume=82 |year=2010 |journal= Reviews of Modern Physics|page=1155 |arxiv=0810.4729 |last1=Clerk |first1=A. A. |last2=Devoret |first2=M. H. |last3=Girvin |first3=S. M. |last4=Marquardt |first4=Florian |last5=Schoelkopf |first5=R. J. |issue=2 |bibcode=2010RvMP...82.1155C |s2cid=119200464 }}</ref> The additional "<math>+1</math>" term in the expression of <math>S_x(\omega)</math> at positive frequencies can also be thought of as linked to [[spontaneous emission]]. An often cited result is also the symmetrized power spectral density

<math display="block">\frac{S_x(\omega)+S_x(-\omega)}{2} = 2\hbar\left[n_\text{BE}(\omega)+\frac{1}{2}\right] \text{Im}\left[\chi(\omega)\right] = \hbar\coth\left(\frac{\hbar\omega}{2k_BT}\right)\text{Im}\left[\chi(\omega)\right].</math>

The "<math>+1/2</math>" can be thought of as linked to [[quantum fluctuation]]s, or to [[Zero-point energy|zero-point motion]] of the observable <math>\hat{x}</math>. At high enough temperatures, <math>n_\text{BE}\approx (\beta\hbar\omega)^{-1}\gg 1</math>, i.e. the quantum contribution is negligible, and we recover the classical version.

==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]] <math>T_\text{g}</math>, [[Spin glass|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.

<!-- Deleted image removed: [[File:Violation of FDT1.png|600px|thumb|Violation of the fluctuation-dissipation theorem (FDT) in the <math>D=3</math> Edwards-Anderson system with temperature <math>T</math>, magnetic susceptibility <math>\chi</math>, and spin-temporal correlation function <math>C</math>. Figure replotted from <ref name="Parisi2017">{{cite journal |title=A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements |doi=10.1073/pnas.1621242114 |url=https://www.pnas.org/content/114/8/1838/tab-article-info | volume=114 |year=2017 |journal=Proceedings of the National Academy of Sciences |pages=1838–1843 |author=Baity-Jesi Marco, Calore Enrico, Cruz Andres, Antonio Fernandez Luis, Miguel Gil-Narvión José, Gordillo-Guerrero Antonio, Iñiguez David, Maiorano Andrea, Marinari Enzo, Martin-Mayor Victor, Monforte-Garcia Jorge, Muñoz Sudupe Antonio, Navarro Denis, Parisi Giorgio, Perez-Gaviro Sergio, Ricci-Tersenghi Federico, Jesus Ruiz-Lorenzo Juan, Fabio Schifano Sebastiano, Seoane Beatriz, Tarancón Alfonso, Tripiccione Raffaele, Yllanes David}}</ref>.]] -->
To study the violation of the fluctuation-dissipation relation in glassy systems, particularly [[spin glasses]], researchers have performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional [[Edwards-Anderson model]] using supercomputers.<ref name="Parisi2017">{{cite journal |title=A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements | doi=10.1073/pnas.1621242114 | volume=114 | year=2017 |journal=Proceedings of the National Academy of Sciences |pages=1838–1843 |author=Baity-Jesi Marco, Calore Enrico, Cruz Andres, Antonio Fernandez Luis, Miguel Gil-Narvión José, Gordillo-Guerrero Antonio, Iñiguez David, Maiorano Andrea, Marinari Enzo, Martin-Mayor Victor, Monforte-Garcia Jorge, Muñoz Sudupe Antonio, Navarro Denis, Parisi Giorgio, Perez-Gaviro Sergio, Ricci-Tersenghi Federico, Jesus Ruiz-Lorenzo Juan, Fabio Schifano Sebastiano, Seoane Beatriz, Tarancón Alfonso, Tripiccione Raffaele, Yllanes David|issue=8 |pmid=28174274 |pmc=5338409 |arxiv=1610.01418 |bibcode=2017PNAS..114.1838B |doi-access=free }}</ref> In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature <math>T=0.64 T_\text{g}</math> below the glass temperature <math>T_\text{g}</math>, and left to equilibrate for a very long time <math>t_\text{w}</math> under a magnetic field <math>H</math>. Then, at a later time <math>t + t_\text{w}</math>, two dynamical observables are probed, namely the [[response function]]
<math display="block">\chi(t+t_\text{w},t_\text{w})\equiv\left.\frac{\partial m(t+t_\text{w})}{\partial H}\right|_{H=0}</math>
and the spin-temporal [[correlation function]]
<math display="block">C(t+t_\text{w},t_\text{w})\equiv \frac{1}{V}\left.\sum_{x}\langle S_x(t_\text{w}) S_x(t+t_\text{w})\rangle\right|_{H=0}</math>
where <math>S_x = \pm 1</math> is the spin living on the node <math>x</math> of the cubic lattice of volume <math>V</math>, and <math display="inline">m(t) \equiv \frac{1}{V} \sum_{x} \langle S_{x}(t) \rangle</math> is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as
<math display="block">T\chi(t+t_\text{w}, t_\text{w}) = 1-C(t+t_\text{w}, t_\text{w})</math>

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.<ref>{{cite journal | title=Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model |doi=10.1103/PhysRevLett.71.173 | volume=71 |year=1993 |journal=Physical Review Letters |pages=173–176 |author=Cugliandolo L. F.|authorlink= Leticia Cugliandolo|author2= Kurchan J.|issue=1 |pmid=10054401 |arxiv=cond-mat/9303036 |bibcode=1993PhRvL..71..173C |s2cid=8591240 }}</ref> This relation is proposed to hold in glassy systems beyond the models for which it was initially found.

==Nonequilibrium driven systems==
In systems subjected to an external driving force, which could be an electromagnetic field or a mechanical shear flow, the standard fluctuation-dissipation theorem gets modified because the statistics of the bath is influenced by the driving field. As a result, the thermal noise becomes biased and the fluctuation-dissipation relation becomes intrinsically non-Markovian, typically with a memory related to the time-autocorrelation of the external field (for the case of a time-dependent external drive). These modified fluctuation-dissipation relations can be derived from a Caldeira-Leggett Hamiltonian for a particle interacting with a thermal bath, where both the particle and the bath respond to the external field.<ref>{{cite journal | title=Generalized Langevin equation and fluctuation-dissipation theorem for particle-bath systems in external oscillating fields |doi=10.1103/PhysRevE.97.060102 | volume=97 |year=2018 |journal=Physical Review E |pages=060102(R) |author=Cui B.|author2= Zaccone A.|issue= |pmid= |arxiv=1802.09848 |bibcode= |s2cid= }}</ref><ref>{{cite journal | title=Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian |doi=10.1103/PhysRevE.107.064102| volume=107 |year=2023 |journal=Physical Review E |pages=064102 |author=Pelargonio S.|author2= Zaccone A.|issue= |pmid= |arxiv=2302.03982 |bibcode= |s2cid= }}</ref>

==See also==
* [[Non-equilibrium thermodynamics]]
* [[Green–Kubo relations]]
* [[Onsager reciprocal relations]]
* [[Equipartition theorem]]
* [[Boltzmann distribution]]
* [[Dissipative system]]

==Notes==
{{Reflist|1}}

==References==
{{refbegin}}
* {{cite journal|author=H. B. Callen, T. A. Welton |year=1951 |title=Irreversibility and Generalized Noise |volume=83 |pages=34–40 |doi=10.1103/PhysRev.83.34|journal=Physical Review|issue=1 |bibcode = 1951PhRv...83...34C }}
* {{cite book |author=L. D. Landau, E. M. Lifshitz |title=Statistical Physics |series=[[Course of Theoretical Physics]] |volume=5 |edition=3 |year=1980 }}
* {{cite journal|author1=Umberto Marini Bettolo Marconi|author2=Andrea Puglisi|author3=Lamberto Rondoni|author4=Angelo Vulpiani|title=Fluctuation-Dissipation: Response Theory in Statistical Physics|year=2008|journal=[[Physics Reports]]|doi=10.1016/j.physrep.2008.02.002|volume=461|issue=4–6|pages=111–195|arxiv=0803.0719|bibcode = 2008PhR...461..111M |s2cid=118575899}}
{{refend}}

==Further reading==
* [http://physics416.blogspot.com/2005/12/lecture-24-fluctuation-dissipation.html Audio recording] of a lecture by Prof. E. W. Carlson of [[Purdue University]]
* [http://www-f1.ijs.si/~ramsak/km1/kubo.pdf Kubo's famous text: Fluctuation-dissipation theorem]
* {{cite journal | author = Weber J | year = 1956 | title = Fluctuation Dissipation Theorem  | journal = [[Physical Review]] | volume = 101 | issue = 6 | pages = 1620&ndash;1626 | doi = 10.1103/PhysRev.101.1620|bibcode = 1956PhRv..101.1620W | arxiv = 0710.4394 }}
* {{cite journal | author = Felderhof BU | year = 1978 | title = On the derivation of the fluctuation-dissipation theorem | journal = Journal of Physics A | volume = 11 | pages = 921&ndash;927 | doi = 10.1088/0305-4470/11/5/021 | issue = 5|bibcode = 1978JPhA...11..921F }}
*{{cite journal | author = Cristani A, Ritort F | year = 2003 | title = Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence | journal = Journal of Physics A | volume = 36 | pages = R181&ndash;R290 | doi = 10.1088/0305-4470/36/21/201 | issue = 21|bibcode = 2003JPhA...36R.181C |arxiv = cond-mat/0212490 | s2cid = 14144683 }}
* {{cite book | author = Chandler D | year = 1987 | title = Introduction to Modern Statistical Mechanics | publisher = Oxford University Press | isbn = 978-0-19-504277-1 | pages = [https://archive.org/details/introductiontomo0000chan/page/231 231&ndash;265] | url = https://archive.org/details/introductiontomo0000chan/page/231 }}
* {{cite book | author = Reichl LE|authorlink= Linda Reichl | year = 1980 | title = A Modern Course in Statistical Physics | publisher = University of Texas Press | location = Austin TX | isbn = 0-292-75080-3 | pages = 545&ndash;595}}
* {{cite book | author = Plischke M, Bergersen B | year = 1989 | title = Equilibrium Statistical Physics | publisher = Prentice Hall | location = Englewood Cliffs, NJ | isbn = 0-13-283276-3 | pages = 251&ndash;296}}
* {{cite book | author = Pathria RK | authorlink = Raj Pathria|year = 1972 | title = Statistical Mechanics | publisher = Pergamon Press | location = Oxford | isbn = 0-08-018994-6 | pages = 443, 474&ndash;477}}
* {{cite book | author = Huang K | year = 1987 | title = Statistical Mechanics | publisher = John Wiley and Sons | location = New York | isbn = 0-471-81518-7 | pages = 153, 394&ndash;396}}
* {{cite book | author = Callen HB | year = 1985 | title = Thermodynamics and an Introduction to Thermostatistics | publisher = John Wiley and Sons | location = New York | isbn = 0-471-86256-8 | pages = 307&ndash;325}}
*{{cite journal |first=Oleg |last=Mazonka |title=Easy as Pi: The Fluctuation-Dissipation Relation |journal=Journal of Reference |volume=16 |year=2016 |url=http://jrxv.net/x/16/fdt.pdf}}

{{DEFAULTSORT:Fluctuation-Dissipation Theorem}}
[[Category:Statistical mechanics]]
[[Category:Non-equilibrium thermodynamics]]
[[Category:Physics theorems]]
[[Category:Statistical mechanics theorems]]