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| The '''fluctuation-dissipation theorem''' ('''FDT''') is a powerful tool in [[statistical physics]] for predicting the behavior of [[non-equilibrium thermodynamics|non-equilibrium thermodynamical]] systems. These systems involve the [[irreversibility|irreversible]] [[dissipation]] of energy into [[heat]] from their [[reversible process (thermodynamics)|reversible]] [[thermal fluctuations]] at [[thermodynamic equilibrium]]. The fluctuation-dissipation theorem applies both to [[classical physics|classical]] and [[quantum mechanics|quantum mechanical]] systems.
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| The fluctuation-dissipation theorem relies on the assumption that the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation. Therefore, the theorem connects the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium.<ref>
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| {{cite book
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| |author = David Chandler
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| |year=1987
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| |title=Introduction to Modern Statistical Mechanics
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| |page=255
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| |publisher=[[Oxford University Press]]
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| |isbn=978-0-19-504277-1
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| }}</ref> Often the linear response takes the form of one or more exponential decays.
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| The fluctuation-dissipation theorem was originally formulated by [[Harry Nyquist]] in 1928,<ref>
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| {{cite journal
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| | author = Nyquist H
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| | year = 1928
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| | title = Thermal Agitation of Electric Charge in Conductors
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| | journal = [[Physical Review]]
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| | volume = 32 | pages = 110–113
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| | bibcode = 1928PhRv...32..110N
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| | doi = 10.1103/PhysRev.32.110
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| }}</ref> and later proven by [[Herbert Callen]] and [[Theodore A. Welton]] in 1951.<ref>
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| {{cite journal
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| |author = H.B. Callen, T.A. Welton
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| | year = 1951
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| | title = Irreversibility and Generalized Noise
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| | journal = [[Physical Review]]
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| | volume = 83 | pages = 34–40
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| | bibcode = 1951PhRv...83...34C
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| | doi = 10.1103/PhysRev.83.34
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| }}</ref>
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| ==General applicability==
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| The fluctuation-dissipation theorem is a general result of [[statistical thermodynamics]] that quantifies the relation between the fluctuations in a system at [[thermal equilibrium]] and the response of the system to applied perturbations.
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| The model thus allows, for example, the use of molecular models to predict material properties in the context of linear response theory. The theorem assumes that applied perturbations, e.g., mechanical forces or electric fields, are weak enough that rates of [[Relaxation time|relaxation]] remain unchanged.
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| ===Brownian motion===
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| 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.
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| From this observation Einstein was able to use [[statistical mechanics]] to derive a previously unexpected connection, the [[Einstein relation (kinetic theory)|Einstein-Smoluchowski relation]]:
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| :<math> D = {\mu_p \, k_B T} </math>
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| linking ''D'', the [[Fick's law of diffusion|diffusion constant]], and ''μ'', the mobility of the particles. (''μ'' is the ratio of the particle's terminal drift velocity to an applied force, ''μ = v<sub>d</sub> / F''). ''k''<sub>''B''</sub> ≈ 1.38065 × 10<sup>−23</sup> m<sup>2</sup> kg s<sup>−2</sup> ''K''<sup>−1</sup> is the [[Boltzmann constant]], and ''T'' is the [[absolute temperature]].
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| ===Thermal noise in a resistor===
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| 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'', <math>k_BT</math>, and the bandwidth <math>\Delta\nu</math> over which the voltage is measured:
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| : <math> \langle V^2 \rangle = 4Rk_BT\,\Delta\nu. </math>
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| ==General formulation==
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| The fluctuation-dissipation theorem can be formulated in many ways; one particularly useful form is the following:{{Citation needed|date=October 2010}}
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| 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.
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| The observable <math>x(t)</math> will fluctuate around its mean value <math>\langle x\rangle_0</math>
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| with fluctuations characterized by a [[power spectrum]] <math>S_x(\omega) = \hat{x}(\omega)\hat{x}^*(\omega)</math>.
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| Suppose that we can switch on a scalar field <math>f(t)</math> which alters the Hamiltonian
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| to <math>H(x)=H_0(x)+fx</math>.
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| The response of the observable <math>x(t)</math> to a time-dependent field <math>f(t)</math> is
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| characterized to first order by the [[susceptibility]] or [[linear response function]]
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| <math>\chi(t)</math> of the system
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| : <math> \langle x(t) \rangle = \langle x \rangle_0 + \int\limits_{-\infty}^{t} \! f(\tau) \chi(t-\tau)\,d\tau, </math>
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| where the perturbation is adiabatically switched on at <math>\tau =-\infty</math>.
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| The fluctuation-dissipation theorem relates the two-sided power spectrum of <math>x</math> to the imaginary part of the [[Fourier transform]] <math>\hat{\chi}(\omega)</math> of the susceptibility <math>\chi(t)</math>:
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| : <math>S_x(\omega) = \frac{2 k_\mathrm{B} T}{\omega} \mathrm{Im}\,\hat{\chi}(\omega).</math>
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| 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>.
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| This is the classical form of the theorem; quantum fluctuations are taken into account by
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| replacing <math>2 k_\mathrm{B} T/\omega</math> with <math>{\hbar}\,\coth(\hbar\omega/2k_\mathrm{B}T)</math> (whose limit for <math>\hbar\to 0</math> is <math>2 k_\mathrm{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}}
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| 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.{{H. B. Callen and T. A. Welton, Phys. Rev. 83, (1951) 34|date=April 2013}}
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| ==Derivation==
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| We derive the fluctuation-dissipation theorem in the form given above, using the same notation.
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| Consider the following test case: The field ''f'' has been on for infinite time and is switched off at ''t''=0
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| : <math> f(t)=f_0 \theta(-t) . </math>
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| We can express the expectation value of ''x'' by the probability distribution ''W''(''x'',0) and the transition probability <math> P(x',t | x,0) </math>
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| : <math> \langle x(t) \rangle = \int dx' \int dx \, x' P(x',t|x,0) W(x,0) . </math>
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| The probability distribution function ''W''(''x'',0) is an equilibrium distribution and hence
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| given by the [[Boltzmann distribution]] for the Hamiltonian <math> H(x)=H_0(x) + x f_0 </math>
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| : <math> W(x,0)= \frac{\exp(-\beta H(x))}{\int dx' \, \exp(-\beta H(x'))} \;, </math>
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| where <math>\beta^{-1} = k_{\rm B}T</math>.
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| For a weak field <math> \beta x f_0 \ll 1 </math>, we can expand the right-hand side
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| : <math> W(x,0) \approx W_0(x) (1-\beta f_0 x), </math>
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| here <math> W_0(x) </math> is the equilibrium distribution in the absence of a field.
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| Plugging this approximation in the formula for <math> \langle x(t) \rangle </math> yields
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| {{NumBlk|:|<math>\langle x(t) \rangle = \langle x \rangle_0 - \beta f_0 A(t),</math>|*}}
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| where ''A''(''t'') is the auto-correlation function of ''x'' in the absence of a field:
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| : <math> A(t)=\langle x(t) x(0) \rangle_0. </math>
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| Note that in the absence of a field the system is invariant under time-shifts.
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| We can rewrite <math> \langle x(t) \rangle - \langle x \rangle_0 </math> using the susceptibility
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| of the system and hence find with the above equation '''(*)'''
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| : <math> f_0 \int_0^{\infty} d\tau \, \chi(\tau) \theta(\tau-t) = \beta f_0 A(t) </math> | |
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| Consequently,
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| {{NumBlk|:|<math>-\chi(t) = \beta {\operatorname{d}A(t)\over\operatorname{d}t}
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| \theta(t) . </math>|**}}
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| 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
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| : <math> -\hat\chi(\omega) = i\omega\beta \int\limits_0^\infty \mathrm{e}^{-i\omega t} A(t)\, dt.</math>
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| Since <math>A(t)</math> is real and symmetric, it follows that
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| :<math> 2\,\mathrm{Im}[\hat\chi(\omega)] = \omega\beta \hat A(\omega).</math>
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| 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:
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| : <math> S_x(\omega) = \hat{A}(\omega).</math>
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| Therefore, it follows that
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| :<math> S_x(\omega) = \frac{2k_\text{B} T}{\omega} \,\mathrm{Im}[\hat\chi(\omega)].</math>
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| ==Violations in glassy systems==
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| While the fluctuation-dissipation theorem provides a general relation between the response of equilibrium systems to small external perturbations and their spontaneous fluctuations, no general relation is known for systems out of equilibrium. Glassy systems at low temperatures, as well as real glasses, are characterized by slow approaches to equilibrium states. Thus these systems require large time-scales to be studied while they remain in disequilibrium.
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| In the mid 1990s, in the study of non-equilibrium dynamics of [[spin glass]] models, a generalization of the fluctuation-dissipation theorem was discovered{{Citation needed|date=September 2011}} 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.
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| This relation is proposed to hold in glassy systems beyond the models for which it was initially found.
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| ==See also==
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| * [[Non-equilibrium thermodynamics]]
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| * [[Green-Kubo relations]]
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| * [[Onsager reciprocal relations]]
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| * [[Equipartition theorem]]
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| * [[Boltzmann factor]]
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| * [[Dissipative system]]
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| ==Notes==
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| {{Reflist|1}}
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| ==References==
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| * {{cite journal|author=H. B. Callen, T. A. Welton |year=1951 |title=Irreversibility and Generalized Noise |volume=83 |pages=34 |doi=10.1103/PhysRev.83.34|journal=Physical Review|bibcode = 1951PhRv...83...34C }}
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| * {{cite book |author=L. D. Landau, E. M. Lifshitz |title=Physique Statistique |series=[[Course of Theoretical Physics|Cours de physique théorique]] |volume=5 |publisher=Mir}}
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| * {{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 }}
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| ==Further reading==
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| * [http://physics416.blogspot.com/2005/12/lecture-24-fluctuation-dissipation.html Audio recording] of a lecture by Prof. E. W. Carlson of [[Purdue University]]
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| * [http://www-f1.ijs.si/~ramsak/km1/kubo.pdf Kubo's famous text: Fluctuation-dissipation theorem]
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| * {{cite journal | author = Weber J | year = 1956 | title = Fluctuation Dissipation Theorem | journal = [[Physical Review]] | volume = 101 | issue = 6 | pages = 1620–1626 | doi = 10.1103/PhysRev.101.1620|bibcode = 1956PhRv..101.1620W }}
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| * {{cite journal | author = Felderhof BU | year = 1978 | title = On the derivation of the fluctuation-dissipation theorem | journal = Journal of Mathematical Physics A | volume = 11 | pages = 921–927 | doi = 10.1088/0305-4470/11/5/021 | issue = 5|bibcode = 1978JPhA...11..921F }}
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| *{{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: Mathematical and General | volume = 36 | pages = R181–R290 | doi = 10.1088/0305-4470/36/21/201 | issue = 21|bibcode = 2003JPhA...36..R181 }}
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| * {{cite book | author = Chandler D | year = 1987 | title = Introduction to Modern Statistical Mechanics | publisher = Oxford University Press | isbn =978-0-19-504277-1 | pages = 231–265}}
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| * {{cite book | author = Reichl LE | year = 1980 | title = A Modern Course in Statistical Physics | publisher = University of Texas Press | location = Austin TX | isbn = 0-292-75080-3 | pages = 545–595}}
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| * {{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 = 251–296}}
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| * {{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, 474–477}}
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| * {{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, 394–396}}
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| * {{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 = 307–325}}
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| {{DEFAULTSORT:Fluctuation-Dissipation Theorem}}
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| [[Category:Statistical mechanics]]
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| [[Category:Non-equilibrium thermodynamics]]
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| [[Category:Physics theorems]]
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| [[Category:Statistical mechanics theorems]]
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