Fluctuation-dissipation theorem

The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951[1] and expanded by Ryogo Kubo. There are antecedents to the general theorem, including Einstein's explanation of Brownian motion[2] during his annus mirabilis and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors.[3] Conteúdo 1 Qualitative overview and examples 2 Examples in detail 2.1 movimento browniano 2.2 Thermal noise in a resistor 3 General formulation 4 Derivação 4.1 Classical version 4.2 Quantum version 5 Violations in glassy systems 6 Quantum version 7 Veja também 8 Notas 9 Referências 10 Further reading Qualitative overview and examples The fluctuation–dissipation theorem says that when there is a process that dissipates energy, turning it into heat (por exemplo., friction), there is a reverse process related to thermal fluctuations. This is best understood by considering some examples: Drag and Brownian motion If an object is moving through a fluid, it experiences 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. 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. Light absorption and thermal radiation When light impinges on an object, some fraction of the light is absorbed, making the object hotter. Nesse caminho, light absorption turns light energy into heat. The corresponding fluctuation is thermal radiation (por exemplo., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. De fato, 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. Em outras palavras, 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={dentro ,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. kB 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 {estilo de exibição R} , {estilo de exibição 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, por exemplo, a simple circuit consisting of a resistor with a resistance {estilo de exibição R} and a capacitor with a small capacitance {estilo de exibição C} . Kirchhoff's law yields {displaystyle V=-R{fratura {dQ}{dt}}+{fratura {Q}{C}}} and so the response function for this circuit is {chi de estilo de exibição (ómega )equivalente {fratura {Q(ómega )}{V(ómega )}}={fratura {1}{{fratura {1}{C}}-iomega R}}} In the low-frequency limit {displaystyle omega ll (RC)^{-1}} , its imaginary part is simply {estilo de exibição {texto{Eu estou}}deixei[chi (ómega )certo]approx omega RC^{2}} which then can be linked to the power spectral density function {estilo de exibição S_{V}(ómega )} of the voltage via the fluctuation-dissipation theorem {estilo de exibição S_{V}(ómega )={fratura {S_{Q}(ómega )}{C^{2}}}Aproximadamente {fratura {2k_{rm {B}}T}{C^{2}ómega }}{texto{Eu estou}}deixei[chi (ómega )certo]=2Rk_{rm {B}}T} The Johnson–Nyquist voltage noise {displaystyle langle V^{2}chocalho } was observed within a small frequency bandwidth {displaystyle Delta nu =Delta omega /(2pi )} centered around {displaystyle omega =pm omega _{0}} . Por isso {displaystyle langle V^{2}rangle approx S_{V}(ómega )times 2Delta nu approx 4Rk_{rm {B}}TDelta nu } General formulation The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:[citação necessária].

Deixar {estilo de exibição x(t)} be an observable of a dynamical system with Hamiltonian {estilo de exibição H_{0}(x)} subject to thermal fluctuations. The observable {estilo de exibição x(t)} will fluctuate around its mean value {displaystyle langle xrangle _{0}} with fluctuations characterized by a power spectrum {estilo de exibição S_{x}(ómega )=ângulo {chapéu {x}}(ómega ){chapéu {x}}^{*}(ómega )chocalho } . Suppose that we can switch on a time-varying, spatially constant field {estilo de exibição f(t)} which alters the Hamiltonian to {estilo de exibição H(x)=H_{0}(x)-f(t)x} . The response of the observable {estilo de exibição x(t)} to a time-dependent field {estilo de exibição f(t)} is characterized to first order by the susceptibility or linear response function {chi de estilo de exibição (t)} of the system {estilo de exibição lang x(t)rangle =langle xrangle _{0}+int_{-infty }^{t}!f(sim )chi (t-tau ),dtau ,} where the perturbation is adiabatically (very slowly) switched on at {displaystyle tau =-infty } .

The fluctuation–dissipation theorem relates the two-sided power spectrum (ou seja. both positive and negative frequencies) do {estilo de exibição x} to the imaginary part of the Fourier transform {estilo de exibição {chapéu {chi }}(ómega )} of the susceptibility {chi de estilo de exibição (t)} : {estilo de exibição S_{x}(ómega )=-{fratura {2k_{matemática {B} }T}{ómega }}nome do operador {Eu estou} {chapéu {chi }}(ómega ).} Which holds under the Fourier transform convention {estilo de exibição f(ómega )=int_{-infty }^{infty }f(t)e^{-iomega t},dt} . The left-hand side describes fluctuations in {estilo de exibição x} , the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field {estilo de exibição f(t)=Fsin(omega t+phi )} .

This is the classical form of the theorem; quantum fluctuations are taken into account by replacing {displaystyle 2k_{matemática {B} }T/omega } com {estilo de exibição hbar ,coth(hbar omega /2k_{matemática {B} }T)} (whose limit for {displaystyle hbar to 0} é {displaystyle 2k_{matemática {B} }T/omega } ). A proof can be found by means of the LSZ reduction, an identity from quantum field theory.[citação necessária] 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.[1] A special case in which the fluctuating quantity is the energy itself is the fluctuation-dissipation theorem for the frequency-dependent specific heat.[5] 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 f has been on for infinite time and is switched off at t=0 {estilo de exibição f(t)=f_{0}teta (-t),} Onde {estilo de exibição teta (t)} is the Heaviside function. We can express the expectation value of {estilo de exibição x} by the probability distribution W(x,0) and the transition probability {estilo de exibição P(x',t|x,0)} {estilo de exibição lang x(t)rangle =int dx'int dx,x'P(x',t|x,0)C(x,0).} The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution for the Hamiltonian {estilo de exibição H(x)=H_{0}(x)-xf_{0}} {estilo de exibição W.(x,0)={fratura {exp(-beta H(x))}{int dx',exp(-beta H(x'))}},,} Onde {displaystyle beta ^{-1}=k_{rm {B}}T} . For a weak field {displaystyle beta xf_{0}ll 1} , we can expand the right-hand side {estilo de exibição W.(x,0)approx W_{0}(x)[1+beta f_{0}(x(0)-langle xrangle _{0})],} aqui {estilo de exibição W_{0}(x)} is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for {estilo de exibição lang x(t)chocalho } yields {estilo de exibição lang x(t)rangle =langle xrangle _{0}+beta f_{0}UMA(t),} (*) onde um(t) is the auto-correlation function of x in the absence of a field: {estilo de exibição A(t)=ângulo [x(t)-langle xrangle _{0}][x(0)-langle xrangle _{0}]rangle _{0}.} Note that in the absence of a field the system is invariant under time-shifts. We can rewrite {estilo de exibição lang x(t)rangle -langle xrangle _{0}} using the susceptibility of the system and hence find with the above equation (*) {estilo de exibição f_{0}int_{0}^{infty }dtau ,chi (sim )teta (tau -t)=beta f_{0}UMA(t)} Consequentemente, {displaystyle -chi (t)=beta {dA(t) over dt}teta (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 {estilo de exibição -{chapéu {chi }}(ómega )=iomega beta int _{0}^{infty }e^{-iomega t}UMA(t),dt-beta A(0).} Desde {estilo de exibição A(t)} is real and symmetric, segue que {displaystyle 2operatorname {Eu estou} [{chapéu {chi }}(ómega )]=-omega beta {chapéu {UMA}}(ómega ).} Finalmente, 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: {estilo de exibição S_{x}(ómega )={chapéu {UMA}}(ómega ).} Portanto, segue que {estilo de exibição S_{x}(ómega )=-{fratura {2k_{texto{B}}T}{ómega }}nome do operador {Eu estou} [{chapéu {chi }}(ómega )].} Quantum version The fluctuation-dissipation theorem relates the correlation function of the observable of interest {idioma do estilo de exibição {chapéu {x}}(t){chapéu {x}}(0)chocalho } (a measure of fluctuation) to the imaginary part of the response function {estilo de exibição {texto{Eu estou}}deixei[chi (ómega )certo]= esquerda[chi (ómega )-chi ^{*}(ómega )certo]/2eu} in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called Kubo formula[6] {chi de estilo de exibição (t-t')={fratura {eu}{hbar }}teta (t-t')ângulo [{chapéu {x}}(t),{chapéu {x}}(t')]chocalho } which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable {idioma do estilo de exibição {chapéu {x}}(t)chocalho } in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as {estilo de exibição {texto{Eu estou}}deixei[chi (ómega )certo]={fratura {1}{2hbar }}int_{-infty }^{+infty }ângulo {chapéu {x}}(t){chapéu {x}}(0)-{chapéu {x}}(0){chapéu {x}}(t)rangle e^{iomega t}dt.} In the canonical ensemble, the second term can be re-expressed as {idioma do estilo de exibição {chapéu {x}}(0){chapéu {x}}(t)chocalho ={texto{Tr }}e^{-beta {chapéu {H}}}{chapéu {x}}(0){chapéu {x}}(t)={texto{Tr }}{chapéu {x}}(t)e^{-beta {chapéu {H}}}{chapéu {x}}(0)={texto{Tr }}e^{-beta {chapéu {H}}}underbrace {e^{beta {chapéu {H}}}{chapéu {x}}(t)e^{-beta {chapéu {H}}}} _{{chapéu {x}}(t-ihbar beta )}{chapéu {x}}(0)=ângulo {chapéu {x}}(t-ihbar beta ){chapéu {x}}(0)chocalho } where in the second equality we re-positioned {estilo de exibição {chapéu {x}}(t)} using the cyclic property of trace. Próximo, in the third equality, we inserted {estilo de exibição e^{-beta {chapéu {H}}}e^{beta {chapéu {H}}}} next to the trace and interpreted {estilo de exibição e^{-beta {chapéu {H}}}} as a time evolution operator {estilo de exibição e^{-{fratura {eu}{hbar }}{chapéu {H}}Delta t}} with imaginary time interval {displaystyle Delta t=-ihbar beta } . The imaginary time shift turns into a {estilo de exibição e^{-beta hbar omega }} factor after Fourier transform {estilo de exibição int _{-infty }^{+infty }ângulo {chapéu {x}}(t-ihbar beta ){chapéu {x}}(0)rangle e^{iomega t}dt=e^{-beta hbar omega }int_{-infty }^{+infty }ângulo {chapéu {x}}(t){chapéu {x}}(0)rangle e^{iomega t}dt} and thus the expression for {estilo de exibição {texto{Eu estou}}deixei[chi (ómega )certo]} can be easily rewritten as the quantum fluctuation-dissipation relation [7] {estilo de exibição S_{x}(ómega )=2hbar left[n_{rm {SER}}(ómega )+1certo]{texto{Eu estou}}deixei[chi (ómega )certo]} where the power spectral density {estilo de exibição S_{x}(ómega )} is the Fourier transform of the auto-correlation {idioma do estilo de exibição {chapéu {x}}(t){chapéu {x}}(0)chocalho } e {estilo de exibição n_{rm {SER}}(ómega )= esquerda(e^{beta hbar omega }-1certo)^{-1}} is the Bose-Einstein distribution function. The same calculation also yields {estilo de exibição S_{x}(-ómega )=e^{-beta hbar omega }S_{x}(ómega )=2hbar left[n_{rm {SER}}(ómega )certo]{texto{Eu estou}}deixei[chi (ómega )certo]neq S_{x}(+ómega )} portanto, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, {idioma do estilo de exibição {chapéu {x}}(t){chapéu {x}}(0)chocalho } has an imaginary part originating from the commutation rules of operators.[8] The additional " {estilo de exibição +1} " term in the expression of {estilo de exibição S_{x}(ómega )} at positive frequencies can also be thought of as linked to spontaneous emission. An often cited result is also the symmetrized power spectral density {estilo de exibição {fratura {S_{x}(ómega )+S_{x}(-ómega )}{2}}=2hbar left[n_{rm {SER}}(ómega )+{fratura {1}{2}}certo]{texto{Eu estou}}deixei[chi (ómega )certo]=hbar coth left({fratura {hbar omega }{2k_{B}T}}certo){texto{Eu estou}}deixei[chi (ómega )certo].} o " {estilo de exibição +1/2} " can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable {estilo de exibição {chapéu {x}}} . At high enough temperatures, {estilo de exibição n_{rm {SER}}Aproximadamente (beta hbar omega )^{-1}gg 1} , ou seja. 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 {estilo de exibição 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.

To study the violation of the fluctuation-dissipation relation in glassy systems, particularly spin glasses, Ref.[9] performed numerical simulations of macroscopic systems (ou seja. 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 {displaystyle T=0.64T_{rm {g}}} below the glass temperature {estilo de exibição T_{g}} , and left to equilibrate for a very long time {estilo de exibição t_{rm {W}}} under a magnetic field {estilo de exibição H} . Então, at a later time {displaystyle t+t_{rm {W}}} , two dynamical observables are probed, namely the response function {chi de estilo de exibição (t+t_{rm {W}},t_{rm {W}})equiv left.{fratura {partial m(t+t_{rm {W}})}{partial H}}certo|_{H=0}} and the spin-temporal correlation function {estilo de exibição C(t+t_{rm {W}},t_{rm {W}})equivalente {fratura {1}{V}}left.sum _{x}langle S_{x}(t_{rm {W}})S_{x}(t+t_{rm {W}})rangle right|_{H=0}} Onde {estilo de exibição S_{x}=pm 1} is the spin living on the node {estilo de exibição x} of the cubic lattice of volume {estilo de exibição V} , e {textstyle m(t)equivalente {fratura {1}{V}}soma _{x}langle S_{x}(t)chocalho } is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as {displaystyle Tchi (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 [10] 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.

Quantum version The Rényi entropy as well as von Neumann entropy in quantum physics are not observables since they depend nonlinearly on the density matrix. Recently, Mohammad H. Ansari and Yuli V. Nazarov proved an exact correspondence that reveals the physical meaning of the Rényi entropy flow in time. This correspondence is similar to the fluctuation-dissipation theorem in spirit and allows the measurement of quantum entropy using the full counting statistics (FCS) of energy transfers.[11][12][13] See also Non-equilibrium thermodynamics Green–Kubo relations Onsager reciprocal relations Equipartition theorem Boltzmann distribution Dissipative system Notes ^ Jump up to: a b H.B. Callen; T.A. Welton (1951). "Irreversibility and Generalized Noise". Revisão Física. 83 (1): 34-40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34. ^ Einstein, Alberto (Poderia 1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik. 322 (8): 549-560. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806. ^ Nyquist H (1928). "Thermal Agitation of Electric Charge in Conductors". Revisão Física. 32 (1): 110-113. Bibcode:1928PhRv...32..110N. doi:10.1103/PhysRev.32.110. ^ Blundell, Stephen J.; Blundell, Katherine M. (2009). Concepts in thermal physics. OUP Oxford. ^ Nielsen, Johannes K.; Dyre, Jeppe C. (1996-12-01). "Fluctuation-dissipation theorem for frequency-dependent specific heat". Revisão Física B. 54 (22): 15754–15761. doi:10.1103/PhysRevB.54.15754. ISSN 0163-1829. ^ Kubo R (1966). "The fluctuation-dissipation theorem". Relatórios sobre o Progresso na Física. 29 (1): 255–284. Bibcode:1966RPPh...29..255K. doi:10.1088/0034-4885/29/1/306. ^ Hänggi Peter, Ingold Gert-Ludwig (2005). "Fundamental aspects of quantum Brownian motion". Chaos: An Interdisciplinary Journal of Nonlinear Science. 15 (2): 026105. arXiv:quant-ph/0412052. Bibcode:2005Chaos..15b6105H. doi:10.1063/1.1853631. PMID 16035907. S2CID 9787833. ^ Clerk, UMA. UMA.; Devoret, M. H.; Girvin, S. M.; Marquardt, Florian; Schoelkopf, R. J. (2010). "Introduction to Quantum Noise, Measurement and Amplification". Comentários de Física Moderna. 82 (2): 1155. arXiv:0810.4729. Bibcode:2010RvMP...82.1155C. doi:10.1103/RevModPhys.82.1155. S2CID 119200464. ^ 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 (2017). "A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements". Anais da Academia Nacional de Ciências. 114 (8): 1838–1843. arXiv:1610.01418. Bibcode:2017PNAS..114.1838B. doi:10.1073/pnas.1621242114. PMC 5338409. PMID 28174274. ^ Cugliandolo L. F.; Kurchan J. (1993). "Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model". Cartas de Revisão Física. 71 (1): 173-176. arXiv:cond-mat/9303036. Bibcode:1993PhRvL..71..173C. doi:10.1103/PhysRevLett.71.173. PMID 10054401. S2CID 8591240. ^ Ansari Nazarov (2016) ^ Ansari Nazarov (2015uma) ^ Ansari Nazarov (2015b) References H. B. Callen, T. UMA. Welton (1951). "Irreversibility and Generalized Noise". Revisão Física. 83 (1): 34-40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34. eu. D. Landau, E. M. Lifshitz (1980). Statistical Physics. Course of Theoretical Physics. Volume. 5 (3 ed.). Umberto Marini Bettolo Marconi; Andrea Puglisi; Lamberto Rondoni; Angelo Vulpiani (2008). "Fluctuation-Dissipation: Response Theory in Statistical Physics". Physics Reports. 461 (4–6): 111-195. arXiv:0803.0719. Bibcode:2008PhR...461..111M. doi:10.1016/j.physrep.2008.02.002. S2CID 118575899. Further reading Audio recording of a lecture by Prof. E. C. Carlson of Purdue University Kubo's famous text: Fluctuation-dissipation theorem Weber J (1956). "Fluctuation Dissipation Theorem". Revisão Física. 101 (6): 1620–1626. Bibcode:1956PhRv..101.1620W. doi:10.1103/PhysRev.101.1620. Felderhof BU (1978). "On the derivation of the fluctuation-dissipation theorem". Revista de Física A. 11 (5): 921–927. Bibcode:1978JPhA...11..921F. doi:10.1088/0305-4470/11/5/021. Cristani A, Ritort F (2003). "Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence". Revista de Física A. 36 (21): R181–R290. arXiv:cond-mat/0212490. Bibcode:2003JPhA...36R.181C. doi:10.1088/0305-4470/36/21/201. S2CID 14144683. Chandler D (1987). Introduction to Modern Statistical Mechanics. imprensa da Universidade de Oxford. pp. 231–265. ISBN 978-0-19-504277-1. Reichl LE (1980). A Modern Course in Statistical Physics. Austin TX: University of Texas Press. pp. 545–595. ISBN 0-292-75080-3. Plischke M, Bergersen B (1989). Equilibrium Statistical Physics. Englewood Cliffs, Nova Jersey: Prentice Hall. pp. 251–296. ISBN 0-13-283276-3. Pathria RK (1972). Statistical Mechanics. Oxford: Pergamon Press. pp. 443, 474–477. ISBN 0-08-018994-6. Huang K (1987). Statistical Mechanics. Nova york: John Wiley e Filhos. pp. 153, 394–396. ISBN 0-471-81518-7. Callen HB (1985). Thermodynamics and an Introduction to Thermostatistics. Nova york: John Wiley e Filhos. pp. 307-325. ISBN 0-471-86256-8. Mazonka, Oleg (2016). "Easy as Pi: The Fluctuation-Dissipation Relation" (PDF). Journal of Reference. 16. Ansari, Mohammad H.; Nazarov, Yuli V. (2015). "Exact correspondence between Rényi entropy flows and physical flows". Revisão Física B. 91 (17): 174307. arXiv:1502.08020. Bibcode:2015PhRvB..91q4307A. doi:10.1103/PhysRevB.91.174307. S2CID 36847902. Categorias: Statistical mechanicsNon-equilibrium thermodynamicsPhysics theoremsStatistical mechanics theorems

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