Fluctuation-dissipation theorem

Fluctuation-dissipation theorem (Redirected from Fluctuation dissipation theorem) Jump to navigation Jump to search 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 thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance (to be intended in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, eccetera.), e viceversa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.

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] Contenuti 1 Qualitative overview and examples 2 Examples in detail 2.1 Moto browniano 2.2 Thermal noise in a resistor 3 General formulation 4 Derivazione 4.1 Classical version 4.2 Quantum version 5 Violations in glassy systems 6 Quantum version 7 Guarda anche 8 Appunti 9 Riferimenti 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 (per esempio., 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. In questo modo, light absorption turns light energy into heat. The corresponding fluctuation is thermal radiation (per esempio., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. Infatti, 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 altre parole, 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 {stile di visualizzazione D={in ,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, Giovanni B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance {stile di visualizzazione R} , {stile di visualizzazione 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, Per esempio, a simple circuit consisting of a resistor with a resistance {stile di visualizzazione R} and a capacitor with a small capacitance {stile di visualizzazione C} . Kirchhoff's law yields {displaystyle V=-R{frac {dQ}{dt}}+{frac {Q}{C}}} and so the response function for this circuit is {stile di visualizzazione chi (omega )equivalente {frac {Q(omega )}{V(omega )}}={frac {1}{{frac {1}{C}}-iomega R}}} In the low-frequency limit {displaystyle omega ll (RC)^{-1}} , its imaginary part is simply {stile di visualizzazione {testo{Io sono}}sinistra[chi (omega )Giusto]approx omega RC^{2}} which then can be linked to the power spectral density function {stile di visualizzazione S_{V}(omega )} of the voltage via the fluctuation-dissipation theorem {stile di visualizzazione S_{V}(omega )={frac {S_{Q}(omega )}{C^{2}}}ca {frac {2K_{rm {B}}T}{C^{2}omega }}{testo{Io sono}}sinistra[chi (omega )Giusto]=2Rk_{rm {B}}T} The Johnson–Nyquist voltage noise {displaystyle langle V^{2}sonaglio } was observed within a small frequency bandwidth {displaystyle Delta nu =Delta omega /(2pi )} centered around {displaystyle omega =pm omega _{0}} . Quindi {displaystyle langle V^{2}rangle approx S_{V}(omega )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:[citazione necessaria].

Permettere {stile di visualizzazione x(t)} be an observable of a dynamical system with Hamiltonian {stile di visualizzazione H_{0}(X)} subject to thermal fluctuations. The observable {stile di visualizzazione x(t)} will fluctuate around its mean value {displaystyle langle xrangle _{0}} with fluctuations characterized by a power spectrum {stile di visualizzazione S_{X}(omega )= angolo {cappello {X}}(omega ){cappello {X}}^{*}(omega )sonaglio } . Suppose that we can switch on a time-varying, spatially constant field {stile di visualizzazione f(t)} which alters the Hamiltonian to {stile di visualizzazione H(X)=H_{0}(X)-f(t)X} . The response of the observable {stile di visualizzazione x(t)} to a time-dependent field {stile di visualizzazione f(t)} is characterized to first order by the susceptibility or linear response function {stile di visualizzazione chi (t)} of the system {angolo dello stile di visualizzazione x(t)rangle =langle xrangle _{0}+int _{-infty }^{t}!f(sì )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 (cioè. both positive and negative frequencies) di {stile di visualizzazione x} to the imaginary part of the Fourier transform {stile di visualizzazione {cappello {chi }}(omega )} of the susceptibility {stile di visualizzazione chi (t)} : {stile di visualizzazione S_{X}(omega )=-{frac {2K_{matematica {B} }T}{omega }}nome operatore {Io sono} {cappello {chi }}(omega ).} Which holds under the Fourier transform convention {stile di visualizzazione f(omega )=int _{-infty }^{infty }f(t)e^{-iomega t},dt} . The left-hand side describes fluctuations in {stile di visualizzazione x} , the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field {stile di visualizzazione f(t)=Fsin(omega t+phi )} .

This is the classical form of the theorem; quantum fluctuations are taken into account by replacing {displaystyle 2k_{matematica {B} }T/omega } insieme a {barra dello stile di visualizzazione ,coth(hbar omega /2k_{matematica {B} }T)} (whose limit for {displaystyle hbar to 0} è {displaystyle 2k_{matematica {B} }T/omega } ). A proof can be found by means of the LSZ reduction, an identity from quantum field theory.[citazione necessaria] 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 {stile di visualizzazione f(t)=f_{0}teta (-t),} dove {stile di visualizzazione theta (t)} is the Heaviside function. We can express the expectation value of {stile di visualizzazione x} by the probability distribution W(X,0) and the transition probability {stile di visualizzazione P(X',t|X,0)} {angolo dello stile di visualizzazione 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 {stile di visualizzazione H(X)=H_{0}(X)-xf_{0}} {stile di visualizzazione W.(X,0)={frac {esp(-beta H(X))}{int dx',esp(-beta H(X'))}},,} dove {displaystyle beta ^{-1}=k_{rm {B}}T} . For a weak field {displaystyle beta xf_{0}ll 1} , we can expand the right-hand side {stile di visualizzazione W.(X,0)approx W_{0}(X)[1+beta f_{0}(X(0)-langle xrangle _{0})],} qui {stile di visualizzazione W_{0}(X)} is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for {angolo dello stile di visualizzazione x(t)sonaglio } yields {angolo dello stile di visualizzazione x(t)rangle =langle xrangle _{0}+beta f_{0}UN(t),} (*) dove un(t) is the auto-correlation function of x in the absence of a field: {stile di visualizzazione A(t)= angolo [X(t)-langle xrangle _{0}][X(0)-langle xrangle _{0}]sonaglio _{0}.} Note that in the absence of a field the system is invariant under time-shifts. We can rewrite {angolo dello stile di visualizzazione x(t)rangle -langle xrangle _{0}} using the susceptibility of the system and hence find with the above equation (*) {stile di visualizzazione f_{0}int _{0}^{infty }dtau ,chi (sì )teta (tau -t)=beta f_{0}UN(t)} Di conseguenza, {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 {stile di visualizzazione -{cappello {chi }}(omega )=iomega beta int _{0}^{infty }e^{-iomega t}UN(t),dt-beta A(0).} Da {stile di visualizzazione A(t)} is real and symmetric, ne consegue che {displaystyle 2operatorname {Io sono} [{cappello {chi }}(omega )]=-omega beta {cappello {UN}}(omega ).} Infine, 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: {stile di visualizzazione S_{X}(omega )={cappello {UN}}(omega ).} Perciò, ne consegue che {stile di visualizzazione S_{X}(omega )=-{frac {2K_{testo{B}}T}{omega }}nome operatore {Io sono} [{cappello {chi }}(omega )].} Quantum version The fluctuation-dissipation theorem relates the correlation function of the observable of interest {angolo dello stile di visualizzazione {cappello {X}}(t){cappello {X}}(0)sonaglio } (a measure of fluctuation) to the imaginary part of the response function {stile di visualizzazione {testo{Io sono}}sinistra[chi (omega )Giusto]= sinistra[chi (omega )-chi ^{*}(omega )Giusto]/2io} in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called Kubo formula[6] {stile di visualizzazione chi (t-t')={frac {io}{hbar }}teta (t-t')angolo [{cappello {X}}(t),{cappello {X}}(t')]sonaglio } which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable {angolo dello stile di visualizzazione {cappello {X}}(t)sonaglio } in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as {stile di visualizzazione {testo{Io sono}}sinistra[chi (omega )Giusto]={frac {1}{2hbar }}int _{-infty }^{+infty }angolo {cappello {X}}(t){cappello {X}}(0)-{cappello {X}}(0){cappello {X}}(t)rangle e^{iomega t}dt.} In the canonical ensemble, the second term can be re-expressed as {angolo dello stile di visualizzazione {cappello {X}}(0){cappello {X}}(t)sonaglio ={testo{tr }}e^{-beta {cappello {H}}}{cappello {X}}(0){cappello {X}}(t)={testo{tr }}{cappello {X}}(t)e^{-beta {cappello {H}}}{cappello {X}}(0)={testo{tr }}e^{-beta {cappello {H}}}underbrace {e^{beta {cappello {H}}}{cappello {X}}(t)e^{-beta {cappello {H}}}} _{{cappello {X}}(t-ihbar beta )}{cappello {X}}(0)= angolo {cappello {X}}(t-ihbar beta ){cappello {X}}(0)sonaglio } where in the second equality we re-positioned {stile di visualizzazione {cappello {X}}(t)} using the cyclic property of trace. Prossimo, in the third equality, we inserted {stile di visualizzazione e^{-beta {cappello {H}}}e^{beta {cappello {H}}}} next to the trace and interpreted {stile di visualizzazione e^{-beta {cappello {H}}}} as a time evolution operator {stile di visualizzazione e^{-{frac {io}{hbar }}{cappello {H}}Delta t}} with imaginary time interval {displaystyle Delta t=-ihbar beta } . The imaginary time shift turns into a {stile di visualizzazione e^{-beta hbar omega }} factor after Fourier transform {displaystyle int _{-infty }^{+infty }angolo {cappello {X}}(t-ihbar beta ){cappello {X}}(0)rangle e^{iomega t}dt=e^{-beta hbar omega }int _{-infty }^{+infty }angolo {cappello {X}}(t){cappello {X}}(0)rangle e^{iomega t}dt} and thus the expression for {stile di visualizzazione {testo{Io sono}}sinistra[chi (omega )Giusto]} can be easily rewritten as the quantum fluctuation-dissipation relation [7] {stile di visualizzazione S_{X}(omega )=2hbar left[n_{rm {ESSERE}}(omega )+1Giusto]{testo{Io sono}}sinistra[chi (omega )Giusto]} where the power spectral density {stile di visualizzazione S_{X}(omega )} is the Fourier transform of the auto-correlation {angolo dello stile di visualizzazione {cappello {X}}(t){cappello {X}}(0)sonaglio } e {stile di visualizzazione n_{rm {ESSERE}}(omega )= sinistra(e^{beta hbar omega }-1Giusto)^{-1}} is the Bose-Einstein distribution function. The same calculation also yields {stile di visualizzazione S_{X}(-omega )=e^{-beta hbar omega }S_{X}(omega )=2hbar left[n_{rm {ESSERE}}(omega )Giusto]{testo{Io sono}}sinistra[chi (omega )Giusto]neq S_{X}(+omega )} così, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, {angolo dello stile di visualizzazione {cappello {X}}(t){cappello {X}}(0)sonaglio } has an imaginary part originating from the commutation rules of operators.[8] The additional " {stile di visualizzazione +1} " term in the expression of {stile di visualizzazione 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 {stile di visualizzazione {frac {S_{X}(omega )+S_{X}(-omega )}{2}}=2hbar left[n_{rm {ESSERE}}(omega )+{frac {1}{2}}Giusto]{testo{Io sono}}sinistra[chi (omega )Giusto]=hbar coth left({frac {hbar omega }{2K_{B}T}}Giusto){testo{Io sono}}sinistra[chi (omega )Giusto].} Il " {stile di visualizzazione +1/2} " can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable {stile di visualizzazione {cappello {X}}} . At high enough temperatures, {stile di visualizzazione n_{rm {ESSERE}}ca (beta hbar omega )^{-1}gg 1} , cioè. 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 {stile di visualizzazione 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 (cioè. 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 {stile di visualizzazione T_{g}} , and left to equilibrate for a very long time {stile di visualizzazione t_{rm {w}}} under a magnetic field {stile di visualizzazione H} . Quindi, at a later time {displaystyle t+t_{rm {w}}} , two dynamical observables are probed, namely the response function {stile di visualizzazione chi (t+t_{rm {w}},t_{rm {w}})equiv left.{frac {partial m(t+t_{rm {w}})}{parziale h}}Giusto|_{H=0}} and the spin-temporal correlation function {stile di visualizzazione C(t+t_{rm {w}},t_{rm {w}})equivalente {frac {1}{V}}left.sum _{X}langle S_{X}(t_{rm {w}})S_{X}(t+t_{rm {w}})suona bene|_{H=0}} dove {stile di visualizzazione S_{X}= pm 1} is the spin living on the node {stile di visualizzazione x} of the cubic lattice of volume {stile di visualizzazione V} , e {textstyle m(t)equivalente {frac {1}{V}}somma _{X}langle S_{X}(t)sonaglio } 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". Revisione fisica. 83 (1): 34–40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34. ^ Einstein, Alberto (Maggio 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". Revisione fisica. 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". Revisione fisica B. 54 (22): 15754–15761. doi:10.1103/PhysRevB.54.15754. ISSN 0163-1829. ^ Kubo R (1966). "The fluctuation-dissipation theorem". Reports on Progress in Physics. 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". Caos: 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, UN. UN.; Devoret, M. H.; Girvin, S. M.; Marquardt, Florian; Schoelkopf, R. J. (2010). "Introduction to Quantum Noise, Measurement and Amplification". Reviews of Modern Physics. 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". Atti dell'Accademia Nazionale delle Scienze. 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". Lettere di revisione fisica. 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 (2015un) ^ Ansari Nazarov (2015b) References H. B. Callen, T. UN. Welton (1951). "Irreversibility and Generalized Noise". Revisione fisica. 83 (1): 34–40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34. l. D. Landau, e. M. Lifshitz (1980). Statistical Physics. Course of Theoretical Physics. vol. 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. w. Carlson of Purdue University Kubo's famous text: Fluctuation-dissipation theorem Weber J (1956). "Fluctuation Dissipation Theorem". Revisione fisica. 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". Giornale di fisica 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". Giornale di fisica 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. la stampa dell'università di 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, NJ: Sala dell'Apprendista. 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. New York: John Wiley e figli. pp. 153, 394–396. ISBN 0-471-81518-7. Callen HB (1985). Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley e figli. 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". Revisione fisica B. 91 (17): 174307. arXiv:1502.08020. Bibcode:2015PhRvB..91q4307A. doi:10.1103/PhysRevB.91.174307. S2CID 36847902. Categorie: Statistical mechanicsNon-equilibrium thermodynamicsPhysics theoremsStatistical mechanics theorems

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