# Fluctuation-dissipation theorem The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951 and expanded by Ryogo Kubo. There are antecedents to the general theorem, including Einstein's explanation of Brownian motion during his annus mirabilis and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors. Inhalt 1 Qualitative overview and examples 2 Examples in detail 2.1 Brownsche Bewegung 2.2 Thermal noise in a resistor 3 General formulation 4 Ableitung 4.1 Classical version 4.2 Quantum version 5 Violations in glassy systems 6 Quantum version 7 Siehe auch 8 Anmerkungen 9 Verweise 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 (z.B., 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. Auf diese Weise, light absorption turns light energy into heat. The corresponding fluctuation is thermal radiation (z.B., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. In der Tat, 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. Mit anderen Worten, 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={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, Johannes B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance {Anzeigestil R} , {Anzeigestil k_{rm {B}}T} , and the bandwidth {displaystyle Delta nu } over which the voltage is measured: {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. Nehmen, zum Beispiel, a simple circuit consisting of a resistor with a resistance {Anzeigestil R} and a capacitor with a small capacitance {Anzeigestil C} . Kirchhoff's law yields {displaystyle V=-R{frac {dQ}{dt}}+{frac {Q}{C}}} and so the response function for this circuit is {Displaystil Chi (Omega )Äquiv {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 {Anzeigestil {Text{Ich bin}}links[chi (Omega )Rechts]approx omega RC^{2}} which then can be linked to the power spectral density function {Anzeigestil S_{v}(Omega )} of the voltage via the fluctuation-dissipation theorem {Anzeigestil S_{v}(Omega )={frac {S_{Q}(Omega )}{C^{2}}}ca {frac {2k_{rm {B}}T}{C^{2}Omega }}{Text{Ich bin}}links[chi (Omega )Rechts]=2Rk_{rm {B}}T} The Johnson–Nyquist voltage noise {displaystyle langle V^{2}Rassel } was observed within a small frequency bandwidth {displaystyle Delta nu =Delta omega /(2Pi )} centered around {displaystyle omega =pm omega _{0}} . Somit {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:[Zitat benötigt].

Lassen {Anzeigestil x(t)} be an observable of a dynamical system with Hamiltonian {Anzeigestil H_{0}(x)} subject to thermal fluctuations. The observable {Anzeigestil x(t)} will fluctuate around its mean value {displaystyle langle xrangle _{0}} with fluctuations characterized by a power spectrum {Anzeigestil S_{x}(Omega )=Winkel {Hut {x}}(Omega ){Hut {x}}^{*}(Omega )Rassel } . Suppose that we can switch on a time-varying, spatially constant field {Anzeigestil f(t)} which alters the Hamiltonian to {Anzeigestil H(x)=H_{0}(x)-f(t)x} . The response of the observable {Anzeigestil x(t)} to a time-dependent field {Anzeigestil f(t)} is characterized to first order by the susceptibility or linear response function {Displaystil Chi (t)} of the system {Anzeigestil Sprache x(t)rangle =langle xrangle _{0}+int _{-unendlich }^{t}!f(Jawohl )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 (d.h. both positive and negative frequencies) von {Anzeigestil x} to the imaginary part of the Fourier transform {Anzeigestil {Hut {chi }}(Omega )} of the susceptibility {Displaystil Chi (t)} : {Anzeigestil S_{x}(Omega )=-{frac {2k_{Mathrm {B} }T}{Omega }}Name des Bedieners {Ich bin} {Hut {chi }}(Omega ).} Which holds under the Fourier transform convention {Anzeigestil f(Omega )=int _{-unendlich }^{unendlich }f(t)e^{-iomega t},dt} . The left-hand side describes fluctuations in {Anzeigestil x} , the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field {Anzeigestil f(t)=Fsin(omega t+phi )} .

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 {Anzeigestil 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. performed numerical simulations of macroscopic systems (d.h. 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 {Anzeigestil T_{g}} , and left to equilibrate for a very long time {Anzeigestil t_{rm {w}}} under a magnetic field {Anzeigestil H} . Dann, at a later time {displaystyle t+t_{rm {w}}} , two dynamical observables are probed, namely the response function {Displaystil Chi (t+t_{rm {w}},t_{rm {w}})equiv left.{frac {partial m(t+t_{rm {w}})}{partial H}}Rechts|_{H=0}} and the spin-temporal correlation function {Anzeigestil C(t+t_{rm {w}},t_{rm {w}})Äquiv {frac {1}{v}}left.sum _{x}langle S_{x}(t_{rm {w}})S_{x}(t+t_{rm {w}})rangle right|_{H=0}} wo {Anzeigestil S_{x}=pm 1} is the spin living on the node {Anzeigestil x} of the cubic lattice of volume {Anzeigestil V} , und {textstyle m(t)Äquiv {frac {1}{v}}Summe _{x}langle S_{x}(t)Rassel } 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  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. 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". Körperliche Überprüfung. 83 (1): 34–40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34. ^ Einstein, Albert (Kann 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". Körperliche Überprüfung. 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". Körperliche Überprüfung 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". 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, EIN. A.; 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". Proceedings of the National Academy of Sciences. 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". Briefe zur körperlichen Überprüfung. 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 (2015a) ^ Ansari Nazarov (2015b) References H. B. Callen, T. EIN. Welton (1951). "Irreversibility and Generalized Noise". Körperliche Überprüfung. 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". Körperliche Überprüfung. 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". Journal of Physics 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". Journal of Physics 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. Oxford University Press. 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: Lehrlingshalle. 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 und Söhne. pp. 153, 394–396. ISBN 0-471-81518-7. Callen HB (1985). Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley und Söhne. 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". Körperliche Überprüfung B. 91 (17): 174307. arXiv:1502.08020. Bibcode:2015PhRvB..91q4307A. doi:10.1103/PhysRevB.91.174307. S2CID 36847902. Kategorien: Statistical mechanicsNon-equilibrium thermodynamicsPhysics theoremsStatistical mechanics theorems

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