5257:
7961:
4816:
5252:{\displaystyle \langle {\hat {x}}(0){\hat {x}}(t)\rangle ={\text{Tr }}e^{-\beta {\hat {H}}}{\hat {x}}(0){\hat {x}}(t)={\text{Tr }}{\hat {x}}(t)e^{-\beta {\hat {H}}}{\hat {x}}(0)={\text{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 }
5773:
4801:
6674:
233:
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
7599:
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
500:
5545:
7326:
7962:
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,
174:. 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.
1053:
7151:
6307:
6861:
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.
2058:
1834:
1277:
3898:
730:
2965:
4605:
3658:
7159:
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2751:
1523:
4542:
4216:
146:. 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.
6125:
7592:
5768:{\displaystyle \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}
3335:
2148:
494:
7479:
4022:
6811:
891:
7030:
2343:
66:(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
5935:
3748:
6375:
6039:
5459:
4293:
3545:
5362:
1964:
844:
4107:
622:
1141:
190:
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
6137:
4597:
2421:
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.
3488:
2412:
2281:
2235:
4398:
5537:
3016:
5501:
3171:
1930:
1723:
1177:
2503:
1185:
289:
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1426:
1089:
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778:
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1621:
637:
6909:
3055:
5820:
2621:
6434:
5974:
1866:
883:
7022:
5298:
401:
119:
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
7362:
6967:
6938:
6855:
2369:
424:
6745:
2535:
3210:
1959:
1715:
1364:
7484:
4796:{\displaystyle {\text{Im}}\left={\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.}
6708:
6669:{\displaystyle {\frac {S_{x}(\omega )+S_{x}(-\omega )}{2}}=2\hbar \left{\text{Im}}\left=\hbar \coth \left({\frac {\hbar \omega }{2k_{B}T}}\right){\text{Im}}\left.}
3930:
1679:
1650:
1552:
1393:
1317:
7596:
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.
3556:
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7402:
7382:
6987:
2559:
2172:
1889:
553:
529:
369:
1434:
6312:
thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently,
8155:
Umberto Marini
Bettolo Marconi; Andrea Puglisi; Lamberto Rondoni; Angelo Vulpiani (2008). "Fluctuation-Dissipation: Response Theory in Statistical Physics".
7963:
Ricci-Tersenghi
Federico, Jesus Ruiz-Lorenzo Juan, Fabio Schifano Sebastiano, Seoane Beatriz, Tarancón Alfonso, Tripiccione Raffaele, Yllanes David (2017).
7909:
Clerk, A. A.; Devoret, M. H.; Girvin, S. M.; Marquardt, Florian; Schoelkopf, R. J. (2010). "Introduction to
Quantum Noise, Measurement and Amplification".
4118:
2840:
7965:"A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements"
6044:
8032:
7600:
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.
7321:{\displaystyle C(t+t_{\rm {w}},t_{\rm {w}})\equiv {\frac {1}{V}}\left.\sum _{x}\langle S_{x}(t_{\rm {w}})S_{x}(t+t_{\rm {w}})\rangle \right|_{H=0}}
4409:
2237:. The spectrum of fluctuations reveal the linear response, because past fluctuations cause future fluctuations via a linear response upon itself.
4599:
in the presence of a perturbing source. Once
Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as
3255:
2065:
17:
2628:
507:
This observation can be understood through the lens of the fluctuation-dissipation theorem. Take, for example, a simple circuit consisting of a
6869:, performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional
432:
8295:
Cristani A, Ritort F (2003). "Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence".
3938:
7407:
2434:
We derive the fluctuation–dissipation theorem in the form given above, using the same notation. Consider the following test case: the field
1048:{\displaystyle S_{V}(\omega )={\frac {S_{Q}(\omega )}{C^{2}}}\approx {\frac {2k_{\rm {B}}T}{C^{2}\omega }}{\text{Im}}\left=2Rk_{\rm {B}}T}
6750:
194:(e.g., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. Indeed,
2286:
166:
in it, the current will rapidly go to zero because of the resistance. Resistance dissipates electrical energy, turning it into heat (
6873:
using supercomputers. In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature
7705:"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
7481:
is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as
8512:
8502:
5828:
3671:
195:
6315:
5979:
5409:
4233:
3496:
5303:
180:
8357:
152:
142:(air resistance or fluid resistance). Drag dissipates kinetic energy, turning it into heat. The corresponding fluctuation is
786:
1871:
The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of
7146:{\displaystyle \chi (t+t_{\rm {w}},t_{\rm {w}})\equiv \left.{\frac {\partial m(t+t_{\rm {w}})}{\partial H}}\right|_{H=0}}
4049:
565:
4400:
in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called
1097:
6302:{\displaystyle S_{x}(-\omega )=e^{-\beta \hbar \omega }S_{x}(\omega )=2\hbar \left{\text{Im}}\left\neq S_{x}(+\omega )}
242:
8458:
8439:
8420:
8398:
8379:
1287:
The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:.
6825:, when detailed balance is violated comparison of fluctuations to dissipation is more complex. Below the so called
4558:
8210:
3371:
2374:
2243:
2181:
2053:{\displaystyle S_{x}(\omega )=-{\frac {2k_{\mathrm {B} }T}{\omega }}\operatorname {Im} {\hat {\chi }}(\omega ).}
1829:{\displaystyle \langle x(t)\rangle =\langle x\rangle _{0}+\int _{-\infty }^{t}\!f(\tau )\chi (t-\tau )\,d\tau ,}
8497:
7609:
4301:
1272:{\displaystyle \langle V^{2}\rangle \approx S_{V}(\omega )\times 2\Delta \nu \approx 4Rk_{\rm {B}}T\Delta \nu }
6821:
While the fluctuation–dissipation theorem provides a general relation between the response of systems obeying
5506:
2973:
8147:
8049:; Kurchan J. (1993). "Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model".
5468:
3063:
1897:
1146:
1092:
3893:{\displaystyle -{\hat {\chi }}(\omega )=i\omega \beta \int _{0}^{\infty }e^{-i\omega t}A(t)\,dt-\beta A(0).}
2448:
250:
8154:
7619:
5367:
1398:
1061:
725:{\displaystyle \chi (\omega )\equiv {\frac {Q(\omega )}{V(\omega )}}={\frac {1}{{\frac {1}{C}}-i\omega R}}}
3766:
To make a statement about frequency dependence, it is necessary to take the
Fourier transform of equation
3215:
3212:
is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for
2771:
738:
4032:
1557:
556:
6876:
3021:
295:
210:
5781:
8507:
348:
108:
6403:
5943:
2240:
This is the classical form of the theorem; quantum fluctuations are taken into account by replacing
1842:
852:
7614:
6992:
5265:
4548:
1686:
1682:
374:
7331:
6943:
6914:
6831:
6128:
2572:
2348:
406:
8349:
8343:
6721:
2511:
198:
confirms that the more effectively an object absorbs light, the more thermal radiation it emits.
7629:
6870:
6865:
To study the violation of the fluctuation-dissipation relation in glassy systems, particularly
2765:
3493:
Note that in the absence of a field the system is invariant under time-shifts. We can rewrite
3179:
1935:
1691:
1333:
8026:
7624:
1328:
340:
238:
7845:
3653:{\displaystyle f_{0}\int _{0}^{\infty }d\tau \,\chi (\tau )\theta (\tau -t)=\beta f_{0}A(t)}
8314:
8275:
8246:
8176:
8122:
8068:
7986:
7928:
7867:
7810:
7760:
7718:
7674:
7154:
6437:
6377:
has an imaginary part originating from the commutation rules of operators. The additional "
4227:
328:
63:
55:
6682:
3906:
1655:
1626:
1528:
1369:
1293:
8:
8046:
6711:
1518:{\displaystyle S_{x}(\omega )=\langle {\hat {x}}(\omega ){\hat {x}}^{*}(\omega )\rangle }
120:
47:
8318:
8279:
8250:
8180:
8126:
8072:
7990:
7932:
7871:
7814:
7764:
7722:
7678:
6380:
8330:
8304:
8236:
8192:
8166:
8092:
8058:
8009:
7976:
7964:
7944:
7918:
7891:
7857:
7826:
7709:
7660:
7634:
7387:
7367:
6972:
4807:
4028:
2544:
2538:
2157:
1874:
538:
514:
354:
320:
82:
8326:
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8394:
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8287:
8214:
8196:
8084:
8014:
7948:
7883:
7830:
7822:
7025:
6826:
6813:, i.e. the quantum contribution is negligible, and we recover the classical version.
6715:
4296:
4211:{\displaystyle S_{x}(\omega )=-{\frac {2k_{\text{B}}T}{\omega }}\operatorname {Im} .}
4040:
1892:
628:
306:
191:
184:
71:
67:
8334:
8322:
8283:
8254:
8184:
8130:
8096:
8076:
8004:
7994:
7936:
7895:
7875:
7818:
7768:
7726:
7682:
6822:
4552:
2960:{\displaystyle W(x,0)={\frac {\exp(-\beta H(x))}{\int dx'\,\exp(-\beta H(x'))}}\,,}
1324:
214:
51:
8188:
8266:
Felderhof BU (1978). "On the derivation of the fluctuation-dissipation theorem".
8227:
8157:
7751:
7700:
7665:
6120:{\displaystyle n_{\rm {BE}}(\omega )=\left(e^{\beta \hbar \omega }-1\right)^{-1}}
230:
226:
143:
132:
99:
94:
90:
7587:{\displaystyle T\chi (t+t_{\rm {w}},t_{\rm {w}})=1-C(t+t_{\rm {w}},t_{\rm {w}})}
8080:
7940:
7656:
5462:
4036:
1429:
309:
139:
128:
78:
7704:
3547:
using the susceptibility of the system and hence find with the above equation
503:
A simple circuit for illustrating
Johnson–Nyquist thermal noise in a resistor.
8491:
8258:
7746:
7731:
5300:
using the cyclic property of trace. Next, in the third equality, we inserted
2415:
351:. With no applied current, the mean-square voltage depends on the resistance
344:
213:
that quantifies the relation between the fluctuations in a system that obeys
171:
167:
156:
104:
8219:
8113:
H. B. Callen, T. A. Welton (1951). "Irreversibility and
Generalized Noise".
7999:
4537:{\displaystyle \chi (t-t')={\frac {i}{\hbar }}\theta (t-t')\langle \rangle }
8367:
8088:
8018:
7887:
7772:
6866:
4401:
3330:{\displaystyle \langle x(t)\rangle =\langle x\rangle _{0}+\beta f_{0}A(t),}
2143:{\displaystyle f(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}
8134:
7686:
2746:{\displaystyle \langle x(t)\rangle =\int dx'\int dx\,x'P(x',t|x,0)W(x,0).}
54:. Given that a system obeys detailed balance, the theorem is a proof that
8408:
8309:
8063:
7862:
5822:
can be easily rewritten as the quantum fluctuation-dissipation relation
1525:. Suppose that we can switch on a time-varying, spatially constant field
489:{\displaystyle \langle V^{2}\rangle \approx 4Rk_{\rm {B}}T\,\Delta \nu .}
7474:{\textstyle m(t)\equiv {\frac {1}{V}}\sum _{x}\langle S_{x}(t)\rangle }
6858:
6440:. An often cited result is also the symmetrized power spectral density
4017:{\displaystyle 2\operatorname {Im} =-\omega \beta {\hat {A}}(\omega ).}
1320:
86:
59:
7879:
499:
1839:
where the perturbation is adiabatically (very slowly) switched on at
532:
7981:
6806:{\displaystyle n_{\rm {BE}}\approx (\beta \hbar \omega )^{-1}\gg 1}
508:
163:
8241:
8171:
7923:
7843:
849:
which then can be linked to the power spectral density function
8469:
2338:{\displaystyle \hbar \,\coth(\hbar \omega /2k_{\mathrm {B} }T)}
58:
in a physical variable predict the response quantified by the
7749:(1928). "Thermal Agitation of Electric Charge in Conductors".
8141:
6436:
at positive frequencies can also be thought of as linked to
8045:
7908:
7220:
7081:
5930:{\displaystyle S_{x}(\omega )=2\hbar \left{\text{Im}}\left}
3743:{\displaystyle -\chi (t)=\beta {dA(t) \over dt}\theta (t).}
234:
one tries to perturb the system in a particular direction.
209:
The fluctuation–dissipation theorem is a general result of
8374:. Austin TX: University of Texas Press. pp. 545–595.
8112:
6370:{\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }
6034:{\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }
5454:{\displaystyle e^{-{\frac {i}{\hbar }}{\hat {H}}\Delta t}}
4288:{\displaystyle \langle {\hat {x}}(t){\hat {x}}(0)\rangle }
3540:{\displaystyle \langle x(t)\rangle -\langle x\rangle _{0}}
2764:,0) is an equilibrium distribution and hence given by the
162:
If electric current is running through a wire loop with a
89:. There are antecedents to the general theorem, including
8393:. Englewood Cliffs, NJ: Prentice Hall. pp. 251–296.
5357:{\displaystyle e^{-\beta {\hat {H}}}e^{\beta {\hat {H}}}}
217:
and the response of the system to applied perturbations.
8434:. New York: John Wiley and Sons. pp. 153, 394–396.
7850:
6131:
distribution function. The same calculation also yields
4295:(a measure of fluctuation) to the imaginary part of the
123:. This is best understood by considering some examples:
8388:
3770:. By integrating by parts, it is possible to show that
2174:, the right-hand side is closely related to the energy
885:
of the voltage via the fluctuation-dissipation theorem
138:
If an object is moving through a fluid, it experiences
8451:
Thermodynamics and an Introduction to Thermostatistics
7801:
Kubo R (1966). "The fluctuation-dissipation theorem".
7655:
7410:
7786:
Blundell, Stephen J.; Blundell, Katherine M. (2009).
7487:
7390:
7370:
7334:
7162:
7033:
6995:
6975:
6946:
6917:
6879:
6834:
6753:
6724:
6685:
6449:
6406:
6383:
6318:
6140:
6047:
5982:
5946:
5831:
5784:
5548:
5509:
5471:
5412:
5370:
5306:
5268:
4819:
4608:
4561:
4412:
4304:
4236:
4121:
4052:
3941:
3909:
3779:
3674:
3559:
3499:
3374:
3258:
3218:
3182:
3066:
3024:
2976:
2843:
2774:
2631:
2575:
2547:
2514:
2451:
2438:
has been on for infinite time and is switched off at
2377:
2351:
2289:
2246:
2184:
2160:
2068:
1967:
1938:
1900:
1877:
1845:
1726:
1694:
1658:
1629:
1560:
1531:
1437:
1401:
1372:
1336:
1296:
1188:
1149:
1100:
1064:
894:
855:
839:{\displaystyle {\text{Im}}\left\approx \omega RC^{2}}
789:
741:
640:
568:
541:
517:
435:
409:
377:
357:
253:
8294:
114:
8453:. New York: John Wiley and Sons. pp. 307–325.
8225:Weber J (1956). "Fluctuation Dissipation Theorem".
8220:
Kubo's famous text: Fluctuation-dissipation theorem
7024:, two dynamical observables are probed, namely the
4102:{\displaystyle S_{x}(\omega )={\hat {A}}(\omega ).}
2062:which holds under the Fourier transform convention
617:{\displaystyle V=-R{\frac {dQ}{dt}}+{\frac {Q}{C}}}
8470:"Easy as Pi: The Fluctuation-Dissipation Relation"
7586:
7473:
7396:
7376:
7356:
7320:
7145:
7016:
6981:
6961:
6932:
6903:
6849:
6805:
6739:
6702:
6668:
6428:
6392:
6369:
6301:
6119:
6033:
5968:
5929:
5814:
5767:
5531:
5495:
5453:
5398:
5356:
5292:
5251:
4795:
4591:
4536:
4392:
4287:
4210:
4101:
4016:
3924:
3892:
3742:
3652:
3539:
3482:
3329:
3239:
3204:
3165:
3049:
3010:
2959:
2827:
2745:
2615:
2553:
2529:
2497:
2406:
2363:
2337:
2275:
2229:
2178:by the system when pumped by an oscillatory field
2166:
2142:
2052:
1953:
1924:
1883:
1860:
1828:
1709:
1673:
1644:
1615:
1546:
1517:
1420:
1387:
1358:
1311:
1271:
1171:
1136:{\displaystyle \Delta \nu =\Delta \omega /(2\pi )}
1135:
1083:
1047:
877:
838:
772:
724:
616:
547:
523:
488:
418:
395:
363:
283:
77:The fluctuation–dissipation theorem was proven by
8415:. Oxford: Pergamon Press. pp. 443, 474–477.
7663:(1951). "Irreversibility and Generalized Noise".
5976:is the Fourier transform of the auto-correlation
4226:The fluctuation-dissipation theorem relates the
1785:
50:for predicting the behavior of systems that obey
8489:
7846:"Fundamental aspects of quantum Brownian motion"
7785:
1366:subject to thermal fluctuations. The observable
7969:Proceedings of the National Academy of Sciences
7651:
7649:
6940:, and left to equilibrate for a very long time
6816:
237:From this observation Einstein was able to use
5262:where in the second equality we re-positioned
334:
4592:{\displaystyle \langle {\hat {x}}(t)\rangle }
8345:Introduction to Modern Statistical Mechanics
8265:
8031:: CS1 maint: multiple names: authors list (
7646:
7468:
7446:
7298:
7233:
6364:
6319:
6028:
5983:
5740:
5695:
5627:
5570:
5246:
5189:
4865:
4820:
4765:
4675:
4586:
4562:
4547:which follows, under the assumptions of the
4531:
4472:
4282:
4237:
3528:
3521:
3515:
3500:
3468:
3455:
3448:
3418:
3411:
3390:
3287:
3280:
3274:
3259:
3234:
3219:
3145:
3138:
2647:
2632:
1755:
1748:
1742:
1727:
1512:
1460:
1409:
1402:
1202:
1189:
1078:
1065:
449:
436:
7779:
8407:
8341:
3483:{\displaystyle A(t)=\langle \rangle _{0}.}
2541:. We can express the expectation value of
2407:{\displaystyle 2k_{\mathrm {B} }T/\omega }
2276:{\displaystyle 2k_{\mathrm {B} }T/\omega }
2230:{\displaystyle f(t)=F\sin(\omega t+\phi )}
8448:
8366:
8308:
8240:
8170:
8062:
8008:
7998:
7980:
7955:
7922:
7861:
7844:Hänggi Peter, Ingold Gert-Ludwig (2005).
7745:
7730:
4810:, the second term can be re-expressed as
4393:{\displaystyle {\text{Im}}\left=\left/2i}
3862:
3591:
2953:
2914:
2676:
2418:, an identity from quantum field theory.
2293:
2133:
1816:
476:
264:
7699:
5532:{\displaystyle e^{-\beta \hbar \omega }}
5503:. The imaginary time shift turns into a
3011:{\displaystyle \beta ^{-1}=k_{\rm {B}}T}
2414:). A proof can be found by means of the
1681:is characterized to first order by the
498:
8467:
8429:
8224:
8213:of a lecture by Prof. E. W. Carlson of
5496:{\displaystyle \Delta t=-i\hbar \beta }
4551:theory, from the time evolution of the
3932:is real and symmetric, it follows that
3166:{\displaystyle W(x,0)\approx W_{0}(x),}
1925:{\displaystyle {\hat {\chi }}(\omega )}
1172:{\displaystyle \omega =\pm \omega _{0}}
14:
8490:
8372:A Modern Course in Statistical Physics
7800:
3361:) is the auto-correlation function of
2756:The probability distribution function
2498:{\displaystyle f(t)=f_{0}\theta (-t),}
1282:
1091:was observed within a small frequency
284:{\displaystyle D={\mu \,k_{\rm {B}}T}}
8142:L. D. Landau, E. M. Lifshitz (1980).
5399:{\displaystyle e^{-\beta {\hat {H}}}}
1428:with fluctuations characterized by a
1421:{\displaystyle \langle x\rangle _{0}}
1395:will fluctuate around its mean value
1084:{\displaystyle \langle V^{2}\rangle }
204:
8348:. Oxford University Press. pp.
3665:
3249:
3240:{\displaystyle \langle x(t)\rangle }
3057:, we can expand the right-hand side
2828:{\displaystyle H(x)=H_{0}(x)-xf_{0}}
2429:
773:{\displaystyle \omega \ll (RC)^{-1}}
426:over which the voltage is measured:
196:Kirchhoff's law of thermal radiation
170:). The corresponding fluctuation is
2569:,0) and the transition probability
1616:{\displaystyle H(x)=H_{0}(x)-f(t)x}
24:
8204:
7575:
7560:
7524:
7509:
7289:
7255:
7196:
7181:
7118:
7107:
7086:
7067:
7052:
7008:
6953:
6924:
6895:
6841:
6763:
6760:
6526:
6523:
6227:
6224:
6057:
6054:
5874:
5871:
5690:
5682:
5565:
5557:
5472:
5443:
5364:next to the trace and interpreted
4670:
4662:
4221:
4043:of the auto-correlation function:
3826:
3580:
2999:
2387:
2323:
2256:
2097:
2092:
2005:
1855:
1775:
1263:
1254:
1233:
1110:
1101:
1058:The Johnson–Nyquist voltage noise
1036:
967:
477:
467:
410:
384:
271:
220:
25:
8524:
6904:{\displaystyle T=0.64T_{\rm {g}}}
6778:
6710:" can be thought of as linked to
6589:
6509:
6210:
6177:
6089:
5940:where the power spectral density
5857:
5666:
5597:
5521:
5487:
5426:
5216:
5154:
4648:
4444:
3050:{\displaystyle \beta xf_{0}\ll 1}
2303:
1623:. The response of the observable
115:Qualitative overview and examples
8389:Plischke M, Bergersen B (1989).
5815:{\displaystyle {\text{Im}}\left}
2561:by the probability distribution
1554:which alters the Hamiltonian to
40:fluctuation–dissipation relation
8391:Equilibrium Statistical Physics
7384:of the cubic lattice of volume
7364:is the spin living on the node
6747:. At high enough temperatures,
5539:factor after Fourier transform
2150:. The left-hand side describes
780:, its imaginary part is simply
32:fluctuation–dissipation theorem
18:Fluctuation-dissipation theorem
8513:Statistical mechanics theorems
8503:Non-equilibrium thermodynamics
8039:
7902:
7837:
7803:Reports on Progress in Physics
7794:
7739:
7693:
7610:Non-equilibrium thermodynamics
7581:
7545:
7530:
7494:
7465:
7459:
7420:
7414:
7295:
7274:
7261:
7246:
7202:
7166:
7113:
7092:
7073:
7037:
6785:
6772:
6731:
6655:
6649:
6578:
6572:
6538:
6532:
6494:
6485:
6469:
6463:
6429:{\displaystyle S_{x}(\omega )}
6423:
6417:
6361:
6355:
6349:
6340:
6334:
6328:
6296:
6287:
6266:
6260:
6239:
6233:
6201:
6195:
6160:
6151:
6069:
6063:
6025:
6019:
6013:
6004:
5998:
5992:
5969:{\displaystyle S_{x}(\omega )}
5963:
5957:
5919:
5913:
5886:
5880:
5848:
5842:
5804:
5798:
5737:
5731:
5725:
5716:
5710:
5704:
5624:
5618:
5612:
5603:
5585:
5579:
5437:
5388:
5346:
5324:
5287:
5281:
5275:
5243:
5237:
5231:
5222:
5204:
5198:
5183:
5177:
5171:
5160:
5142:
5136:
5115:
5095:
5089:
5083:
5069:
5043:
5015:
5009:
5003:
4989:
4969:
4963:
4957:
4940:
4934:
4928:
4919:
4913:
4907:
4893:
4862:
4856:
4850:
4841:
4835:
4829:
4762:
4756:
4750:
4741:
4735:
4729:
4717:
4711:
4705:
4696:
4690:
4684:
4628:
4622:
4583:
4577:
4571:
4528:
4525:
4514:
4508:
4496:
4490:
4484:
4475:
4469:
4452:
4433:
4416:
4371:
4365:
4349:
4343:
4324:
4318:
4279:
4273:
4267:
4258:
4252:
4246:
4230:of the observable of interest
4202:
4199:
4193:
4187:
4178:
4138:
4132:
4093:
4087:
4081:
4069:
4063:
4008:
4002:
3996:
3975:
3972:
3966:
3960:
3951:
3919:
3913:
3884:
3878:
3859:
3853:
3801:
3795:
3789:
3734:
3728:
3711:
3705:
3687:
3681:
3647:
3641:
3619:
3607:
3601:
3595:
3512:
3506:
3464:
3442:
3436:
3430:
3427:
3405:
3399:
3393:
3384:
3378:
3321:
3315:
3271:
3265:
3231:
3225:
3199:
3193:
3157:
3154:
3129:
3107:
3104:
3098:
3082:
3070:
2947:
2944:
2933:
2921:
2895:
2892:
2886:
2874:
2859:
2847:
2806:
2800:
2784:
2778:
2737:
2725:
2719:
2706:
2688:
2644:
2638:
2610:
2597:
2579:
2524:
2518:
2489:
2480:
2461:
2455:
2355:
2332:
2300:
2224:
2209:
2194:
2188:
2111:
2105:
2078:
2072:
2044:
2038:
2032:
1984:
1978:
1948:
1942:
1919:
1913:
1907:
1861:{\displaystyle \tau =-\infty }
1813:
1801:
1795:
1789:
1739:
1733:
1704:
1698:
1668:
1662:
1639:
1633:
1607:
1601:
1592:
1586:
1570:
1564:
1541:
1535:
1509:
1503:
1491:
1481:
1475:
1469:
1454:
1448:
1382:
1376:
1353:
1347:
1306:
1300:
1224:
1218:
1130:
1121:
1013:
1007:
936:
930:
911:
905:
878:{\displaystyle S_{V}(\omega )}
872:
866:
809:
803:
758:
748:
682:
676:
668:
662:
650:
644:
305:, the ratio of the particle's
243:Einstein–Smoluchowski relation
13:
1:
8189:10.1016/j.physrep.2008.02.002
8148:Course of Theoretical Physics
8106:
7017:{\displaystyle t+t_{\rm {w}}}
5406:as a time evolution operator
5293:{\displaystyle {\hat {x}}(t)}
2424:
1891:to the imaginary part of the
7620:Onsager reciprocal relations
6911:below the glass temperature
6817:Violations in glassy systems
6400:" term in the expression of
5778:and thus the expression for
3756:
396:{\displaystyle k_{\rm {B}}T}
7:
8327:10.1088/0305-4470/36/21/201
8150:. Vol. 5 (3 ed.).
7788:Concepts in thermal physics
7603:
7357:{\displaystyle S_{x}=\pm 1}
6962:{\displaystyle t_{\rm {w}}}
6933:{\displaystyle T_{\rm {g}}}
6850:{\displaystyle T_{\rm {g}}}
4112:Therefore, it follows that
3365:in the absence of a field:
3343:
2616:{\displaystyle P(x',t|x,0)}
2364:{\displaystyle \hbar \to 0}
735:In the low-frequency limit
419:{\displaystyle \Delta \nu }
335:Thermal noise in a resistor
229:noted in his 1905 paper on
27:Statistical physics theorem
10:
8529:
8288:10.1088/0305-4470/11/5/021
8081:10.1103/PhysRevLett.71.173
7941:10.1103/RevModPhys.82.1155
7823:10.1088/0034-4885/29/1/306
6740:{\displaystyle {\hat {x}}}
4035:states that the two-sided
2530:{\displaystyle \theta (t)}
1652:to a time-dependent field
301:and the particle mobility
211:statistical thermodynamics
107:'s explanation in 1928 of
56:thermodynamic fluctuations
7911:Reviews of Modern Physics
535:with a small capacitance
111:in electrical resistors.
85:in 1951 and expanded by
8259:10.1103/PhysRev.101.1620
7732:10.1002/andp.19053220806
7640:
6989:. Then, at a later time
3205:{\displaystyle W_{0}(x)}
1954:{\displaystyle \chi (t)}
1710:{\displaystyle \chi (t)}
1687:linear response function
1359:{\displaystyle H_{0}(x)}
46:) is a powerful tool in
8051:Physical Review Letters
8000:10.1073/pnas.1621242114
6969:under a magnetic field
4033:Wiener–Khinchin theorem
8468:Mazonka, Oleg (2016).
7773:10.1103/PhysRev.32.110
7630:Boltzmann distribution
7588:
7475:
7398:
7378:
7358:
7322:
7153:and the spin-temporal
7147:
7018:
6983:
6963:
6934:
6905:
6871:Edwards-Anderson model
6851:
6807:
6741:
6704:
6670:
6430:
6394:
6371:
6303:
6121:
6035:
5970:
5931:
5816:
5769:
5533:
5497:
5455:
5400:
5358:
5294:
5253:
4797:
4593:
4538:
4394:
4289:
4212:
4103:
4018:
3926:
3894:
3744:
3654:
3541:
3484:
3331:
3241:
3206:
3167:
3051:
3012:
2961:
2829:
2766:Boltzmann distribution
2747:
2617:
2555:
2531:
2499:
2408:
2365:
2339:
2277:
2231:
2168:
2144:
2054:
1955:
1932:of the susceptibility
1926:
1885:
1862:
1830:
1711:
1675:
1646:
1617:
1548:
1519:
1422:
1389:
1360:
1313:
1273:
1173:
1137:
1085:
1049:
879:
840:
774:
726:
618:
549:
525:
504:
490:
420:
397:
365:
285:
8498:Statistical mechanics
8432:Statistical Mechanics
8413:Statistical Mechanics
8135:10.1103/PhysRev.83.34
7687:10.1103/PhysRev.83.34
7625:Equipartition theorem
7589:
7476:
7399:
7379:
7359:
7323:
7148:
7019:
6984:
6964:
6935:
6906:
6852:
6808:
6742:
6705:
6671:
6431:
6395:
6372:
6304:
6122:
6036:
5971:
5932:
5817:
5770:
5534:
5498:
5456:
5401:
5359:
5295:
5254:
4798:
4594:
4539:
4395:
4290:
4213:
4104:
4019:
3927:
3895:
3745:
3655:
3542:
3485:
3332:
3242:
3207:
3168:
3052:
3013:
2962:
2830:
2748:
2618:
2556:
2532:
2500:
2409:
2366:
2340:
2278:
2232:
2169:
2145:
2055:
1956:
1927:
1886:
1863:
1831:
1712:
1676:
1647:
1618:
1549:
1520:
1423:
1390:
1361:
1314:
1274:
1174:
1138:
1086:
1050:
880:
841:
775:
727:
619:
550:
526:
502:
491:
421:
398:
366:
349:Johnson–Nyquist noise
312:to an applied force.
286:
239:statistical mechanics
8477:Journal of Reference
8297:Journal of Physics A
8268:Journal of Physics A
7615:Green–Kubo relations
7485:
7408:
7388:
7368:
7332:
7160:
7155:correlation function
7031:
6993:
6973:
6944:
6915:
6877:
6832:
6751:
6722:
6712:quantum fluctuations
6703:{\displaystyle +1/2}
6683:
6447:
6438:spontaneous emission
6404:
6381:
6316:
6138:
6045:
5980:
5944:
5829:
5782:
5546:
5507:
5469:
5410:
5368:
5304:
5266:
4817:
4606:
4559:
4410:
4302:
4234:
4228:correlation function
4119:
4050:
4029:stationary processes
3939:
3925:{\displaystyle A(t)}
3907:
3777:
3672:
3557:
3497:
3372:
3256:
3216:
3180:
3064:
3022:
2974:
2841:
2772:
2768:for the Hamiltonian
2629:
2573:
2545:
2512:
2449:
2375:
2349:
2287:
2244:
2182:
2158:
2066:
1965:
1936:
1898:
1875:
1843:
1724:
1692:
1674:{\displaystyle f(t)}
1656:
1645:{\displaystyle x(t)}
1627:
1558:
1547:{\displaystyle f(t)}
1529:
1435:
1399:
1388:{\displaystyle x(t)}
1370:
1334:
1312:{\displaystyle x(t)}
1294:
1186:
1147:
1098:
1062:
892:
853:
787:
739:
638:
631:for this circuit is
566:
539:
515:
433:
407:
403:, and the bandwidth
375:
355:
329:absolute temperature
251:
121:thermal fluctuations
8342:Chandler D (1987).
8319:2003JPhA...36R.181C
8280:1978JPhA...11..921F
8251:1956PhRv..101.1620W
8181:2008PhR...461..111M
8144:Statistical Physics
8127:1951PhRv...83...34C
8073:1993PhRvL..71..173C
7991:2017PNAS..114.1838B
7933:2010RvMP...82.1155C
7872:2005Chaos..15b6105H
7815:1966RPPh...29..255K
7765:1928PhRv...32..110N
7723:1905AnP...322..549E
7679:1951PhRv...83...34C
5694:
5569:
4674:
3830:
3584:
3018:. For a weak field
2101:
1784:
1283:General formulation
559:voltage law yields
294:which connects the
48:statistical physics
8449:Callen HB (1985).
7710:Annalen der Physik
7635:Dissipative system
7584:
7471:
7445:
7394:
7374:
7354:
7318:
7232:
7143:
7014:
6979:
6959:
6930:
6901:
6847:
6803:
6737:
6718:of the observable
6700:
6666:
6426:
6393:{\displaystyle +1}
6390:
6367:
6299:
6117:
6031:
5966:
5927:
5812:
5765:
5674:
5549:
5529:
5493:
5451:
5396:
5354:
5290:
5249:
5164:
5127:
4808:canonical ensemble
4793:
4654:
4589:
4555:of the observable
4534:
4390:
4285:
4208:
4099:
4014:
3922:
3890:
3816:
3740:
3650:
3570:
3537:
3480:
3327:
3237:
3202:
3163:
3047:
3008:
2957:
2825:
2743:
2613:
2551:
2539:Heaviside function
2527:
2495:
2404:
2361:
2335:
2273:
2227:
2164:
2140:
2084:
2050:
1951:
1922:
1881:
1858:
1826:
1767:
1707:
1671:
1642:
1613:
1544:
1515:
1418:
1385:
1356:
1309:
1269:
1169:
1133:
1081:
1045:
875:
836:
770:
722:
614:
545:
521:
511:with a resistance
505:
486:
416:
393:
361:
321:Boltzmann constant
296:diffusion constant
281:
205:Examples in detail
93:'s explanation of
72:quantum mechanical
8359:978-0-19-504277-1
8303:(21): R181–R290.
8215:Purdue University
8047:Cugliandolo L. F.
7880:10.1063/1.1853631
7436:
7434:
7397:{\displaystyle V}
7377:{\displaystyle x}
7223:
7216:
7125:
7026:response function
6982:{\displaystyle H}
6827:glass temperature
6734:
6716:zero-point motion
6639:
6630:
6562:
6552:
6501:
6352:
6331:
6250:
6016:
5995:
5903:
5788:
5728:
5707:
5615:
5582:
5440:
5429:
5391:
5349:
5327:
5278:
5234:
5201:
5174:
5139:
5118:
5086:
5072:
5053:
5051:
5046:
5024:
5006:
4992:
4960:
4949:
4931:
4910:
4896:
4874:
4853:
4832:
4753:
4732:
4708:
4687:
4652:
4612:
4574:
4511:
4487:
4447:
4308:
4297:response function
4270:
4249:
4190:
4170:
4160:
4084:
4041:Fourier transform
3999:
3963:
3792:
3764:
3763:
3723:
3351:
3350:
2951:
2554:{\displaystyle x}
2430:Classical version
2345:(whose limit for
2167:{\displaystyle x}
2035:
2018:
1910:
1893:Fourier transform
1884:{\displaystyle x}
1494:
1472:
997:
992:
950:
793:
720:
705:
686:
629:response function
612:
599:
548:{\displaystyle C}
524:{\displaystyle R}
364:{\displaystyle R}
192:thermal radiation
185:thermal radiation
16:(Redirected from
8520:
8508:Physics theorems
8484:
8474:
8464:
8445:
8430:Huang K (1987).
8426:
8404:
8385:
8363:
8338:
8312:
8310:cond-mat/0212490
8291:
8262:
8244:
8235:(6): 1620–1626.
8200:
8174:
8165:(4–6): 111–195.
8151:
8138:
8101:
8100:
8066:
8064:cond-mat/9303036
8043:
8037:
8036:
8030:
8022:
8012:
8002:
7984:
7975:(8): 1838–1843.
7959:
7953:
7952:
7926:
7906:
7900:
7899:
7865:
7863:quant-ph/0412052
7841:
7835:
7834:
7798:
7792:
7791:
7783:
7777:
7776:
7743:
7737:
7736:
7734:
7701:Einstein, Albert
7697:
7691:
7690:
7653:
7593:
7591:
7590:
7585:
7580:
7579:
7578:
7565:
7564:
7563:
7529:
7528:
7527:
7514:
7513:
7512:
7480:
7478:
7477:
7472:
7458:
7457:
7444:
7435:
7427:
7403:
7401:
7400:
7395:
7383:
7381:
7380:
7375:
7363:
7361:
7360:
7355:
7344:
7343:
7327:
7325:
7324:
7319:
7317:
7316:
7305:
7301:
7294:
7293:
7292:
7273:
7272:
7260:
7259:
7258:
7245:
7244:
7231:
7217:
7209:
7201:
7200:
7199:
7186:
7185:
7184:
7152:
7150:
7149:
7144:
7142:
7141:
7130:
7126:
7124:
7116:
7112:
7111:
7110:
7084:
7072:
7071:
7070:
7057:
7056:
7055:
7023:
7021:
7020:
7015:
7013:
7012:
7011:
6988:
6986:
6985:
6980:
6968:
6966:
6965:
6960:
6958:
6957:
6956:
6939:
6937:
6936:
6931:
6929:
6928:
6927:
6910:
6908:
6907:
6902:
6900:
6899:
6898:
6856:
6854:
6853:
6848:
6846:
6845:
6844:
6823:detailed balance
6812:
6810:
6809:
6804:
6796:
6795:
6768:
6767:
6766:
6746:
6744:
6743:
6738:
6736:
6735:
6727:
6709:
6707:
6706:
6701:
6696:
6675:
6673:
6672:
6667:
6662:
6658:
6640:
6637:
6635:
6631:
6629:
6625:
6624:
6611:
6603:
6585:
6581:
6563:
6560:
6558:
6554:
6553:
6545:
6531:
6530:
6529:
6502:
6497:
6484:
6483:
6462:
6461:
6451:
6435:
6433:
6432:
6427:
6416:
6415:
6399:
6397:
6396:
6391:
6376:
6374:
6373:
6368:
6354:
6353:
6345:
6333:
6332:
6324:
6308:
6306:
6305:
6300:
6286:
6285:
6273:
6269:
6251:
6248:
6246:
6242:
6232:
6231:
6230:
6194:
6193:
6184:
6183:
6150:
6149:
6126:
6124:
6123:
6118:
6116:
6115:
6107:
6103:
6096:
6095:
6062:
6061:
6060:
6040:
6038:
6037:
6032:
6018:
6017:
6009:
5997:
5996:
5988:
5975:
5973:
5972:
5967:
5956:
5955:
5936:
5934:
5933:
5928:
5926:
5922:
5904:
5901:
5899:
5895:
5879:
5878:
5877:
5841:
5840:
5821:
5819:
5818:
5813:
5811:
5807:
5789:
5786:
5774:
5772:
5771:
5766:
5758:
5757:
5730:
5729:
5721:
5709:
5708:
5700:
5693:
5685:
5673:
5672:
5645:
5644:
5617:
5616:
5608:
5584:
5583:
5575:
5568:
5560:
5538:
5536:
5535:
5530:
5528:
5527:
5502:
5500:
5499:
5494:
5460:
5458:
5457:
5452:
5450:
5449:
5442:
5441:
5433:
5430:
5422:
5405:
5403:
5402:
5397:
5395:
5394:
5393:
5392:
5384:
5363:
5361:
5360:
5355:
5353:
5352:
5351:
5350:
5342:
5331:
5330:
5329:
5328:
5320:
5299:
5297:
5296:
5291:
5280:
5279:
5271:
5258:
5256:
5255:
5250:
5236:
5235:
5227:
5203:
5202:
5194:
5176:
5175:
5167:
5163:
5141:
5140:
5132:
5128:
5123:
5122:
5121:
5120:
5119:
5111:
5088:
5087:
5079:
5076:
5075:
5074:
5073:
5065:
5050:
5049:
5048:
5047:
5039:
5025:
5022:
5008:
5007:
4999:
4996:
4995:
4994:
4993:
4985:
4962:
4961:
4953:
4950:
4947:
4933:
4932:
4924:
4912:
4911:
4903:
4900:
4899:
4898:
4897:
4889:
4875:
4872:
4855:
4854:
4846:
4834:
4833:
4825:
4802:
4800:
4799:
4794:
4783:
4782:
4755:
4754:
4746:
4734:
4733:
4725:
4710:
4709:
4701:
4689:
4688:
4680:
4673:
4665:
4653:
4651:
4640:
4635:
4631:
4613:
4610:
4598:
4596:
4595:
4590:
4576:
4575:
4567:
4553:ensemble average
4543:
4541:
4540:
4535:
4524:
4513:
4512:
4504:
4489:
4488:
4480:
4468:
4448:
4440:
4432:
4399:
4397:
4396:
4391:
4383:
4378:
4374:
4364:
4363:
4331:
4327:
4309:
4306:
4294:
4292:
4291:
4286:
4272:
4271:
4263:
4251:
4250:
4242:
4217:
4215:
4214:
4209:
4192:
4191:
4183:
4171:
4166:
4162:
4161:
4158:
4148:
4131:
4130:
4108:
4106:
4105:
4100:
4086:
4085:
4077:
4062:
4061:
4039:is equal to the
4037:spectral density
4023:
4021:
4020:
4015:
4001:
4000:
3992:
3965:
3964:
3956:
3931:
3929:
3928:
3923:
3899:
3897:
3896:
3891:
3849:
3848:
3829:
3824:
3794:
3793:
3785:
3758:
3749:
3747:
3746:
3741:
3724:
3722:
3714:
3697:
3666:
3659:
3657:
3656:
3651:
3637:
3636:
3583:
3578:
3569:
3568:
3546:
3544:
3543:
3538:
3536:
3535:
3489:
3487:
3486:
3481:
3476:
3475:
3463:
3462:
3426:
3425:
3345:
3336:
3334:
3333:
3328:
3311:
3310:
3295:
3294:
3250:
3246:
3244:
3243:
3238:
3211:
3209:
3208:
3203:
3192:
3191:
3172:
3170:
3169:
3164:
3153:
3152:
3128:
3127:
3097:
3096:
3056:
3054:
3053:
3048:
3040:
3039:
3017:
3015:
3014:
3009:
3004:
3003:
3002:
2989:
2988:
2966:
2964:
2963:
2958:
2952:
2950:
2943:
2913:
2898:
2866:
2834:
2832:
2831:
2826:
2824:
2823:
2799:
2798:
2752:
2750:
2749:
2744:
2709:
2698:
2684:
2666:
2622:
2620:
2619:
2614:
2600:
2589:
2560:
2558:
2557:
2552:
2536:
2534:
2533:
2528:
2504:
2502:
2501:
2496:
2476:
2475:
2413:
2411:
2410:
2405:
2400:
2392:
2391:
2390:
2370:
2368:
2367:
2362:
2344:
2342:
2341:
2336:
2328:
2327:
2326:
2313:
2282:
2280:
2279:
2274:
2269:
2261:
2260:
2259:
2236:
2234:
2233:
2228:
2173:
2171:
2170:
2165:
2149:
2147:
2146:
2141:
2132:
2131:
2100:
2095:
2059:
2057:
2056:
2051:
2037:
2036:
2028:
2019:
2014:
2010:
2009:
2008:
1994:
1977:
1976:
1960:
1958:
1957:
1952:
1931:
1929:
1928:
1923:
1912:
1911:
1903:
1890:
1888:
1887:
1882:
1867:
1865:
1864:
1859:
1835:
1833:
1832:
1827:
1783:
1778:
1763:
1762:
1716:
1714:
1713:
1708:
1680:
1678:
1677:
1672:
1651:
1649:
1648:
1643:
1622:
1620:
1619:
1614:
1585:
1584:
1553:
1551:
1550:
1545:
1524:
1522:
1521:
1516:
1502:
1501:
1496:
1495:
1487:
1474:
1473:
1465:
1447:
1446:
1427:
1425:
1424:
1419:
1417:
1416:
1394:
1392:
1391:
1386:
1365:
1363:
1362:
1357:
1346:
1345:
1325:dynamical system
1318:
1316:
1315:
1310:
1278:
1276:
1275:
1270:
1259:
1258:
1257:
1217:
1216:
1201:
1200:
1178:
1176:
1175:
1170:
1168:
1167:
1143:centered around
1142:
1140:
1139:
1134:
1120:
1090:
1088:
1087:
1082:
1077:
1076:
1054:
1052:
1051:
1046:
1041:
1040:
1039:
1020:
1016:
998:
995:
993:
991:
987:
986:
976:
972:
971:
970:
956:
951:
949:
948:
939:
929:
928:
918:
904:
903:
884:
882:
881:
876:
865:
864:
845:
843:
842:
837:
835:
834:
816:
812:
794:
791:
779:
777:
776:
771:
769:
768:
731:
729:
728:
723:
721:
719:
706:
698:
692:
687:
685:
671:
657:
623:
621:
620:
615:
613:
605:
600:
598:
590:
582:
554:
552:
551:
546:
530:
528:
527:
522:
495:
493:
492:
487:
472:
471:
470:
448:
447:
425:
423:
422:
417:
402:
400:
399:
394:
389:
388:
387:
370:
368:
367:
362:
290:
288:
287:
282:
280:
276:
275:
274:
215:detailed balance
181:Light absorption
52:detailed balance
21:
8528:
8527:
8523:
8522:
8521:
8519:
8518:
8517:
8488:
8487:
8472:
8461:
8442:
8423:
8401:
8382:
8360:
8228:Physical Review
8211:Audio recording
8207:
8205:Further reading
8158:Physics Reports
8115:Physical Review
8109:
8104:
8044:
8040:
8024:
8023:
7960:
7956:
7907:
7903:
7842:
7838:
7799:
7795:
7784:
7780:
7752:Physical Review
7744:
7740:
7698:
7694:
7666:Physical Review
7654:
7647:
7643:
7606:
7574:
7573:
7569:
7559:
7558:
7554:
7523:
7522:
7518:
7508:
7507:
7503:
7486:
7483:
7482:
7453:
7449:
7440:
7426:
7409:
7406:
7405:
7389:
7386:
7385:
7369:
7366:
7365:
7339:
7335:
7333:
7330:
7329:
7306:
7288:
7287:
7283:
7268:
7264:
7254:
7253:
7249:
7240:
7236:
7227:
7222:
7219:
7218:
7208:
7195:
7194:
7190:
7180:
7179:
7175:
7161:
7158:
7157:
7131:
7117:
7106:
7105:
7101:
7085:
7083:
7080:
7079:
7066:
7065:
7061:
7051:
7050:
7046:
7032:
7029:
7028:
7007:
7006:
7002:
6994:
6991:
6990:
6974:
6971:
6970:
6952:
6951:
6947:
6945:
6942:
6941:
6923:
6922:
6918:
6916:
6913:
6912:
6894:
6893:
6889:
6878:
6875:
6874:
6840:
6839:
6835:
6833:
6830:
6829:
6819:
6788:
6784:
6759:
6758:
6754:
6752:
6749:
6748:
6726:
6725:
6723:
6720:
6719:
6692:
6684:
6681:
6680:
6645:
6641:
6636:
6620:
6616:
6612:
6604:
6602:
6598:
6568:
6564:
6559:
6544:
6522:
6521:
6517:
6516:
6512:
6479:
6475:
6457:
6453:
6452:
6450:
6448:
6445:
6444:
6411:
6407:
6405:
6402:
6401:
6382:
6379:
6378:
6344:
6343:
6323:
6322:
6317:
6314:
6313:
6281:
6277:
6256:
6252:
6247:
6223:
6222:
6218:
6217:
6213:
6189:
6185:
6170:
6166:
6145:
6141:
6139:
6136:
6135:
6108:
6085:
6081:
6080:
6076:
6075:
6053:
6052:
6048:
6046:
6043:
6042:
6008:
6007:
5987:
5986:
5981:
5978:
5977:
5951:
5947:
5945:
5942:
5941:
5909:
5905:
5900:
5870:
5869:
5865:
5864:
5860:
5836:
5832:
5830:
5827:
5826:
5794:
5790:
5785:
5783:
5780:
5779:
5747:
5743:
5720:
5719:
5699:
5698:
5686:
5678:
5659:
5655:
5634:
5630:
5607:
5606:
5574:
5573:
5561:
5553:
5547:
5544:
5543:
5514:
5510:
5508:
5505:
5504:
5470:
5467:
5466:
5432:
5431:
5421:
5417:
5413:
5411:
5408:
5407:
5383:
5382:
5375:
5371:
5369:
5366:
5365:
5341:
5340:
5336:
5332:
5319:
5318:
5311:
5307:
5305:
5302:
5301:
5270:
5269:
5267:
5264:
5263:
5226:
5225:
5193:
5192:
5166:
5165:
5131:
5130:
5129:
5110:
5109:
5102:
5098:
5078:
5077:
5064:
5063:
5059:
5055:
5054:
5052:
5038:
5037:
5030:
5026:
5021:
4998:
4997:
4984:
4983:
4976:
4972:
4952:
4951:
4946:
4923:
4922:
4902:
4901:
4888:
4887:
4880:
4876:
4871:
4845:
4844:
4824:
4823:
4818:
4815:
4814:
4772:
4768:
4745:
4744:
4724:
4723:
4700:
4699:
4679:
4678:
4666:
4658:
4644:
4639:
4618:
4614:
4609:
4607:
4604:
4603:
4566:
4565:
4560:
4557:
4556:
4549:linear response
4517:
4503:
4502:
4479:
4478:
4461:
4439:
4425:
4411:
4408:
4407:
4379:
4359:
4355:
4339:
4335:
4314:
4310:
4305:
4303:
4300:
4299:
4262:
4261:
4241:
4240:
4235:
4232:
4231:
4224:
4222:Quantum version
4182:
4181:
4157:
4153:
4149:
4147:
4126:
4122:
4120:
4117:
4116:
4076:
4075:
4057:
4053:
4051:
4048:
4047:
3991:
3990:
3955:
3954:
3940:
3937:
3936:
3908:
3905:
3904:
3835:
3831:
3825:
3820:
3784:
3783:
3778:
3775:
3774:
3715:
3698:
3696:
3673:
3670:
3669:
3632:
3628:
3579:
3574:
3564:
3560:
3558:
3555:
3554:
3531:
3527:
3498:
3495:
3494:
3471:
3467:
3458:
3454:
3421:
3417:
3373:
3370:
3369:
3306:
3302:
3290:
3286:
3257:
3254:
3253:
3217:
3214:
3213:
3187:
3183:
3181:
3178:
3177:
3148:
3144:
3123:
3119:
3092:
3088:
3065:
3062:
3061:
3035:
3031:
3023:
3020:
3019:
2998:
2997:
2993:
2981:
2977:
2975:
2972:
2971:
2936:
2906:
2899:
2867:
2865:
2842:
2839:
2838:
2819:
2815:
2794:
2790:
2773:
2770:
2769:
2705:
2691:
2677:
2659:
2630:
2627:
2626:
2596:
2582:
2574:
2571:
2570:
2546:
2543:
2542:
2513:
2510:
2509:
2471:
2467:
2450:
2447:
2446:
2432:
2427:
2396:
2386:
2385:
2381:
2376:
2373:
2372:
2350:
2347:
2346:
2322:
2321:
2317:
2309:
2288:
2285:
2284:
2265:
2255:
2254:
2250:
2245:
2242:
2241:
2183:
2180:
2179:
2159:
2156:
2155:
2118:
2114:
2096:
2088:
2067:
2064:
2063:
2027:
2026:
2004:
2003:
1999:
1995:
1993:
1972:
1968:
1966:
1963:
1962:
1937:
1934:
1933:
1902:
1901:
1899:
1896:
1895:
1876:
1873:
1872:
1844:
1841:
1840:
1779:
1771:
1758:
1754:
1725:
1722:
1721:
1693:
1690:
1689:
1657:
1654:
1653:
1628:
1625:
1624:
1580:
1576:
1559:
1556:
1555:
1530:
1527:
1526:
1497:
1486:
1485:
1484:
1464:
1463:
1442:
1438:
1436:
1433:
1432:
1412:
1408:
1400:
1397:
1396:
1371:
1368:
1367:
1341:
1337:
1335:
1332:
1331:
1295:
1292:
1291:
1285:
1253:
1252:
1248:
1212:
1208:
1196:
1192:
1187:
1184:
1183:
1163:
1159:
1148:
1145:
1144:
1116:
1099:
1096:
1095:
1072:
1068:
1063:
1060:
1059:
1035:
1034:
1030:
1003:
999:
994:
982:
978:
977:
966:
965:
961:
957:
955:
944:
940:
924:
920:
919:
917:
899:
895:
893:
890:
889:
860:
856:
854:
851:
850:
830:
826:
799:
795:
790:
788:
785:
784:
761:
757:
740:
737:
736:
697:
696:
691:
672:
658:
656:
639:
636:
635:
604:
591:
583:
581:
567:
564:
563:
540:
537:
536:
516:
513:
512:
466:
465:
461:
443:
439:
434:
431:
430:
408:
405:
404:
383:
382:
378:
376:
373:
372:
356:
353:
352:
343:discovered and
341:John B. Johnson
337:
318:
270:
269:
265:
260:
252:
249:
248:
231:Brownian motion
227:Albert Einstein
223:
221:Brownian motion
207:
144:Brownian motion
133:Brownian motion
117:
100:annus mirabilis
95:Brownian motion
83:Theodore Welton
28:
23:
22:
15:
12:
11:
5:
8526:
8516:
8515:
8510:
8505:
8500:
8486:
8485:
8465:
8459:
8446:
8440:
8427:
8421:
8405:
8399:
8386:
8380:
8364:
8358:
8339:
8292:
8274:(5): 921–927.
8263:
8222:
8217:
8206:
8203:
8202:
8201:
8152:
8139:
8108:
8105:
8103:
8102:
8057:(1): 173–176.
8038:
7954:
7901:
7836:
7809:(1): 255–284.
7793:
7778:
7759:(1): 110–113.
7738:
7717:(8): 549–560.
7692:
7644:
7642:
7639:
7638:
7637:
7632:
7627:
7622:
7617:
7612:
7605:
7602:
7583:
7577:
7572:
7568:
7562:
7557:
7553:
7550:
7547:
7544:
7541:
7538:
7535:
7532:
7526:
7521:
7517:
7511:
7506:
7502:
7499:
7496:
7493:
7490:
7470:
7467:
7464:
7461:
7456:
7452:
7448:
7443:
7439:
7433:
7430:
7425:
7422:
7419:
7416:
7413:
7393:
7373:
7353:
7350:
7347:
7342:
7338:
7315:
7312:
7309:
7304:
7300:
7297:
7291:
7286:
7282:
7279:
7276:
7271:
7267:
7263:
7257:
7252:
7248:
7243:
7239:
7235:
7230:
7226:
7221:
7215:
7212:
7207:
7204:
7198:
7193:
7189:
7183:
7178:
7174:
7171:
7168:
7165:
7140:
7137:
7134:
7129:
7123:
7120:
7115:
7109:
7104:
7100:
7097:
7094:
7091:
7088:
7082:
7078:
7075:
7069:
7064:
7060:
7054:
7049:
7045:
7042:
7039:
7036:
7010:
7005:
7001:
6998:
6978:
6955:
6950:
6926:
6921:
6897:
6892:
6888:
6885:
6882:
6859:glassy systems
6843:
6838:
6818:
6815:
6802:
6799:
6794:
6791:
6787:
6783:
6780:
6777:
6774:
6771:
6765:
6762:
6757:
6733:
6730:
6699:
6695:
6691:
6688:
6677:
6676:
6665:
6661:
6657:
6654:
6651:
6648:
6644:
6634:
6628:
6623:
6619:
6615:
6610:
6607:
6601:
6597:
6594:
6591:
6588:
6584:
6580:
6577:
6574:
6571:
6567:
6557:
6551:
6548:
6543:
6540:
6537:
6534:
6528:
6525:
6520:
6515:
6511:
6508:
6505:
6500:
6496:
6493:
6490:
6487:
6482:
6478:
6474:
6471:
6468:
6465:
6460:
6456:
6425:
6422:
6419:
6414:
6410:
6389:
6386:
6366:
6363:
6360:
6357:
6351:
6348:
6342:
6339:
6336:
6330:
6327:
6321:
6310:
6309:
6298:
6295:
6292:
6289:
6284:
6280:
6276:
6272:
6268:
6265:
6262:
6259:
6255:
6245:
6241:
6238:
6235:
6229:
6226:
6221:
6216:
6212:
6209:
6206:
6203:
6200:
6197:
6192:
6188:
6182:
6179:
6176:
6173:
6169:
6165:
6162:
6159:
6156:
6153:
6148:
6144:
6114:
6111:
6106:
6102:
6099:
6094:
6091:
6088:
6084:
6079:
6074:
6071:
6068:
6065:
6059:
6056:
6051:
6030:
6027:
6024:
6021:
6015:
6012:
6006:
6003:
6000:
5994:
5991:
5985:
5965:
5962:
5959:
5954:
5950:
5938:
5937:
5925:
5921:
5918:
5915:
5912:
5908:
5898:
5894:
5891:
5888:
5885:
5882:
5876:
5873:
5868:
5863:
5859:
5856:
5853:
5850:
5847:
5844:
5839:
5835:
5810:
5806:
5803:
5800:
5797:
5793:
5776:
5775:
5764:
5761:
5756:
5753:
5750:
5746:
5742:
5739:
5736:
5733:
5727:
5724:
5718:
5715:
5712:
5706:
5703:
5697:
5692:
5689:
5684:
5681:
5677:
5671:
5668:
5665:
5662:
5658:
5654:
5651:
5648:
5643:
5640:
5637:
5633:
5629:
5626:
5623:
5620:
5614:
5611:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5581:
5578:
5572:
5567:
5564:
5559:
5556:
5552:
5526:
5523:
5520:
5517:
5513:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5463:imaginary time
5448:
5445:
5439:
5436:
5428:
5425:
5420:
5416:
5390:
5387:
5381:
5378:
5374:
5348:
5345:
5339:
5335:
5326:
5323:
5317:
5314:
5310:
5289:
5286:
5283:
5277:
5274:
5260:
5259:
5248:
5245:
5242:
5239:
5233:
5230:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5200:
5197:
5191:
5188:
5185:
5182:
5179:
5173:
5170:
5162:
5159:
5156:
5153:
5150:
5147:
5144:
5138:
5135:
5126:
5117:
5114:
5108:
5105:
5101:
5097:
5094:
5091:
5085:
5082:
5071:
5068:
5062:
5058:
5045:
5042:
5036:
5033:
5029:
5020:
5017:
5014:
5011:
5005:
5002:
4991:
4988:
4982:
4979:
4975:
4971:
4968:
4965:
4959:
4956:
4945:
4942:
4939:
4936:
4930:
4927:
4921:
4918:
4915:
4909:
4906:
4895:
4892:
4886:
4883:
4879:
4870:
4867:
4864:
4861:
4858:
4852:
4849:
4843:
4840:
4837:
4831:
4828:
4822:
4804:
4803:
4792:
4789:
4786:
4781:
4778:
4775:
4771:
4767:
4764:
4761:
4758:
4752:
4749:
4743:
4740:
4737:
4731:
4728:
4722:
4719:
4716:
4713:
4707:
4704:
4698:
4695:
4692:
4686:
4683:
4677:
4672:
4669:
4664:
4661:
4657:
4650:
4647:
4643:
4638:
4634:
4630:
4627:
4624:
4621:
4617:
4588:
4585:
4582:
4579:
4573:
4570:
4564:
4545:
4544:
4533:
4530:
4527:
4523:
4520:
4516:
4510:
4507:
4501:
4498:
4495:
4492:
4486:
4483:
4477:
4474:
4471:
4467:
4464:
4460:
4457:
4454:
4451:
4446:
4443:
4438:
4435:
4431:
4428:
4424:
4421:
4418:
4415:
4389:
4386:
4382:
4377:
4373:
4370:
4367:
4362:
4358:
4354:
4351:
4348:
4345:
4342:
4338:
4334:
4330:
4326:
4323:
4320:
4317:
4313:
4284:
4281:
4278:
4275:
4269:
4266:
4260:
4257:
4254:
4248:
4245:
4239:
4223:
4220:
4219:
4218:
4207:
4204:
4201:
4198:
4195:
4189:
4186:
4180:
4177:
4174:
4169:
4165:
4156:
4152:
4146:
4143:
4140:
4137:
4134:
4129:
4125:
4110:
4109:
4098:
4095:
4092:
4089:
4083:
4080:
4074:
4071:
4068:
4065:
4060:
4056:
4025:
4024:
4013:
4010:
4007:
4004:
3998:
3995:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3962:
3959:
3953:
3950:
3947:
3944:
3921:
3918:
3915:
3912:
3901:
3900:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3861:
3858:
3855:
3852:
3847:
3844:
3841:
3838:
3834:
3828:
3823:
3819:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3791:
3788:
3782:
3762:
3761:
3752:
3750:
3739:
3736:
3733:
3730:
3727:
3721:
3718:
3713:
3710:
3707:
3704:
3701:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3663:Consequently,
3661:
3660:
3649:
3646:
3643:
3640:
3635:
3631:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3590:
3587:
3582:
3577:
3573:
3567:
3563:
3534:
3530:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3491:
3490:
3479:
3474:
3470:
3466:
3461:
3457:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3424:
3420:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3349:
3348:
3339:
3337:
3326:
3323:
3320:
3317:
3314:
3309:
3305:
3301:
3298:
3293:
3289:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3236:
3233:
3230:
3227:
3224:
3221:
3201:
3198:
3195:
3190:
3186:
3174:
3173:
3162:
3159:
3156:
3151:
3147:
3143:
3140:
3137:
3134:
3131:
3126:
3122:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3095:
3091:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3046:
3043:
3038:
3034:
3030:
3027:
3007:
3001:
2996:
2992:
2987:
2984:
2980:
2968:
2967:
2956:
2949:
2946:
2942:
2939:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2912:
2909:
2905:
2902:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2822:
2818:
2814:
2811:
2808:
2805:
2802:
2797:
2793:
2789:
2786:
2783:
2780:
2777:
2754:
2753:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2708:
2704:
2701:
2697:
2694:
2690:
2687:
2683:
2680:
2675:
2672:
2669:
2665:
2662:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2612:
2609:
2606:
2603:
2599:
2595:
2592:
2588:
2585:
2581:
2578:
2550:
2526:
2523:
2520:
2517:
2506:
2505:
2494:
2491:
2488:
2485:
2482:
2479:
2474:
2470:
2466:
2463:
2460:
2457:
2454:
2431:
2428:
2426:
2423:
2403:
2399:
2395:
2389:
2384:
2380:
2360:
2357:
2354:
2334:
2331:
2325:
2320:
2316:
2312:
2308:
2305:
2302:
2299:
2296:
2292:
2272:
2268:
2264:
2258:
2253:
2249:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2163:
2139:
2136:
2130:
2127:
2124:
2121:
2117:
2113:
2110:
2107:
2104:
2099:
2094:
2091:
2087:
2083:
2080:
2077:
2074:
2071:
2049:
2046:
2043:
2040:
2034:
2031:
2025:
2022:
2017:
2013:
2007:
2002:
1998:
1992:
1989:
1986:
1983:
1980:
1975:
1971:
1950:
1947:
1944:
1941:
1921:
1918:
1915:
1909:
1906:
1880:
1857:
1854:
1851:
1848:
1837:
1836:
1825:
1822:
1819:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1782:
1777:
1774:
1770:
1766:
1761:
1757:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1717:of the system
1706:
1703:
1700:
1697:
1683:susceptibility
1670:
1667:
1664:
1661:
1641:
1638:
1635:
1632:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1543:
1540:
1537:
1534:
1514:
1511:
1508:
1505:
1500:
1493:
1490:
1483:
1480:
1477:
1471:
1468:
1462:
1459:
1456:
1453:
1450:
1445:
1441:
1430:power spectrum
1415:
1411:
1407:
1404:
1384:
1381:
1378:
1375:
1355:
1352:
1349:
1344:
1340:
1308:
1305:
1302:
1299:
1284:
1281:
1280:
1279:
1268:
1265:
1262:
1256:
1251:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1215:
1211:
1207:
1204:
1199:
1195:
1191:
1166:
1162:
1158:
1155:
1152:
1132:
1129:
1126:
1123:
1119:
1115:
1112:
1109:
1106:
1103:
1080:
1075:
1071:
1067:
1056:
1055:
1044:
1038:
1033:
1029:
1026:
1023:
1019:
1015:
1012:
1009:
1006:
1002:
990:
985:
981:
975:
969:
964:
960:
954:
947:
943:
938:
935:
932:
927:
923:
916:
913:
910:
907:
902:
898:
874:
871:
868:
863:
859:
847:
846:
833:
829:
825:
822:
819:
815:
811:
808:
805:
802:
798:
767:
764:
760:
756:
753:
750:
747:
744:
733:
732:
718:
715:
712:
709:
704:
701:
695:
690:
684:
681:
678:
675:
670:
667:
664:
661:
655:
652:
649:
646:
643:
625:
624:
611:
608:
603:
597:
594:
589:
586:
580:
577:
574:
571:
544:
520:
497:
496:
485:
482:
479:
475:
469:
464:
460:
457:
454:
451:
446:
442:
438:
415:
412:
392:
386:
381:
360:
336:
333:
316:
310:drift velocity
292:
291:
279:
273:
268:
263:
259:
256:
241:to derive the
222:
219:
206:
203:
202:
201:
200:
199:
177:
176:
175:
149:
148:
147:
116:
113:
79:Herbert Callen
26:
9:
6:
4:
3:
2:
8525:
8514:
8511:
8509:
8506:
8504:
8501:
8499:
8496:
8495:
8493:
8482:
8478:
8471:
8466:
8462:
8460:0-471-86256-8
8456:
8452:
8447:
8443:
8441:0-471-81518-7
8437:
8433:
8428:
8424:
8422:0-08-018994-6
8418:
8414:
8410:
8406:
8402:
8400:0-13-283276-3
8396:
8392:
8387:
8383:
8381:0-292-75080-3
8377:
8373:
8369:
8365:
8361:
8355:
8351:
8347:
8346:
8340:
8336:
8332:
8328:
8324:
8320:
8316:
8311:
8306:
8302:
8298:
8293:
8289:
8285:
8281:
8277:
8273:
8269:
8264:
8260:
8256:
8252:
8248:
8243:
8238:
8234:
8230:
8229:
8223:
8221:
8218:
8216:
8212:
8209:
8208:
8198:
8194:
8190:
8186:
8182:
8178:
8173:
8168:
8164:
8160:
8159:
8153:
8149:
8145:
8140:
8136:
8132:
8128:
8124:
8120:
8116:
8111:
8110:
8098:
8094:
8090:
8086:
8082:
8078:
8074:
8070:
8065:
8060:
8056:
8052:
8048:
8042:
8034:
8028:
8020:
8016:
8011:
8006:
8001:
7996:
7992:
7988:
7983:
7978:
7974:
7970:
7966:
7958:
7950:
7946:
7942:
7938:
7934:
7930:
7925:
7920:
7916:
7912:
7905:
7897:
7893:
7889:
7885:
7881:
7877:
7873:
7869:
7864:
7859:
7856:(2): 026105.
7855:
7851:
7847:
7840:
7832:
7828:
7824:
7820:
7816:
7812:
7808:
7804:
7797:
7790:. OUP Oxford.
7789:
7782:
7774:
7770:
7766:
7762:
7758:
7754:
7753:
7748:
7742:
7733:
7728:
7724:
7720:
7716:
7712:
7711:
7706:
7702:
7696:
7688:
7684:
7680:
7676:
7672:
7668:
7667:
7662:
7658:
7652:
7650:
7645:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7611:
7608:
7607:
7601:
7597:
7594:
7570:
7566:
7555:
7551:
7548:
7542:
7539:
7536:
7533:
7519:
7515:
7504:
7500:
7497:
7491:
7488:
7462:
7454:
7450:
7441:
7437:
7431:
7428:
7423:
7417:
7411:
7391:
7371:
7351:
7348:
7345:
7340:
7336:
7313:
7310:
7307:
7302:
7284:
7280:
7277:
7269:
7265:
7250:
7241:
7237:
7228:
7224:
7213:
7210:
7205:
7191:
7187:
7176:
7172:
7169:
7163:
7156:
7138:
7135:
7132:
7127:
7121:
7102:
7098:
7095:
7089:
7076:
7062:
7058:
7047:
7043:
7040:
7034:
7027:
7003:
6999:
6996:
6976:
6948:
6919:
6890:
6886:
6883:
6880:
6872:
6868:
6863:
6860:
6836:
6828:
6824:
6814:
6800:
6797:
6792:
6789:
6781:
6775:
6769:
6755:
6728:
6717:
6713:
6697:
6693:
6689:
6686:
6663:
6659:
6652:
6646:
6642:
6632:
6626:
6621:
6617:
6613:
6608:
6605:
6599:
6595:
6592:
6586:
6582:
6575:
6569:
6565:
6555:
6549:
6546:
6541:
6535:
6518:
6513:
6506:
6503:
6498:
6491:
6488:
6480:
6476:
6472:
6466:
6458:
6454:
6443:
6442:
6441:
6439:
6420:
6412:
6408:
6387:
6384:
6358:
6346:
6337:
6325:
6293:
6290:
6282:
6278:
6274:
6270:
6263:
6257:
6253:
6243:
6236:
6219:
6214:
6207:
6204:
6198:
6190:
6186:
6180:
6174:
6171:
6167:
6163:
6157:
6154:
6146:
6142:
6134:
6133:
6132:
6130:
6129:Bose-Einstein
6112:
6109:
6104:
6100:
6097:
6092:
6086:
6082:
6077:
6072:
6066:
6049:
6022:
6010:
6001:
5989:
5960:
5952:
5948:
5923:
5916:
5910:
5906:
5896:
5892:
5889:
5883:
5866:
5861:
5854:
5851:
5845:
5837:
5833:
5825:
5824:
5823:
5808:
5801:
5795:
5791:
5762:
5759:
5754:
5751:
5748:
5744:
5734:
5722:
5713:
5701:
5687:
5679:
5675:
5669:
5663:
5660:
5656:
5652:
5649:
5646:
5641:
5638:
5635:
5631:
5621:
5609:
5600:
5594:
5591:
5588:
5576:
5562:
5554:
5550:
5542:
5541:
5540:
5524:
5518:
5515:
5511:
5490:
5484:
5481:
5478:
5475:
5464:
5446:
5434:
5423:
5418:
5414:
5385:
5379:
5376:
5372:
5343:
5337:
5333:
5321:
5315:
5312:
5308:
5284:
5272:
5240:
5228:
5219:
5213:
5210:
5207:
5195:
5186:
5180:
5168:
5157:
5151:
5148:
5145:
5133:
5124:
5112:
5106:
5103:
5099:
5092:
5080:
5066:
5060:
5056:
5040:
5034:
5031:
5027:
5018:
5012:
5000:
4986:
4980:
4977:
4973:
4966:
4954:
4943:
4937:
4925:
4916:
4904:
4890:
4884:
4881:
4877:
4868:
4859:
4847:
4838:
4826:
4813:
4812:
4811:
4809:
4790:
4787:
4784:
4779:
4776:
4773:
4769:
4759:
4747:
4738:
4726:
4720:
4714:
4702:
4693:
4681:
4667:
4659:
4655:
4645:
4641:
4636:
4632:
4625:
4619:
4615:
4602:
4601:
4600:
4580:
4568:
4554:
4550:
4521:
4518:
4505:
4499:
4493:
4481:
4465:
4462:
4458:
4455:
4449:
4441:
4436:
4429:
4426:
4422:
4419:
4413:
4406:
4405:
4404:
4403:
4387:
4384:
4380:
4375:
4368:
4360:
4356:
4352:
4346:
4340:
4336:
4332:
4328:
4321:
4315:
4311:
4298:
4276:
4264:
4255:
4243:
4229:
4205:
4196:
4184:
4175:
4172:
4167:
4163:
4154:
4150:
4144:
4141:
4135:
4127:
4123:
4115:
4114:
4113:
4096:
4090:
4078:
4072:
4066:
4058:
4054:
4046:
4045:
4044:
4042:
4038:
4034:
4030:
4027:Finally, for
4011:
4005:
3993:
3987:
3984:
3981:
3978:
3969:
3957:
3948:
3945:
3942:
3935:
3934:
3933:
3916:
3910:
3887:
3881:
3875:
3872:
3869:
3866:
3863:
3856:
3850:
3845:
3842:
3839:
3836:
3832:
3821:
3817:
3813:
3810:
3807:
3804:
3798:
3786:
3780:
3773:
3772:
3771:
3769:
3760:
3753:
3751:
3737:
3731:
3725:
3719:
3716:
3708:
3702:
3699:
3693:
3690:
3684:
3678:
3675:
3668:
3667:
3664:
3644:
3638:
3633:
3629:
3625:
3622:
3616:
3613:
3610:
3604:
3598:
3592:
3588:
3585:
3575:
3571:
3565:
3561:
3553:
3552:
3551:
3550:
3532:
3524:
3518:
3509:
3503:
3477:
3472:
3459:
3451:
3445:
3439:
3433:
3422:
3414:
3408:
3402:
3396:
3387:
3381:
3375:
3368:
3367:
3366:
3364:
3360:
3356:
3347:
3340:
3338:
3324:
3318:
3312:
3307:
3303:
3299:
3296:
3291:
3283:
3277:
3268:
3262:
3252:
3251:
3248:
3228:
3222:
3196:
3188:
3184:
3160:
3149:
3141:
3135:
3132:
3124:
3120:
3116:
3113:
3110:
3101:
3093:
3089:
3085:
3079:
3076:
3073:
3067:
3060:
3059:
3058:
3044:
3041:
3036:
3032:
3028:
3025:
3005:
2994:
2990:
2985:
2982:
2978:
2954:
2940:
2937:
2930:
2927:
2924:
2918:
2915:
2910:
2907:
2903:
2900:
2889:
2883:
2880:
2877:
2871:
2868:
2862:
2856:
2853:
2850:
2844:
2837:
2836:
2835:
2820:
2816:
2812:
2809:
2803:
2795:
2791:
2787:
2781:
2775:
2767:
2763:
2759:
2740:
2734:
2731:
2728:
2722:
2716:
2713:
2710:
2702:
2699:
2695:
2692:
2685:
2681:
2678:
2673:
2670:
2667:
2663:
2660:
2656:
2653:
2650:
2641:
2635:
2625:
2624:
2623:
2607:
2604:
2601:
2593:
2590:
2586:
2583:
2576:
2568:
2564:
2548:
2540:
2521:
2515:
2492:
2486:
2483:
2477:
2472:
2468:
2464:
2458:
2452:
2445:
2444:
2443:
2441:
2437:
2422:
2419:
2417:
2416:LSZ reduction
2401:
2397:
2393:
2382:
2378:
2358:
2352:
2329:
2318:
2314:
2310:
2306:
2297:
2294:
2290:
2270:
2266:
2262:
2251:
2247:
2238:
2221:
2218:
2215:
2212:
2206:
2203:
2200:
2197:
2191:
2185:
2177:
2161:
2153:
2137:
2134:
2128:
2125:
2122:
2119:
2115:
2108:
2102:
2089:
2085:
2081:
2075:
2069:
2060:
2047:
2041:
2029:
2023:
2020:
2015:
2011:
2000:
1996:
1990:
1987:
1981:
1973:
1969:
1945:
1939:
1916:
1904:
1894:
1878:
1869:
1852:
1849:
1846:
1823:
1820:
1817:
1810:
1807:
1804:
1798:
1792:
1786:
1780:
1772:
1768:
1764:
1759:
1751:
1745:
1736:
1730:
1720:
1719:
1718:
1701:
1695:
1688:
1684:
1665:
1659:
1636:
1630:
1610:
1604:
1598:
1595:
1589:
1581:
1577:
1573:
1567:
1561:
1538:
1532:
1506:
1498:
1488:
1478:
1466:
1457:
1451:
1443:
1439:
1431:
1413:
1405:
1379:
1373:
1350:
1342:
1338:
1330:
1326:
1322:
1303:
1297:
1288:
1266:
1260:
1249:
1245:
1242:
1239:
1236:
1230:
1227:
1221:
1213:
1209:
1205:
1197:
1193:
1182:
1181:
1180:
1164:
1160:
1156:
1153:
1150:
1127:
1124:
1117:
1113:
1107:
1104:
1094:
1073:
1069:
1042:
1031:
1027:
1024:
1021:
1017:
1010:
1004:
1000:
988:
983:
979:
973:
962:
958:
952:
945:
941:
933:
925:
921:
914:
908:
900:
896:
888:
887:
886:
869:
861:
857:
831:
827:
823:
820:
817:
813:
806:
800:
796:
783:
782:
781:
765:
762:
754:
751:
745:
742:
716:
713:
710:
707:
702:
699:
693:
688:
679:
673:
665:
659:
653:
647:
641:
634:
633:
632:
630:
609:
606:
601:
595:
592:
587:
584:
578:
575:
572:
569:
562:
561:
560:
558:
542:
534:
518:
510:
501:
483:
480:
473:
462:
458:
455:
452:
444:
440:
429:
428:
427:
413:
390:
379:
358:
350:
346:
345:Harry Nyquist
342:
332:
330:
326:
322:
315:
311:
308:
304:
300:
297:
277:
266:
261:
257:
254:
247:
246:
245:
244:
240:
235:
232:
228:
225:For example,
218:
216:
212:
197:
193:
189:
188:
187:
186:
182:
178:
173:
172:Johnson noise
169:
168:Joule heating
165:
161:
160:
159:
158:
157:Johnson noise
154:
150:
145:
141:
137:
136:
135:
134:
130:
126:
125:
124:
122:
112:
110:
109:Johnson noise
106:
105:Harry Nyquist
102:
101:
96:
92:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
8480:
8476:
8450:
8431:
8412:
8390:
8371:
8344:
8300:
8296:
8271:
8267:
8232:
8226:
8162:
8156:
8143:
8121:(1): 34–40.
8118:
8114:
8054:
8050:
8041:
8027:cite journal
7972:
7968:
7957:
7914:
7910:
7904:
7853:
7849:
7839:
7806:
7802:
7796:
7787:
7781:
7756:
7750:
7741:
7714:
7708:
7703:(May 1905).
7695:
7673:(1): 34–40.
7670:
7664:
7598:
7595:
6867:spin glasses
6864:
6820:
6678:
6311:
5939:
5777:
5261:
4805:
4546:
4402:Kubo formula
4225:
4111:
4026:
3902:
3767:
3765:
3754:
3662:
3548:
3492:
3362:
3358:
3354:
3352:
3341:
3175:
2969:
2761:
2757:
2755:
2566:
2562:
2507:
2439:
2435:
2433:
2420:
2239:
2175:
2152:fluctuations
2151:
2061:
1870:
1838:
1289:
1286:
1057:
848:
734:
626:
506:
338:
324:
313:
302:
298:
293:
236:
224:
208:
179:
151:
127:
118:
98:
76:
43:
39:
35:
31:
29:
7917:(2): 1155.
7661:T.A. Welton
7657:H.B. Callen
1329:Hamiltonian
627:and so the
557:Kirchhoff's
97:during his
8492:Categories
8409:Pathria RK
8107:References
7982:1610.01418
2425:Derivation
2176:dissipated
1321:observable
347:explained
153:Resistance
87:Ryogo Kubo
60:admittance
8368:Reichl LE
8242:0710.4394
8197:118575899
8172:0803.0719
7949:119200464
7924:0810.4729
7831:250892844
7747:Nyquist H
7540:−
7492:χ
7469:⟩
7447:⟨
7438:∑
7424:≡
7349:±
7299:⟩
7234:⟨
7225:∑
7206:≡
7119:∂
7087:∂
7077:≡
7035:χ
6798:≫
6790:−
6782:ω
6779:ℏ
6776:β
6770:≈
6732:^
6653:ω
6647:χ
6609:ω
6606:ℏ
6596:
6590:ℏ
6576:ω
6570:χ
6536:ω
6510:ℏ
6492:ω
6489:−
6467:ω
6421:ω
6365:⟩
6350:^
6329:^
6320:⟨
6294:ω
6275:≠
6264:ω
6258:χ
6237:ω
6211:ℏ
6199:ω
6181:ω
6178:ℏ
6175:β
6172:−
6158:ω
6155:−
6110:−
6098:−
6093:ω
6090:ℏ
6087:β
6067:ω
6029:⟩
6014:^
5993:^
5984:⟨
5961:ω
5917:ω
5911:χ
5884:ω
5858:ℏ
5846:ω
5802:ω
5796:χ
5752:ω
5741:⟩
5726:^
5705:^
5696:⟨
5691:∞
5683:∞
5680:−
5676:∫
5670:ω
5667:ℏ
5664:β
5661:−
5639:ω
5628:⟩
5613:^
5601:β
5598:ℏ
5592:−
5580:^
5571:⟨
5566:∞
5558:∞
5555:−
5551:∫
5525:ω
5522:ℏ
5519:β
5516:−
5491:β
5488:ℏ
5482:−
5473:Δ
5465:interval
5444:Δ
5438:^
5427:ℏ
5419:−
5389:^
5380:β
5377:−
5347:^
5338:β
5325:^
5316:β
5313:−
5276:^
5247:⟩
5232:^
5220:β
5217:ℏ
5211:−
5199:^
5190:⟨
5172:^
5158:β
5155:ℏ
5149:−
5137:^
5125:⏟
5116:^
5107:β
5104:−
5084:^
5070:^
5061:β
5044:^
5035:β
5032:−
5004:^
4990:^
4981:β
4978:−
4958:^
4929:^
4908:^
4894:^
4885:β
4882:−
4866:⟩
4851:^
4830:^
4821:⟨
4777:ω
4766:⟩
4751:^
4730:^
4721:−
4706:^
4685:^
4676:⟨
4671:∞
4663:∞
4660:−
4656:∫
4649:ℏ
4626:ω
4620:χ
4587:⟩
4572:^
4563:⟨
4532:⟩
4509:^
4485:^
4473:⟨
4459:−
4450:θ
4445:ℏ
4423:−
4414:χ
4369:ω
4361:∗
4357:χ
4353:−
4347:ω
4341:χ
4322:ω
4316:χ
4283:⟩
4268:^
4247:^
4238:⟨
4197:ω
4188:^
4185:χ
4176:
4168:ω
4145:−
4136:ω
4091:ω
4082:^
4067:ω
4006:ω
3997:^
3988:β
3985:ω
3982:−
3970:ω
3961:^
3958:χ
3949:
3873:β
3870:−
3843:ω
3837:−
3827:∞
3818:∫
3814:β
3811:ω
3799:ω
3790:^
3787:χ
3781:−
3726:θ
3694:β
3679:χ
3676:−
3626:β
3614:−
3611:τ
3605:θ
3599:τ
3593:χ
3589:τ
3581:∞
3572:∫
3529:⟩
3522:⟨
3519:−
3516:⟩
3501:⟨
3469:⟩
3456:⟩
3449:⟨
3446:−
3419:⟩
3412:⟨
3409:−
3391:⟨
3300:β
3288:⟩
3281:⟨
3275:⟩
3260:⟨
3235:⟩
3220:⟨
3146:⟩
3139:⟨
3136:−
3117:β
3086:≈
3042:≪
3026:β
2983:−
2979:β
2928:β
2925:−
2919:
2901:∫
2881:β
2878:−
2872:
2810:−
2668:∫
2654:∫
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