25:
671:
complexions that is compatible with a given macroscopic state. It is this quantity that Planck referred to as a 'thermodynamic' probability. It differs from a classical probability inasmuch as it cannot be normalized; that is, its integral over all energies diverges—but it diverges as a power of the energy and not faster. Since its integral over all energies is infinite, we might try to consider its
Laplace transform
1119:
1458:
2823:
1754:
2149:
will belong to the family of exponential distributions known as gamma densities. Consequently, the canonical probability density falls under the jurisdiction of the local law of large numbers which asserts that a sequence of independent and identically distributed random variables tends to the normal
670:
has its legs in two worlds: (i) the macroscopic one in which it is considered a function of the energy, and the other extensive variables, like the volume, that have been held constant in the differentiation of the phase volume, and (ii) the microscopic world where it represents the number of
80:
on the surface of a crystal. The shaking of the atoms is an example of thermal fluctuations. Likewise, thermal fluctuations provide the energy necessary for the atoms to occasionally hop from one site to a neighboring one. For simplicity, the thermal fluctuations of the blue atoms are not
949:
2465:
1260:
580:
2628:
1991:
93:
are random deviations of an atomic system from its average state, that occur in a system at equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches
3924:
3399:
1620:
905:
761:
2144:
4052:
3517:
3748:
3585:
1114:{\displaystyle \langle E\rangle =-{\frac {\partial \ln {\mathcal {Z}}}{\partial \beta }},\qquad \ \langle (E-\langle E\rangle )^{2}\rangle =(\Delta E)^{2}={\frac {\partial ^{2}\ln {\mathcal {Z}}}{\partial \beta ^{2}}},}
1881:
400:
2619:
1812:
1453:{\displaystyle f(E;\beta )={\frac {e^{-\beta E}}{{\mathcal {Z}}(\beta )}}\Omega (E)\approx {\frac {\exp\{-(E-\langle E\rangle )^{2}/2\langle (\Delta E)^{2}\rangle \}}{\sqrt {2\pi (\Delta E)^{2}}}}.}
2350:
2326:
3256:
813:. Most of the contribution to the integral will come from an immediate neighborhood about this value of the energy. This enables the definition of a proper probability density according to
3013:
2869:
943:, which is referred to as the partition function, or generating function. The latter name is due to the fact that the derivatives of its logarithm generate the central moments, namely,
474:
636:
2818:{\displaystyle w=\prod _{i,j=1\ldots n}{\frac {1}{\left(2\pi \right)^{n/2}{\sqrt {\langle x_{i}x_{j}\rangle }}}}\exp \left(-{\frac {x_{i}x_{j}}{2\langle x_{i}x_{j}\rangle }}\right),}
1199:
941:
2501:
1612:
105:
of systems: A system at nonzero temperature does not stay in its equilibrium microscopic state, but instead randomly samples all possible states, with probabilities given by the
1889:
1225:
134:, likewise undergo thermal fluctuations. For example, for a system that has an equilibrium pressure, the system pressure fluctuates to some extent about the equilibrium value.
3788:
319:
2039:
218:
3960:
3621:
3680:
3310:
1749:{\displaystyle \Omega (\langle E\rangle )={\frac {e^{\beta (\langle E\rangle )\langle E\rangle }{\mathcal {Z}}(\beta (\langle E\rangle ))}{\sqrt {2\pi (\Delta E)^{2}}}}.}
1580:
1496:
285:
2906:
1463:
This is the
Gaussian, or normal, distribution, which is defined by its first two moments. In general, one would need all the moments to specify the probability density,
1252:
1154:
811:
668:
3841:
3316:
3835:
3647:
3425:
3202:
3176:
3152:
3128:
3104:
1516:
443:
2219:
784:
3071:
3040:
2946:
1550:
466:
241:
3981:
3809:
3446:
3277:
2966:
2259:
2239:
2184:
423:
261:
819:
677:
2047:
113:
74:
263:
and the momentum space volume. Since the energy is a quadratic form of the momenta for a non-relativistic system, the radius of momentum space will be
3987:
3452:
3686:
3523:
786:
is a positive parameter, will overpower the rapidly increasing surface area so that an enormously sharp peak will develop at a certain energy
4075:
1817:
327:
2514:
2041:, and if the structure function retains the same functional dependency for all values of the energy, the canonical probability density,
1883:. Introducing these two expressions into the expression of the structure function evaluated at the mean value of the energy leads to
4224:
4201:
4178:
60:
1762:
2460:{\displaystyle w(x)={\frac {1}{\sqrt {2\pi \langle x^{2}\rangle }}}\exp \left(-{\frac {x^{2}}{2\langle x^{2}\rangle }}\right).}
177:
1156:
increases no faster than a power of the energy ensures that these moments will be finite. Therefore, we can expand the factor
4244:
2268:
3208:
1582:. Rearranging the normal distribution so that it becomes an expression for the structure function and evaluating it at
1254:
for
Gaussian fluctuations (i.e. average and most probable values coincide), and retaining lowest order terms result in
2968:
in any small part of a body. The small part must still be large enough, however, to have negligible quantum effects.
33:
575:{\displaystyle \Omega (E)={\frac {\partial {\mathcal {V}}}{\partial E}}={\frac {C^{m}\cdot E^{m-1}}{\Gamma (m)}},}
2975:
2831:
2158:
The expressions given below are for systems that are close to equilibrium and have negligible quantum effects.
588:
1159:
913:
150:
2473:
1986:{\displaystyle \Omega (\langle E\rangle )={\frac {\langle E\rangle ^{m-1}m}{{\sqrt {2\pi m}}m^{m}e^{-m}}}}
127:
106:
1585:
4100:
2340:
1519:
86:
1204:
3754:
290:
2000:
468:, which is the usual case in thermodynamics, essentially all the volume will lie near to the surface
199:
39:
4065:
3930:
3591:
2336:
1124:
and so on, where the first term is the mean energy and the second one is the dispersion in energy.
73:
3919:{\displaystyle {\frac {k_{\rm {B}}}{C_{V}}}T^{2}\left({\frac {\partial P}{\partial T}}\right)_{V}}
3653:
3394:{\displaystyle {\frac {k_{\rm {B}}}{C_{V}}}T^{2}\left({\frac {\partial P}{\partial T}}\right)_{V}}
3283:
4193:
4170:
4080:
1555:
1466:
266:
2874:
1498:, which is referred to as the canonical, or posterior, density in contrast to the prior density
1230:
1130:
789:
644:
160:
in many systems. The random forces that give rise to thermal fluctuations are a source of both
4220:
4197:
4174:
3817:
3629:
3407:
3184:
3158:
3134:
3110:
3086:
185:
181:
1501:
428:
2332:
2189:
769:
77:
3049:
3018:
2919:
1528:
4070:
2511:
The above expression has a straightforward generalization to the probability distribution
2916:
In the table below are given the mean square fluctuations of the thermodynamic variables
445:
is the Gamma function. In the case that this hypersphere has a very high dimensionality,
448:
223:
137:
Only the 'control variables' of statistical ensembles (such as the number of particules
4216:
3966:
3794:
3431:
3262:
2951:
2244:
2224:
2169:
766:
which can be given a physical interpretation. The exponential decreasing factor, where
408:
246:
4238:
4163:
3074:
3043:
900:{\displaystyle f(E;\beta )={\frac {e^{-\beta E}}{{\mathcal {Z}}(\beta )}}\Omega (E),}
756:{\displaystyle {\mathcal {Z}}(\beta )=\int _{0}^{\infty }e^{-\beta E}\Omega (E)\,dE,}
169:
95:
176:). The competing effects of random drift and resistance to drift are related by the
2139:{\displaystyle f(E;\beta )=\beta {\frac {(\beta E)^{m-1}}{\Gamma (m)}}e^{-\beta E}}
165:
102:
910:
whose integral over all energies is unity on the strength of the definition of
4047:{\displaystyle -k_{\rm {B}}T\left({\frac {\partial P}{\partial V}}\right)_{S}}
3512:{\displaystyle -k_{\rm {B}}T\left({\frac {\partial V}{\partial P}}\right)_{T}}
16:
Random temperature-influenced deviations of particles from their average state
3743:{\displaystyle k_{\rm {B}}T\left({\frac {\partial V}{\partial T}}\right)_{P}}
3580:{\displaystyle k_{\rm {B}}T\left({\frac {\partial V}{\partial T}}\right)_{P}}
173:
161:
425:
is a constant depending upon the specific properties of the system and
131:
1876:{\displaystyle \langle (\Delta E)^{2}\rangle =\langle E\rangle ^{2}/m}
395:{\displaystyle {\mathcal {V}}={\frac {(C\cdot E)^{m}}{\Gamma (m+1)}},}
2614:{\displaystyle w(x_{1},x_{2},\ldots ,x_{n})dx_{1}dx_{2}\ldots dx_{n}}
117:
157:
121:
72:
1518:, which is referred to as the 'structure' function. This is the
1552:, its Laplace transform, the partition function, will vary as
243:
degrees of freedom is the product of the configuration volume
45:
You can help by providing page numbers for existing citations.
18:
1807:{\displaystyle \beta (\langle E\rangle )=m/\langle E\rangle }
1684:
1306:
1085:
982:
919:
865:
683:
501:
333:
205:
2186:
is a thermodynamic variable. The probability distribution
2912:
Fluctuations of the fundamental thermodynamic quantities
1997:
The denominator is exactly
Stirling's approximation for
1759:
It follows from the expression of the first moment that
101:
Thermal fluctuations are a basic manifestation of the
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202:
2321:{\displaystyle w(x)\propto \exp \left(S(x)\right).}
4162:
4103:they are often simply referred to as fluctuations.
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1985:
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212:
4165:Mathematical Foundations of Statistical Mechanics
3251:{\displaystyle {\frac {k_{\rm {B}}}{C_{V}}}T^{2}}
287:so that the volume of a hypersphere will vary as
124:), random electronic excitations, and so forth.
4144:
112:Thermal fluctuations generally affect all the
4190:Statistical Physics: A Probabilistic Approach
2150:law as the sequence increases without limit.
116:of a system: There can be random vibrations (
8:
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1023:
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1005:
959:
953:
180:. Thermal fluctuations play a major role in
4111:
4109:
3008:{\displaystyle \langle x_{i}x_{j}\rangle }
2864:{\displaystyle \langle x_{i}x_{j}\rangle }
4038:
4014:
3999:
3998:
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3875:
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3852:
3851:
3845:
3843:
3819:
3796:
3774:
3763:
3762:
3756:
3734:
3710:
3695:
3694:
3688:
3662:
3661:
3655:
3631:
3603:
3602:
3593:
3571:
3547:
3532:
3531:
3525:
3503:
3479:
3464:
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3433:
3409:
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3361:
3350:
3338:
3327:
3326:
3320:
3318:
3292:
3291:
3285:
3264:
3242:
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3219:
3218:
3212:
3210:
3186:
3160:
3136:
3112:
3088:
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3026:
3020:
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2560:
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2352:
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2124:
2091:
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2002:
1971:
1961:
1944:
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1914:
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1837:
1819:
1790:
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1734:
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1099:
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1083:
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1064:
1055:
1030:
981:
980:
968:
951:
918:
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915:
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850:
844:
821:
797:
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771:
743:
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682:
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341:
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225:
204:
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201:
61:Learn how and when to remove this message
4140:
4138:
4136:
4134:
4115:
2970:
2335:about its maximum (corresponding to the
1814:, while from the second central moment,
1522:as it applies to thermodynamic systems.
631:{\displaystyle m\Gamma (m)=\Gamma (m+1)}
4211:Landau, L. D.; Lifshitz, E. M. (1985).
4126:
4092:
4076:Stochastic Eulerian Lagrangian methods
1194:{\displaystyle e^{-\beta E}\Omega (E)}
936:{\displaystyle {\mathcal {Z}}(\beta )}
3814:
3626:
3404:
3181:
3081:
2496:{\displaystyle \langle x^{2}\rangle }
156:Thermal fluctuations are a source of
7:
585:where we used the recursion formula
130:, such as pressure, temperature, or
2339:state), the lowest order term is a
1607:{\displaystyle E=\langle E\rangle }
4025:
4017:
4000:
3943:
3897:
3889:
3853:
3821:
3764:
3721:
3713:
3696:
3663:
3633:
3604:
3558:
3550:
3533:
3490:
3482:
3465:
3411:
3372:
3364:
3328:
3293:
3220:
3188:
3162:
3138:
3114:
3090:
2105:
2013:
1893:
1827:
1724:
1624:
1505:
1428:
1395:
1323:
1179:
1134:
1092:
1068:
1045:
989:
971:
882:
731:
710:
648:
610:
595:
554:
508:
496:
478:
432:
368:
14:
1525:If the phase volume increases as
1220:{\displaystyle \langle E\rangle }
3783:{\displaystyle k_{\rm {B}}C_{P}}
2503:is the mean square fluctuation.
314:{\displaystyle {\sqrt {E}}^{2m}}
23:
3015:of thermodynamic fluctuations.
2034:{\displaystyle m!=\Gamma (m+1)}
1001:
178:fluctuation-dissipation theorem
2566:
2521:
2363:
2357:
2307:
2301:
2281:
2275:
2202:
2196:
2154:Distribution about equilibrium
2114:
2108:
2088:
2078:
2066:
2054:
2028:
2016:
1908:
1896:
1834:
1824:
1781:
1769:
1731:
1721:
1710:
1707:
1695:
1689:
1668:
1656:
1639:
1627:
1485:
1473:
1435:
1425:
1402:
1392:
1372:
1353:
1332:
1326:
1317:
1311:
1279:
1267:
1188:
1182:
1143:
1137:
1052:
1042:
1027:
1008:
930:
924:
891:
885:
876:
870:
838:
826:
740:
734:
694:
688:
657:
651:
625:
613:
604:
598:
563:
557:
487:
481:
383:
371:
357:
344:
213:{\displaystyle {\mathcal {V}}}
1:
3955:{\displaystyle -k_{\rm {B}}T}
3616:{\displaystyle -k_{\rm {B}}T}
2241:is determined by the entropy
3675:{\displaystyle k_{\rm {B}}T}
3305:{\displaystyle k_{\rm {B}}T}
4213:Statistical Physics, Part 1
1575:{\displaystyle \beta ^{-m}}
1491:{\displaystyle f(E;\beta )}
1227:, which will coincide with
280:{\displaystyle {\sqrt {E}}}
4261:
4145:Landau & Lifshitz 1985
2901:{\displaystyle x_{i}x_{j}}
1247:{\displaystyle E^{\star }}
1149:{\displaystyle \Omega (E)}
806:{\displaystyle E^{\star }}
663:{\displaystyle \Omega (E)}
220:, occupied by a system of
196:The volume of phase space
321:giving a phase volume of
4161:Khinchin, A. I. (1949).
3830:{\displaystyle \Delta P}
3642:{\displaystyle \Delta S}
3420:{\displaystyle \Delta V}
3197:{\displaystyle \Delta T}
3171:{\displaystyle \Delta P}
3147:{\displaystyle \Delta S}
3123:{\displaystyle \Delta V}
3099:{\displaystyle \Delta T}
145:and the internal energy
4188:Lavenda, B. H. (1991).
1511:{\displaystyle \Omega }
438:{\displaystyle \Gamma }
151:microcanonical ensemble
128:Thermodynamic variables
32:This article cites its
4048:
3977:
3956:
3920:
3831:
3805:
3784:
3744:
3676:
3643:
3617:
3581:
3513:
3442:
3421:
3395:
3306:
3273:
3252:
3198:
3172:
3148:
3124:
3100:
3067:
3046:at constant pressure;
3036:
3009:
2962:
2942:
2902:
2865:
2819:
2615:
2497:
2461:
2322:
2255:
2235:
2215:
2214:{\displaystyle w(x)dx}
2180:
2140:
2035:
1987:
1877:
1808:
1750:
1608:
1576:
1546:
1512:
1492:
1454:
1248:
1221:
1195:
1150:
1115:
937:
901:
807:
780:
779:{\displaystyle \beta }
757:
664:
632:
576:
462:
439:
419:
396:
315:
281:
257:
237:
214:
107:Boltzmann distribution
82:
4245:Statistical mechanics
4101:statistical mechanics
4049:
3978:
3957:
3921:
3832:
3806:
3785:
3745:
3677:
3644:
3618:
3582:
3514:
3443:
3422:
3396:
3307:
3274:
3253:
3199:
3173:
3149:
3125:
3101:
3068:
3066:{\displaystyle C_{V}}
3037:
3035:{\displaystyle C_{P}}
3010:
2963:
2943:
2941:{\displaystyle T,V,P}
2903:
2871:is the mean value of
2866:
2820:
2616:
2498:
2462:
2341:Gaussian distribution
2323:
2256:
2236:
2216:
2181:
2141:
2036:
1988:
1878:
1809:
1751:
1609:
1577:
1547:
1545:{\displaystyle E^{m}}
1520:central limit theorem
1513:
1493:
1455:
1249:
1222:
1201:about the mean value
1196:
1151:
1116:
938:
902:
808:
781:
758:
665:
633:
577:
463:
440:
420:
397:
316:
282:
258:
238:
215:
192:Central limit theorem
120:), random rotations (
87:statistical mechanics
76:
4066:Quantum fluctuations
3988:
3967:
3931:
3842:
3818:
3795:
3755:
3687:
3654:
3630:
3592:
3524:
3453:
3432:
3408:
3317:
3284:
3263:
3209:
3185:
3159:
3135:
3111:
3087:
3077:at constant volume.
3050:
3019:
2976:
2952:
2920:
2875:
2832:
2629:
2515:
2474:
2351:
2269:
2245:
2225:
2190:
2170:
2048:
2001:
1890:
1818:
1763:
1621:
1586:
1556:
1529:
1502:
1467:
1261:
1231:
1205:
1160:
1131:
950:
914:
820:
790:
770:
678:
645:
589:
475:
449:
429:
409:
328:
291:
267:
247:
224:
200:
153:) do not fluctuate.
91:thermal fluctuations
3078:
714:
4194:Wiley-Interscience
4171:Dover Publications
4081:Stokesian dynamics
4044:
3973:
3952:
3916:
3827:
3801:
3780:
3740:
3672:
3639:
3613:
3577:
3509:
3438:
3417:
3391:
3302:
3269:
3248:
3194:
3168:
3144:
3120:
3096:
3063:
3032:
3005:
2971:
2958:
2938:
2898:
2861:
2815:
2665:
2611:
2507:Multiple variables
2493:
2457:
2331:If the entropy is
2318:
2251:
2231:
2211:
2176:
2136:
2031:
1983:
1873:
1804:
1746:
1604:
1572:
1542:
1508:
1488:
1450:
1244:
1217:
1191:
1146:
1111:
933:
897:
803:
776:
753:
700:
660:
628:
572:
461:{\displaystyle 2m}
458:
435:
415:
392:
311:
277:
253:
236:{\displaystyle 2m}
233:
210:
114:degrees of freedom
83:
4057:
4056:
4032:
3976:{\displaystyle 0}
3904:
3869:
3804:{\displaystyle 0}
3728:
3565:
3497:
3441:{\displaystyle 0}
3379:
3344:
3272:{\displaystyle 0}
3236:
2961:{\displaystyle S}
2805:
2734:
2731:
2638:
2447:
2398:
2397:
2254:{\displaystyle S}
2234:{\displaystyle x}
2179:{\displaystyle x}
2118:
1981:
1955:
1741:
1740:
1445:
1444:
1321:
1106:
1004:
996:
880:
641:The surface area
567:
515:
418:{\displaystyle C}
387:
300:
275:
256:{\displaystyle V}
186:chemical kinetics
182:phase transitions
71:
70:
63:
38:does not provide
4252:
4230:
4215:(3rd ed.).
4207:
4184:
4168:
4147:
4142:
4129:
4124:
4118:
4113:
4104:
4097:
4053:
4051:
4050:
4045:
4043:
4042:
4037:
4033:
4031:
4023:
4015:
4005:
4004:
4003:
3982:
3980:
3979:
3974:
3961:
3959:
3958:
3953:
3948:
3947:
3946:
3925:
3923:
3922:
3917:
3915:
3914:
3909:
3905:
3903:
3895:
3887:
3880:
3879:
3870:
3868:
3867:
3858:
3857:
3856:
3846:
3836:
3834:
3833:
3828:
3810:
3808:
3807:
3802:
3789:
3787:
3786:
3781:
3779:
3778:
3769:
3768:
3767:
3749:
3747:
3746:
3741:
3739:
3738:
3733:
3729:
3727:
3719:
3711:
3701:
3700:
3699:
3681:
3679:
3678:
3673:
3668:
3667:
3666:
3648:
3646:
3645:
3640:
3622:
3620:
3619:
3614:
3609:
3608:
3607:
3586:
3584:
3583:
3578:
3576:
3575:
3570:
3566:
3564:
3556:
3548:
3538:
3537:
3536:
3518:
3516:
3515:
3510:
3508:
3507:
3502:
3498:
3496:
3488:
3480:
3470:
3469:
3468:
3447:
3445:
3444:
3439:
3426:
3424:
3423:
3418:
3400:
3398:
3397:
3392:
3390:
3389:
3384:
3380:
3378:
3370:
3362:
3355:
3354:
3345:
3343:
3342:
3333:
3332:
3331:
3321:
3311:
3309:
3308:
3303:
3298:
3297:
3296:
3278:
3276:
3275:
3270:
3257:
3255:
3254:
3249:
3247:
3246:
3237:
3235:
3234:
3225:
3224:
3223:
3213:
3203:
3201:
3200:
3195:
3177:
3175:
3174:
3169:
3153:
3151:
3150:
3145:
3129:
3127:
3126:
3121:
3105:
3103:
3102:
3097:
3079:
3072:
3070:
3069:
3064:
3062:
3061:
3041:
3039:
3038:
3033:
3031:
3030:
3014:
3012:
3011:
3006:
3001:
3000:
2991:
2990:
2967:
2965:
2964:
2959:
2947:
2945:
2944:
2939:
2907:
2905:
2904:
2899:
2897:
2896:
2887:
2886:
2870:
2868:
2867:
2862:
2857:
2856:
2847:
2846:
2824:
2822:
2821:
2816:
2811:
2807:
2806:
2804:
2800:
2799:
2790:
2789:
2773:
2772:
2771:
2762:
2761:
2751:
2735:
2733:
2732:
2727:
2726:
2717:
2716:
2704:
2702:
2701:
2697:
2688:
2684:
2667:
2664:
2620:
2618:
2617:
2612:
2610:
2609:
2594:
2593:
2581:
2580:
2565:
2564:
2546:
2545:
2533:
2532:
2502:
2500:
2499:
2494:
2489:
2488:
2466:
2464:
2463:
2458:
2453:
2449:
2448:
2446:
2442:
2441:
2425:
2424:
2415:
2399:
2393:
2392:
2374:
2370:
2327:
2325:
2324:
2319:
2314:
2310:
2260:
2258:
2257:
2252:
2240:
2238:
2237:
2232:
2220:
2218:
2217:
2212:
2185:
2183:
2182:
2177:
2145:
2143:
2142:
2137:
2135:
2134:
2119:
2117:
2103:
2102:
2101:
2076:
2040:
2038:
2037:
2032:
1992:
1990:
1989:
1984:
1982:
1980:
1979:
1978:
1966:
1965:
1956:
1945:
1942:
1938:
1937:
1915:
1882:
1880:
1879:
1874:
1869:
1864:
1863:
1842:
1841:
1813:
1811:
1810:
1805:
1794:
1755:
1753:
1752:
1747:
1742:
1739:
1738:
1714:
1713:
1688:
1687:
1681:
1680:
1646:
1613:
1611:
1610:
1605:
1581:
1579:
1578:
1573:
1571:
1570:
1551:
1549:
1548:
1543:
1541:
1540:
1517:
1515:
1514:
1509:
1497:
1495:
1494:
1489:
1459:
1457:
1456:
1451:
1446:
1443:
1442:
1418:
1417:
1410:
1409:
1385:
1380:
1379:
1339:
1322:
1320:
1310:
1309:
1302:
1301:
1286:
1253:
1251:
1250:
1245:
1243:
1242:
1226:
1224:
1223:
1218:
1200:
1198:
1197:
1192:
1178:
1177:
1155:
1153:
1152:
1147:
1120:
1118:
1117:
1112:
1107:
1105:
1104:
1103:
1090:
1089:
1088:
1076:
1075:
1065:
1060:
1059:
1035:
1034:
1002:
997:
995:
987:
986:
985:
969:
942:
940:
939:
934:
923:
922:
906:
904:
903:
898:
881:
879:
869:
868:
861:
860:
845:
812:
810:
809:
804:
802:
801:
785:
783:
782:
777:
762:
760:
759:
754:
730:
729:
713:
708:
687:
686:
669:
667:
666:
661:
637:
635:
634:
629:
581:
579:
578:
573:
568:
566:
552:
551:
550:
532:
531:
521:
516:
514:
506:
505:
504:
494:
467:
465:
464:
459:
444:
442:
441:
436:
424:
422:
421:
416:
401:
399:
398:
393:
388:
386:
366:
365:
364:
342:
337:
336:
320:
318:
317:
312:
310:
309:
301:
296:
286:
284:
283:
278:
276:
271:
262:
260:
259:
254:
242:
240:
239:
234:
219:
217:
216:
211:
209:
208:
78:Atomic diffusion
66:
59:
55:
52:
46:
27:
26:
19:
4260:
4259:
4255:
4254:
4253:
4251:
4250:
4249:
4235:
4234:
4233:
4227:
4210:
4204:
4187:
4181:
4160:
4156:
4151:
4150:
4143:
4132:
4125:
4121:
4114:
4107:
4098:
4094:
4089:
4071:Brownian Motion
4062:
4024:
4016:
4010:
4009:
3994:
3986:
3985:
3965:
3964:
3937:
3929:
3928:
3896:
3888:
3882:
3881:
3871:
3859:
3847:
3840:
3839:
3816:
3815:
3793:
3792:
3770:
3758:
3753:
3752:
3720:
3712:
3706:
3705:
3690:
3685:
3684:
3657:
3652:
3651:
3628:
3627:
3598:
3590:
3589:
3557:
3549:
3543:
3542:
3527:
3522:
3521:
3489:
3481:
3475:
3474:
3459:
3451:
3450:
3430:
3429:
3406:
3405:
3371:
3363:
3357:
3356:
3346:
3334:
3322:
3315:
3314:
3287:
3282:
3281:
3261:
3260:
3238:
3226:
3214:
3207:
3206:
3183:
3182:
3157:
3156:
3133:
3132:
3109:
3108:
3085:
3084:
3053:
3048:
3047:
3022:
3017:
3016:
2992:
2982:
2974:
2973:
2950:
2949:
2918:
2917:
2914:
2888:
2878:
2873:
2872:
2848:
2838:
2830:
2829:
2791:
2781:
2774:
2763:
2753:
2752:
2746:
2742:
2718:
2708:
2677:
2673:
2672:
2671:
2627:
2626:
2622:
2601:
2585:
2572:
2556:
2537:
2524:
2513:
2512:
2509:
2480:
2472:
2471:
2433:
2426:
2416:
2410:
2406:
2384:
2349:
2348:
2344:
2333:Taylor expanded
2297:
2293:
2267:
2266:
2262:
2243:
2242:
2223:
2222:
2188:
2187:
2168:
2167:
2164:
2162:Single variable
2156:
2120:
2104:
2087:
2077:
2046:
2045:
1999:
1998:
1967:
1957:
1943:
1923:
1916:
1888:
1887:
1855:
1833:
1816:
1815:
1761:
1760:
1730:
1648:
1647:
1619:
1618:
1584:
1583:
1559:
1554:
1553:
1532:
1527:
1526:
1500:
1499:
1465:
1464:
1434:
1401:
1371:
1340:
1303:
1287:
1259:
1258:
1234:
1229:
1228:
1203:
1202:
1163:
1158:
1157:
1129:
1128:
1095:
1091:
1067:
1066:
1051:
1026:
988:
970:
948:
947:
912:
911:
862:
846:
818:
817:
793:
788:
787:
768:
767:
715:
676:
675:
643:
642:
587:
586:
553:
536:
523:
522:
507:
495:
473:
472:
447:
446:
427:
426:
407:
406:
367:
356:
343:
326:
325:
294:
289:
288:
265:
264:
245:
244:
222:
221:
198:
197:
194:
67:
56:
50:
47:
44:
40:page references
28:
24:
17:
12:
11:
5:
4258:
4256:
4248:
4247:
4237:
4236:
4232:
4231:
4225:
4217:Pergamon Press
4208:
4202:
4185:
4179:
4157:
4155:
4152:
4149:
4148:
4130:
4119:
4105:
4091:
4090:
4088:
4085:
4084:
4083:
4078:
4073:
4068:
4061:
4058:
4055:
4054:
4041:
4036:
4030:
4027:
4022:
4019:
4013:
4008:
4002:
3997:
3993:
3983:
3972:
3962:
3951:
3945:
3940:
3936:
3926:
3913:
3908:
3902:
3899:
3894:
3891:
3885:
3878:
3874:
3866:
3862:
3855:
3850:
3837:
3826:
3823:
3812:
3811:
3800:
3790:
3777:
3773:
3766:
3761:
3750:
3737:
3732:
3726:
3723:
3718:
3715:
3709:
3704:
3698:
3693:
3682:
3671:
3665:
3660:
3649:
3638:
3635:
3624:
3623:
3612:
3606:
3601:
3597:
3587:
3574:
3569:
3563:
3560:
3555:
3552:
3546:
3541:
3535:
3530:
3519:
3506:
3501:
3495:
3492:
3487:
3484:
3478:
3473:
3467:
3462:
3458:
3448:
3437:
3427:
3416:
3413:
3402:
3401:
3388:
3383:
3377:
3374:
3369:
3366:
3360:
3353:
3349:
3341:
3337:
3330:
3325:
3312:
3301:
3295:
3290:
3279:
3268:
3258:
3245:
3241:
3233:
3229:
3222:
3217:
3204:
3193:
3190:
3179:
3178:
3167:
3164:
3154:
3143:
3140:
3130:
3119:
3116:
3106:
3095:
3092:
3082:
3060:
3056:
3029:
3025:
3004:
2999:
2995:
2989:
2985:
2981:
2957:
2937:
2934:
2931:
2928:
2925:
2913:
2910:
2895:
2891:
2885:
2881:
2860:
2855:
2851:
2845:
2841:
2837:
2826:
2825:
2814:
2810:
2803:
2798:
2794:
2788:
2784:
2780:
2777:
2770:
2766:
2760:
2756:
2749:
2745:
2741:
2738:
2730:
2725:
2721:
2715:
2711:
2707:
2700:
2696:
2692:
2687:
2683:
2680:
2676:
2670:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2641:
2637:
2634:
2608:
2604:
2600:
2597:
2592:
2588:
2584:
2579:
2575:
2571:
2568:
2563:
2559:
2555:
2552:
2549:
2544:
2540:
2536:
2531:
2527:
2523:
2520:
2508:
2505:
2492:
2487:
2483:
2479:
2468:
2467:
2456:
2452:
2445:
2440:
2436:
2432:
2429:
2423:
2419:
2413:
2409:
2405:
2402:
2396:
2391:
2387:
2383:
2380:
2377:
2373:
2368:
2365:
2362:
2359:
2356:
2329:
2328:
2317:
2313:
2309:
2306:
2303:
2300:
2296:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2250:
2230:
2210:
2207:
2204:
2201:
2198:
2195:
2175:
2163:
2160:
2155:
2152:
2147:
2146:
2133:
2130:
2127:
2123:
2116:
2113:
2110:
2107:
2100:
2097:
2094:
2090:
2086:
2083:
2080:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
1995:
1994:
1977:
1974:
1970:
1964:
1960:
1954:
1951:
1948:
1941:
1936:
1933:
1930:
1926:
1922:
1919:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1872:
1868:
1862:
1858:
1854:
1851:
1848:
1845:
1840:
1836:
1832:
1829:
1826:
1823:
1803:
1800:
1797:
1793:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1757:
1756:
1745:
1737:
1733:
1729:
1726:
1723:
1720:
1717:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1686:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1651:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1603:
1600:
1597:
1594:
1591:
1569:
1566:
1562:
1539:
1535:
1507:
1487:
1484:
1481:
1478:
1475:
1472:
1461:
1460:
1449:
1441:
1437:
1433:
1430:
1427:
1424:
1421:
1416:
1413:
1408:
1404:
1400:
1397:
1394:
1391:
1388:
1384:
1378:
1374:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1337:
1334:
1331:
1328:
1325:
1319:
1316:
1313:
1308:
1300:
1297:
1294:
1290:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1241:
1237:
1216:
1213:
1210:
1190:
1187:
1184:
1181:
1176:
1173:
1170:
1166:
1145:
1142:
1139:
1136:
1127:The fact that
1122:
1121:
1110:
1102:
1098:
1094:
1087:
1082:
1079:
1074:
1070:
1063:
1058:
1054:
1050:
1047:
1044:
1041:
1038:
1033:
1029:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1000:
994:
991:
984:
979:
976:
973:
967:
964:
961:
958:
955:
932:
929:
926:
921:
908:
907:
896:
893:
890:
887:
884:
878:
875:
872:
867:
859:
856:
853:
849:
843:
840:
837:
834:
831:
828:
825:
800:
796:
775:
764:
763:
752:
749:
746:
742:
739:
736:
733:
728:
725:
722:
718:
712:
707:
703:
699:
696:
693:
690:
685:
659:
656:
653:
650:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
583:
582:
571:
565:
562:
559:
556:
549:
546:
543:
539:
535:
530:
526:
519:
513:
510:
503:
498:
492:
489:
486:
483:
480:
457:
454:
434:
414:
403:
402:
391:
385:
382:
379:
376:
373:
370:
363:
359:
355:
352:
349:
346:
340:
335:
308:
305:
299:
274:
252:
232:
229:
207:
193:
190:
69:
68:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4257:
4246:
4243:
4242:
4240:
4228:
4226:0-08-023038-5
4222:
4218:
4214:
4209:
4205:
4203:0-471-54607-0
4199:
4195:
4191:
4186:
4182:
4180:0-486-60147-1
4176:
4172:
4167:
4166:
4159:
4158:
4153:
4146:
4141:
4139:
4137:
4135:
4131:
4128:
4123:
4120:
4117:
4116:Khinchin 1949
4112:
4110:
4106:
4102:
4096:
4093:
4086:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4063:
4059:
4039:
4034:
4028:
4020:
4011:
4006:
3995:
3991:
3984:
3970:
3963:
3949:
3938:
3934:
3927:
3911:
3906:
3900:
3892:
3883:
3876:
3872:
3864:
3860:
3848:
3838:
3824:
3813:
3798:
3791:
3775:
3771:
3759:
3751:
3735:
3730:
3724:
3716:
3707:
3702:
3691:
3683:
3669:
3658:
3650:
3636:
3625:
3610:
3599:
3595:
3588:
3572:
3567:
3561:
3553:
3544:
3539:
3528:
3520:
3504:
3499:
3493:
3485:
3476:
3471:
3460:
3456:
3449:
3435:
3428:
3414:
3403:
3386:
3381:
3375:
3367:
3358:
3351:
3347:
3339:
3335:
3323:
3313:
3299:
3288:
3280:
3266:
3259:
3243:
3239:
3231:
3227:
3215:
3205:
3191:
3180:
3165:
3155:
3141:
3131:
3117:
3107:
3093:
3083:
3080:
3076:
3075:heat capacity
3058:
3054:
3045:
3044:heat capacity
3027:
3023:
2997:
2993:
2987:
2983:
2969:
2955:
2935:
2932:
2929:
2926:
2923:
2911:
2909:
2893:
2889:
2883:
2879:
2853:
2849:
2843:
2839:
2812:
2808:
2796:
2792:
2786:
2782:
2775:
2768:
2764:
2758:
2754:
2747:
2743:
2739:
2736:
2723:
2719:
2713:
2709:
2698:
2694:
2690:
2685:
2681:
2678:
2674:
2668:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2639:
2635:
2632:
2625:
2624:
2623:
2606:
2602:
2598:
2595:
2590:
2586:
2582:
2577:
2573:
2569:
2561:
2557:
2553:
2550:
2547:
2542:
2538:
2534:
2529:
2525:
2518:
2506:
2504:
2485:
2481:
2470:The quantity
2454:
2450:
2438:
2434:
2427:
2421:
2417:
2411:
2407:
2403:
2400:
2389:
2385:
2378:
2375:
2371:
2366:
2360:
2354:
2347:
2346:
2345:
2342:
2338:
2334:
2315:
2311:
2304:
2298:
2294:
2290:
2287:
2284:
2278:
2272:
2265:
2264:
2263:
2248:
2228:
2208:
2205:
2199:
2193:
2173:
2161:
2159:
2153:
2151:
2131:
2128:
2125:
2121:
2111:
2098:
2095:
2092:
2084:
2081:
2072:
2069:
2063:
2060:
2057:
2051:
2044:
2043:
2042:
2025:
2022:
2019:
2010:
2007:
2004:
1975:
1972:
1968:
1962:
1958:
1952:
1949:
1946:
1939:
1934:
1931:
1928:
1920:
1911:
1902:
1886:
1885:
1884:
1870:
1866:
1860:
1852:
1846:
1838:
1830:
1798:
1791:
1787:
1784:
1775:
1766:
1743:
1735:
1727:
1718:
1715:
1701:
1692:
1674:
1662:
1653:
1649:
1642:
1633:
1617:
1616:
1615:
1598:
1592:
1589:
1567:
1564:
1560:
1537:
1533:
1523:
1521:
1482:
1479:
1476:
1470:
1447:
1439:
1431:
1422:
1419:
1406:
1398:
1386:
1382:
1376:
1365:
1359:
1356:
1350:
1344:
1341:
1335:
1329:
1314:
1298:
1295:
1292:
1288:
1282:
1276:
1273:
1270:
1264:
1257:
1256:
1255:
1239:
1235:
1211:
1185:
1174:
1171:
1168:
1164:
1140:
1125:
1108:
1100:
1096:
1080:
1077:
1072:
1061:
1056:
1048:
1039:
1031:
1020:
1014:
1011:
998:
992:
977:
974:
965:
962:
956:
946:
945:
944:
927:
894:
888:
873:
857:
854:
851:
847:
841:
835:
832:
829:
823:
816:
815:
814:
798:
794:
773:
750:
747:
744:
737:
726:
723:
720:
716:
705:
701:
697:
691:
674:
673:
672:
654:
639:
622:
619:
616:
607:
601:
592:
569:
560:
547:
544:
541:
537:
533:
528:
524:
517:
511:
490:
484:
471:
470:
469:
455:
452:
412:
389:
380:
377:
374:
361:
353:
350:
347:
338:
324:
323:
322:
306:
303:
297:
272:
250:
230:
227:
191:
189:
187:
183:
179:
175:
171:
167:
163:
159:
154:
152:
148:
144:
141:, the volume
140:
135:
133:
129:
125:
123:
119:
115:
110:
108:
104:
99:
97:
96:absolute zero
92:
88:
79:
75:
65:
62:
54:
42:
41:
35:
30:
21:
20:
4212:
4189:
4164:
4127:Lavenda 1991
4122:
4095:
2915:
2827:
2510:
2469:
2330:
2165:
2157:
2148:
1996:
1758:
1524:
1462:
1126:
1123:
909:
765:
640:
584:
404:
195:
155:
146:
142:
138:
136:
126:
111:
100:
90:
84:
57:
48:
37:
2337:equilibrium
168:(including
166:dissipation
103:temperature
4154:References
4026:∂
4018:∂
3992:−
3935:−
3898:∂
3890:∂
3822:Δ
3722:∂
3714:∂
3634:Δ
3596:−
3559:∂
3551:∂
3491:∂
3483:∂
3457:−
3412:Δ
3373:∂
3365:∂
3189:Δ
3163:Δ
3139:Δ
3115:Δ
3091:Δ
3003:⟩
2980:⟨
2972:Averages
2859:⟩
2836:⟨
2802:⟩
2779:⟨
2748:−
2740:
2729:⟩
2706:⟨
2682:π
2659:…
2640:∏
2596:…
2551:…
2491:⟩
2478:⟨
2444:⟩
2431:⟨
2412:−
2404:
2395:⟩
2382:⟨
2379:π
2291:
2285:∝
2129:β
2126:−
2106:Γ
2096:−
2082:β
2073:β
2064:β
2014:Γ
1973:−
1950:π
1932:−
1925:⟩
1918:⟨
1906:⟩
1900:⟨
1894:Ω
1857:⟩
1850:⟨
1844:⟩
1828:Δ
1822:⟨
1802:⟩
1796:⟨
1779:⟩
1773:⟨
1767:β
1725:Δ
1719:π
1705:⟩
1699:⟨
1693:β
1678:⟩
1672:⟨
1666:⟩
1660:⟨
1654:β
1637:⟩
1631:⟨
1625:Ω
1602:⟩
1596:⟨
1565:−
1561:β
1506:Ω
1483:β
1429:Δ
1423:π
1412:⟩
1396:Δ
1390:⟨
1369:⟩
1363:⟨
1360:−
1351:−
1345:
1336:≈
1324:Ω
1315:β
1296:β
1293:−
1277:β
1240:⋆
1215:⟩
1209:⟨
1180:Ω
1172:β
1169:−
1135:Ω
1097:β
1093:∂
1081:
1069:∂
1046:Δ
1037:⟩
1024:⟩
1018:⟨
1015:−
1006:⟨
993:β
990:∂
978:
972:∂
966:−
960:⟩
954:⟨
928:β
883:Ω
874:β
855:β
852:−
836:β
799:⋆
774:β
732:Ω
724:β
721:−
711:∞
702:∫
692:β
649:Ω
611:Γ
596:Γ
555:Γ
545:−
534:⋅
509:∂
497:∂
479:Ω
433:Γ
369:Γ
351:⋅
174:viscosity
162:diffusion
51:July 2020
4239:Category
4060:See also
2166:Suppose
3073:is the
3042:is the
170:damping
149:in the
132:entropy
118:phonons
34:sources
4223:
4200:
4177:
2828:where
1003:
405:where
122:rotons
81:shown.
4087:Notes
1614:give
158:noise
4221:ISBN
4198:ISBN
4175:ISBN
2948:and
2221:for
184:and
172:and
164:and
36:but
4099:In
2737:exp
2401:exp
2288:exp
1342:exp
85:In
4241::
4219:.
4196:.
4192:.
4173:.
4169:.
4133:^
4108:^
2908:.
1078:ln
975:ln
638:.
188:.
109:.
98:.
89:,
4229:.
4206:.
4183:.
4040:S
4035:)
4029:V
4021:P
4012:(
4007:T
4001:B
3996:k
3971:0
3950:T
3944:B
3939:k
3912:V
3907:)
3901:T
3893:P
3884:(
3877:2
3873:T
3865:V
3861:C
3854:B
3849:k
3825:P
3799:0
3776:P
3772:C
3765:B
3760:k
3736:P
3731:)
3725:T
3717:V
3708:(
3703:T
3697:B
3692:k
3670:T
3664:B
3659:k
3637:S
3611:T
3605:B
3600:k
3573:P
3568:)
3562:T
3554:V
3545:(
3540:T
3534:B
3529:k
3505:T
3500:)
3494:P
3486:V
3477:(
3472:T
3466:B
3461:k
3436:0
3415:V
3387:V
3382:)
3376:T
3368:P
3359:(
3352:2
3348:T
3340:V
3336:C
3329:B
3324:k
3300:T
3294:B
3289:k
3267:0
3244:2
3240:T
3232:V
3228:C
3221:B
3216:k
3192:T
3166:P
3142:S
3118:V
3094:T
3059:V
3055:C
3028:P
3024:C
2998:j
2994:x
2988:i
2984:x
2956:S
2936:P
2933:,
2930:V
2927:,
2924:T
2894:j
2890:x
2884:i
2880:x
2854:j
2850:x
2844:i
2840:x
2813:,
2809:)
2797:j
2793:x
2787:i
2783:x
2776:2
2769:j
2765:x
2759:i
2755:x
2744:(
2724:j
2720:x
2714:i
2710:x
2699:2
2695:/
2691:n
2686:)
2679:2
2675:(
2669:1
2662:n
2656:1
2653:=
2650:j
2647:,
2644:i
2636:=
2633:w
2621::
2607:n
2603:x
2599:d
2591:2
2587:x
2583:d
2578:1
2574:x
2570:d
2567:)
2562:n
2558:x
2554:,
2548:,
2543:2
2539:x
2535:,
2530:1
2526:x
2522:(
2519:w
2486:2
2482:x
2455:.
2451:)
2439:2
2435:x
2428:2
2422:2
2418:x
2408:(
2390:2
2386:x
2376:2
2372:1
2367:=
2364:)
2361:x
2358:(
2355:w
2343::
2316:.
2312:)
2308:)
2305:x
2302:(
2299:S
2295:(
2282:)
2279:x
2276:(
2273:w
2261::
2249:S
2229:x
2209:x
2206:d
2203:)
2200:x
2197:(
2194:w
2174:x
2132:E
2122:e
2115:)
2112:m
2109:(
2099:1
2093:m
2089:)
2085:E
2079:(
2070:=
2067:)
2061:;
2058:E
2055:(
2052:f
2029:)
2026:1
2023:+
2020:m
2017:(
2011:=
2008:!
2005:m
1993:.
1976:m
1969:e
1963:m
1959:m
1953:m
1947:2
1940:m
1935:1
1929:m
1921:E
1912:=
1909:)
1903:E
1897:(
1871:m
1867:/
1861:2
1853:E
1847:=
1839:2
1835:)
1831:E
1825:(
1799:E
1792:/
1788:m
1785:=
1782:)
1776:E
1770:(
1744:.
1736:2
1732:)
1728:E
1722:(
1716:2
1711:)
1708:)
1702:E
1696:(
1690:(
1685:Z
1675:E
1669:)
1663:E
1657:(
1650:e
1643:=
1640:)
1634:E
1628:(
1599:E
1593:=
1590:E
1568:m
1538:m
1534:E
1486:)
1480:;
1477:E
1474:(
1471:f
1448:.
1440:2
1436:)
1432:E
1426:(
1420:2
1415:}
1407:2
1403:)
1399:E
1393:(
1387:2
1383:/
1377:2
1373:)
1366:E
1357:E
1354:(
1348:{
1333:)
1330:E
1327:(
1318:)
1312:(
1307:Z
1299:E
1289:e
1283:=
1280:)
1274:;
1271:E
1268:(
1265:f
1236:E
1212:E
1189:)
1186:E
1183:(
1175:E
1165:e
1144:)
1141:E
1138:(
1109:,
1101:2
1086:Z
1073:2
1062:=
1057:2
1053:)
1049:E
1043:(
1040:=
1032:2
1028:)
1021:E
1012:E
1009:(
999:,
983:Z
963:=
957:E
931:)
925:(
920:Z
895:,
892:)
889:E
886:(
877:)
871:(
866:Z
858:E
848:e
842:=
839:)
833:;
830:E
827:(
824:f
795:E
751:,
748:E
745:d
741:)
738:E
735:(
727:E
717:e
706:0
698:=
695:)
689:(
684:Z
658:)
655:E
652:(
626:)
623:1
620:+
617:m
614:(
608:=
605:)
602:m
599:(
593:m
570:,
564:)
561:m
558:(
548:1
542:m
538:E
529:m
525:C
518:=
512:E
502:V
491:=
488:)
485:E
482:(
456:m
453:2
413:C
390:,
384:)
381:1
378:+
375:m
372:(
362:m
358:)
354:E
348:C
345:(
339:=
334:V
307:m
304:2
298:E
273:E
251:V
231:m
228:2
206:V
147:E
143:V
139:N
64:)
58:(
53:)
49:(
43:.
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