120:. Correlation functions describe how microscopic variables, such as spin and density, at different positions are related. More specifically, correlation functions measure quantitatively the extent to which microscopic variables fluctuate together, on average, across space and/or time. Keep in mind that correlation doesn’t automatically equate to causation. So, even if there’s a non-zero correlation between two points in space or time, it doesn’t mean there is a direct causal link between them. Sometimes, a correlation can exist without any causal relationship. This could be purely coincidental or due to other underlying factors, known as confounding variables, which cause both points to covary (statistically).
1962:, should be uncorrelated beyond what we would expect from thermodynamic equilibrium, the evolution in time of a correlation function can be viewed from a physical standpoint as the system gradually 'forgetting' the initial conditions placed upon it via the specification of some microscopic variable. There is actually an intuitive connection between the time evolution of correlation functions and the time evolution of macroscopic systems: on average, the correlation function evolves in time in the same manner as if a system was prepared in the conditions specified by the correlation function's initial value and allowed to evolve.
1978:
1893:, it is clear that one can define the random variables used in these correlation functions, such as atomic positions and spins, away from equilibrium. As such, their scalar product is well-defined away from equilibrium. The operation which is no longer well-defined away from equilibrium is the average over the equilibrium ensemble. This averaging process for non-equilibrium system is typically replaced by averaging the scalar product across the entire sample. This is typical in scattering experiments and computer simulations, and is often used to measure the radial distribution functions of glasses.
27:
1845:. One example is in diffusion. A single-phase system at equilibrium has a homogeneous composition macroscopically. However, if one watches the microscopic movement of each atom, fluctuations in composition are constantly occurring due to the quasi-random walks taken by the individual atoms. Statistical mechanics allows one to make insightful statements about the temporal behavior of such fluctuations of equilibrium systems. This is discussed below in the section on the
2487:, must transition continuously from being infinite to finite when the material is heated through its critical temperature. This gives rise to a power-law dependence of the correlation function as a function of distance at the critical point. This is shown in the figure in the left for the case of a ferromagnetic material, with the quantitative details listed in the section on magnetism.
77:. In all cases, correlations are strongest nearest to the origin, indicating that a spin has the strongest influence on its nearest neighbors. All correlations gradually decay as the distance from the spin at the origin increases. Above the Curie temperature, the correlation between spins tends to zero as the distance between the spins gets very large. In contrast, below
104:, the correlation between the spins does not tend toward zero at large distances, but instead decays to a level consistent with the long-range order of the system. The difference in these decay behaviors, where correlations between microscopic random variables become zero versus non-zero at large distances, is one way of defining short- versus long-range order.
2659:), the interaction between the spins will cause them to be correlated. The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures exponentially-decaying correlations are observed with increasing distance, with the correlation function being given asymptotically by
481:
1682:
1160:
1832:
3416:
However, such higher order correlation functions are relatively difficult to interpret and measure. For example, in order to measure the higher-order analogues of pair distribution functions, coherent x-ray sources are needed. Both the theory of such analysis and the experimental measurement of the
2769:
is an exponent, whose value depends on whether the system is in the disordered phase (i.e. above the critical point), or in the ordered phase (i.e. below the critical point). At high temperatures, the correlation decays to zero exponentially with the distance between the spins. The same exponential
2466:
Continuous phase transitions, such as order-disorder transitions in metallic alloys and ferromagnetic-paramagnetic transitions, involve a transition from an ordered to a disordered state. In terms of correlation functions, the equal-time correlation function is non-zero for all lattice points below
123:
A classic example of spatial correlation can be seen in ferromagnetic and antiferromagnetic materials. In these materials, atomic spins tend to align in parallel and antiparallel configurations with their adjacent counterparts, respectively. The figure on the right visually represents this spatial
1923:
as opposed to x-ray scattering. Neutron scattering can also yield information on pair correlations as well. For systems composed of particles larger than about one micrometer, optical microscopy can be used to measure both equal-time and equal-position correlation functions. Optical microscopy is
1345:
818:. However, in statistical mechanics, not all correlation functions are autocorrelation functions. For example, in multicomponent condensed phases, the pair correlation function between different elements is often of interest. Such mixed-element pair correlation functions are an example of
2634:
1914:
Correlation functions are typically measured with scattering experiments. For example, x-ray scattering experiments directly measure electron-electron equal-time correlations. From knowledge of elemental structure factors, one can also measure elemental pair correlation functions. See
3411:
285:
1498:
976:
1690:
1836:
The above assumption may seem non-intuitive at first: how can an ensemble which is time-invariant have a non-uniform temporal correlation function? Temporal correlations remain relevant to talk about in equilibrium systems because a time-invariant,
662:
1207:
1857:
All of the above correlation functions have been defined in the context of equilibrium statistical mechanics. However, it is possible to define correlation functions for systems away from equilibrium. Examining the general definition of
2744:
754:
1351:
is an example of an equal-time correlation function where the uncorrelated reference is generally not subtracted. Other equal-time spin-spin correlation functions are shown on this page for a variety of materials and conditions.
3190:
Higher-order correlation functions involve multiple reference points, and are defined through a generalization of the above correlation function by taking the expected value of the product of more than two random variables:
1686:
Assuming equilibrium (and thus time invariance of the ensemble) and averaging over all sites in the sample gives a simpler expression for the equal-position correlation function as for the equal-time correlation function:
2508:
2467:
the critical temperature, and is non-negligible for only a fairly small radius above the critical temperature. As the phase transition is continuous, the length over which the microscopic variables are correlated,
3197:
2504:
system, the equal-time correlation function is especially well-studied. It describes the canonical ensemble (thermal) average of the scalar product of the spins at two lattice points over all possible orderings:
2202:
2902:
2442:, the spins exhibit spontaneous ordering, i.e. long-range order, and infinite correlation length. Continuous order-disorder transitions can be understood as the process of the correlation length,
2386:
3114:
2636:
Here the brackets mean the above-mentioned thermal average. Schematic plots of this function are shown for a ferromagnetic material below, at, and above its Curie temperature on the left.
476:{\displaystyle C(r,\tau )=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t+\tau )\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t+\tau )\rangle \,.}
3015:
1677:{\displaystyle C(0,\tau )=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R,t+\tau )\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R,t+\tau )\rangle \,.}
1936:
proposed that the regression of microscopic thermal fluctuations at equilibrium follows the macroscopic law of relaxation of small non-equilibrium disturbances. This is known as the
509:
2828:
2982:
2311:
3050:
the correlation length diverges, as the correlation length must transition continuously from a finite value above the phase transition, to infinite below the phase transition:
102:
55:
810:
describe the same variable, such as a spin-spin correlation function, or a particle position-position correlation function in an elemental liquid or a solid (often called a
3041:
2115:
2020:
1155:{\displaystyle C(r,0)=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t)\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t)\rangle \,.}
2949:
2767:
1891:
1827:{\displaystyle C(\tau )=\langle \mathbf {s_{1}} (0)\cdot \mathbf {s_{2}} (\tau )\rangle \ -\langle \mathbf {s_{1}} (0)\rangle \langle \mathbf {s_{2}} (\tau )\rangle \,.}
1467:
572:
2462:, transitioning from being infinite in the low-temperature, ordered state, to infinite at the critical point, and then finite in a high-temperature, disordered state.
1428:
932:
280:
971:
2951:
introduced above. For example, the exact solution of the two-dimensional Ising model (with short-ranged ferromagnetic interactions) gives precisely at criticality
2925:
2795:
2440:
2415:
is distinguished by the extreme non-locality of the spatial correlations between microscopic values of the relevant order parameter without long-range order. Below
2413:
2269:
2074:
2047:
1960:
906:
874:
847:
808:
781:
567:
540:
188:
161:
3137:
2665:
2657:
2485:
2460:
2242:
2222:
75:
1493:
234:
1402:
1382:
1202:
1182:
511:, indicate the above-mentioned thermal average. It is important to note here, however, that while the brackets are called an average, they are calculated as an
254:
208:
667:
2639:
Even in a magnetically disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale
1340:{\displaystyle C(r)=\langle \mathbf {s_{1}} (0)\cdot \mathbf {s_{2}} (r)\rangle \ -\langle \mathbf {s_{1}} (0)\rangle \langle \mathbf {s_{2}} (r)\rangle }
3564:
Altarelli, M.; Kurta, R. P.; Vartanyants, I. A. (2010). "X-ray cross-correlation analysis and local symmetries of disordered systems: General theory".
3812:
2629:{\displaystyle C(r)=\langle \mathbf {s} (R)\cdot \mathbf {s} (R+r)\rangle \ -\langle \mathbf {s} (R)\rangle \langle \mathbf {s} (R+r)\rangle \,.}
3406:{\displaystyle C_{i_{1}i_{2}\cdots i_{n}}(s_{1},s_{2},\cdots ,s_{n})=\langle X_{i_{1}}(s_{1})X_{i_{2}}(s_{2})\cdots X_{i_{n}}(s_{n})\rangle .}
3758:
20:
3174:. The correlation function can be calculated in exactly solvable models (one-dimensional Bose gas, spin chains, Hubbard model) by means of
3829:
2120:
3734:
3462:
3175:
3662:
Wochner, P.; Gutt, C.; Autenrieth, T.; Demmer, T.; Bugaev, V.; Ortiz, A. D.; Duri, A.; Zontone, F.; Grubel, G.; Dosch, H. (2009).
3617:
LehmkĂĽhler, F.; GrĂĽbel, G.; Gutt, C. (2014). "Detecting orientational order in model systems by X-ray cross-correlation methods".
1900:
3151:
2836:
1966:
888:
influence of a given random variable, say the direction of a spin, on its local environment, without considering later times,
3834:
1469:. They are defined analogously to above equal-time correlation functions, but we now neglect spatial dependencies by setting
3426:
2316:
1364:
evolution of microscopic variables. In other words, how the value of a microscopic variable at a given position and time,
3743:
3163:
1916:
1348:
1204:, by assuming equilibrium (and thus time invariance of the ensemble) and averaging over all sample positions, yielding:
811:
30:
Schematic equal-time spin correlation functions for ferromagnetic and antiferromagnetic materials both above and below
3056:
1977:
3182:. In an isotropic XY model, time and temperature correlations were evaluated by Its, Korepin, Izergin & Slavnov.
3839:
1938:
3522:
2987:
815:
515:, not an average value. It is a matter of convention whether one subtracts the uncorrelated average product of
2022:, as a function of radius for a ferromagnetic spin system above, at, and below at its critical temperature,
1896:
One can also define averages over states for systems perturbed slightly from equilibrium. See, for example,
488:
2800:
2954:
2274:
3167:
1965:
Equilibrium fluctuations of the system can be related to its response to external perturbations via the
908:. In this case, we neglect the time evolution of the system, so the above definition is re-written with
109:
3150:, seen in these transitions. All exponents mentioned are independent of temperature. They are in fact
80:
33:
26:
3780:
3675:
3583:
3490:
1347:
where, again, the choice of whether to subtract the uncorrelated variables differs among fields. The
756:, with the convention differing among fields. The most common uses of correlation functions are when
117:
3020:
2079:
1984:
657:{\displaystyle \langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t+\tau )\rangle }
3644:
3626:
3599:
3573:
3147:
1920:
133:
2934:
2752:
1861:
1437:
3803:
3768:
3754:
3730:
3703:
3458:
2928:
876:
represent the average variations in density as a function position for two distinct elements.
819:
2739:{\displaystyle C(r)\approx {\frac {1}{r^{\vartheta }}}\exp {\left(-{\frac {r}{d}}\right)}\,,}
1407:
911:
259:
3788:
3693:
3683:
3636:
3591:
3498:
3047:
2204:. The power-law dependence dominates at distances short relative to the correlation length,
941:
814:
or a pair correlation function). Correlation functions between the same random variable are
2910:
2773:
2418:
2391:
2247:
2052:
2025:
1945:
891:
852:
825:
786:
759:
545:
518:
166:
139:
3171:
3122:
2642:
2470:
2445:
2227:
2207:
1904:
60:
1472:
213:
3784:
3679:
3664:"X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter"
3587:
3494:
749:{\displaystyle \langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t+\tau )\rangle }
3698:
3663:
2501:
1897:
1387:
1367:
1187:
1167:
512:
239:
193:
1919:
for further information. Equal-time spin–spin correlation functions are measured with
3823:
3807:
3603:
3450:
3648:
3179:
3046:
As the temperature is lowered, thermal disordering is lowered, and in a continuous
1933:
1430:(and usually at the same position). Such temporal correlations are quantified via
2749:
where r is the distance between spins, and d is the dimension of the system, and
3725:
Sethna, James P. (2006). "Chapter 10: Correlations, response, and dissipation".
1847:
temporal evolution of correlation functions and
Onsager's regression hypothesis
3595:
3799:
3792:
3640:
3688:
3551:
3143:
2224:, while the exponential dependence dominates at distances large relative to
3707:
3503:
3479:
116:
is a measure of the order in a system, as characterized by a mathematical
2931:, which does not have any simple relation with the non-critical exponent
1404:, influences the value of the same microscopic variable at a later time,
1942:. As the values of microscopic variables separated by large timescales,
3417:
needed X-ray cross-correlation functions are areas of active research.
2117:
exhibits a combined exponential and power-law dependence on distance:
2797:, but with the limit at large distances being the mean magnetization
1924:
thus common for colloidal suspensions, especially in two dimensions.
3631:
3578:
3547:
2830:. Precisely at the critical point, an algebraic behavior is seen
1976:
1973:
The connection between phase transitions and correlation functions
25:
2197:{\displaystyle C(r,\tau =0)\propto r^{-\vartheta }e^{-r/\xi (T)}}
136:(thermal) average of the scalar product of two random variables,
3771:(1974). "Renormalization Group in Theory of Critical Behavior".
3727:
Statistical
Mechanics: Entropy, Order Parameters, and Complexity
3455:
Statistical
Mechanics: Entropy, Order Parameters, and Complexity
3546:
A.R. Its, V.e. Korepin, A.G. Izergin & N.A. Slavnov (2009)
2770:
decay as a function of radial distance is also observed below
132:
The most common definition of a correlation function is the
1846:
2897:{\displaystyle C(r)\approx {\frac {1}{r^{(d-2+\eta )}}}\,,}
1356:
Equilibrium equal-position (temporal) correlation functions
3154:, i.e. found to be the same in a wide variety of systems.
57:
versus the distance normalized by the correlation length,
1853:
Generalization beyond equilibrium correlation functions
1841:
ensemble can still have non-trivial temporal dynamics
880:
Equilibrium equal-time (spatial) correlation functions
3451:"Chapter 10: Correlations, response, and dissipation"
3200:
3125:
3059:
3023:
2990:
2957:
2937:
2913:
2839:
2803:
2776:
2755:
2668:
2645:
2511:
2473:
2448:
2421:
2394:
2319:
2277:
2250:
2230:
2210:
2123:
2082:
2055:
2028:
1987:
1948:
1864:
1693:
1501:
1475:
1440:
1410:
1390:
1370:
1210:
1190:
1170:
979:
944:
914:
894:
855:
828:
789:
762:
670:
575:
548:
521:
491:
288:
262:
242:
216:
196:
169:
142:
83:
63:
36:
3480:"Reciprocal Relations in Irreversible Processes. I."
2381:{\displaystyle C(r,\tau =0)\propto r^{-(d-2+\eta )}}
1898:
http://xbeams.chem.yale.edu/~batista/vaa/node56.html
3405:
3131:
3108:
3035:
3009:
2976:
2943:
2919:
2896:
2822:
2789:
2761:
2738:
2651:
2628:
2479:
2454:
2434:
2407:
2380:
2305:
2263:
2236:
2216:
2196:
2109:
2068:
2041:
2014:
1954:
1885:
1826:
1676:
1487:
1461:
1422:
1396:
1376:
1339:
1196:
1176:
1154:
965:
926:
900:
868:
841:
802:
775:
748:
656:
561:
534:
503:
475:
274:
248:
228:
202:
182:
155:
96:
69:
49:
3109:{\displaystyle \xi \propto |T-T_{c}|^{-\nu }\,,}
3668:Proceedings of the National Academy of Sciences
124:correlation between spins in such materials.
8:
3397:
3292:
2817:
2804:
2619:
2596:
2593:
2576:
2567:
2527:
1817:
1793:
1790:
1766:
1757:
1709:
1667:
1631:
1628:
1598:
1589:
1523:
1334:
1310:
1307:
1283:
1274:
1226:
1145:
1109:
1106:
1076:
1067:
1001:
743:
671:
651:
609:
606:
576:
498:
492:
466:
424:
421:
391:
382:
310:
3535:Conformal Invariance and Critical Phenomena
3751:Statistical Mechanics of Phase Transitions
2313:, resulting in solely power-law behavior:
3697:
3687:
3630:
3577:
3502:
3388:
3373:
3368:
3352:
3337:
3332:
3319:
3304:
3299:
3280:
3261:
3248:
3233:
3220:
3210:
3205:
3199:
3124:
3102:
3093:
3088:
3081:
3066:
3058:
3022:
3010:{\displaystyle \vartheta ={\frac {1}{2}}}
2997:
2989:
2964:
2956:
2936:
2912:
2890:
2864:
2855:
2838:
2811:
2802:
2781:
2775:
2754:
2732:
2716:
2707:
2693:
2684:
2667:
2644:
2622:
2599:
2579:
2547:
2530:
2510:
2472:
2447:
2426:
2420:
2399:
2393:
2351:
2318:
2288:
2276:
2255:
2249:
2229:
2209:
2175:
2168:
2155:
2122:
2081:
2060:
2054:
2033:
2027:
1986:
1947:
1863:
1820:
1801:
1796:
1774:
1769:
1741:
1736:
1717:
1712:
1692:
1670:
1639:
1634:
1606:
1601:
1561:
1556:
1531:
1526:
1500:
1474:
1439:
1409:
1389:
1369:
1318:
1313:
1291:
1286:
1258:
1253:
1234:
1229:
1209:
1189:
1169:
1148:
1117:
1112:
1084:
1079:
1039:
1034:
1009:
1004:
978:
943:
913:
893:
860:
854:
833:
827:
794:
788:
767:
761:
709:
704:
679:
674:
669:
617:
612:
584:
579:
574:
553:
547:
526:
520:
490:
469:
432:
427:
399:
394:
348:
343:
318:
313:
287:
261:
241:
215:
195:
174:
168:
147:
141:
88:
82:
62:
41:
35:
3816:, vol. 1-20 (1972–2001), Academic Press.
3813:Phase Transitions and Critical Phenomena
3548:Temperature Correlation of Quantum Spins
3444:
3442:
3438:
3162:One common correlation function is the
1928:Time evolution of correlation functions
884:Often, one is interested in solely the
504:{\displaystyle \langle \cdot \rangle }
2823:{\displaystyle \langle M^{2}\rangle }
1164:Often, one omits the reference time,
21:Correlation function (disambiguation)
7:
2977:{\displaystyle \eta ={\frac {1}{4}}}
1432:equal-position correlation functions
1360:One might also be interested in the
3146:correlation is responsible for the
2306:{\displaystyle \xi (T_{C})=\infty }
2271:, the correlation length diverges,
3619:Journal of Applied Crystallography
3186:Higher order correlation functions
2300:
1981:Equal-time correlation functions,
14:
3176:Quantum inverse scattering method
2600:
2580:
2548:
2531:
1802:
1798:
1775:
1771:
1742:
1738:
1718:
1714:
1640:
1636:
1607:
1603:
1562:
1558:
1532:
1528:
1319:
1315:
1292:
1288:
1259:
1255:
1235:
1231:
1118:
1114:
1085:
1081:
1040:
1036:
1010:
1006:
710:
706:
680:
676:
618:
614:
585:
581:
433:
429:
400:
396:
349:
345:
319:
315:
97:{\displaystyle T_{\text{Curie}}}
50:{\displaystyle T_{\text{Curie}}}
3753:. Oxford Science Publications.
3519:The Two-dimensional Ising Model
3119:with another critical exponent
1967:Fluctuation-dissipation theorem
1910:Measuring correlation functions
936:equal-time correlation function
3517:B.M. McCoy and T.T. Wu (1973)
3394:
3381:
3358:
3345:
3325:
3312:
3286:
3241:
3089:
3067:
2883:
2865:
2849:
2843:
2678:
2672:
2616:
2604:
2590:
2584:
2564:
2552:
2541:
2535:
2521:
2515:
2373:
2355:
2341:
2323:
2294:
2281:
2189:
2183:
2145:
2127:
2104:
2086:
2009:
1991:
1880:
1868:
1814:
1808:
1787:
1781:
1754:
1748:
1730:
1724:
1703:
1697:
1664:
1646:
1625:
1613:
1586:
1568:
1550:
1538:
1517:
1505:
1456:
1444:
1331:
1325:
1304:
1298:
1271:
1265:
1247:
1241:
1220:
1214:
1142:
1124:
1103:
1091:
1064:
1046:
1028:
1016:
995:
983:
960:
948:
740:
716:
698:
686:
648:
624:
603:
591:
463:
439:
418:
406:
379:
355:
337:
325:
304:
292:
1:
3158:Radial distribution functions
1939:Onsager regression hypothesis
664:from the correlated product,
3744:Radial distribution function
3164:radial distribution function
3036:{\displaystyle \vartheta =2}
2110:{\displaystyle C(r,\tau =0)}
2015:{\displaystyle C(r,\tau =0)}
1917:Radial distribution function
1349:Radial distribution function
812:Radial distribution function
3729:. Oxford University Press.
3457:. Oxford University Press.
820:cross-correlation functions
16:Measure of a system's order
3856:
3830:Covariance and correlation
3596:10.1103/PhysRevB.82.104207
2944:{\displaystyle \vartheta }
2762:{\displaystyle \vartheta }
1886:{\displaystyle C(r,\tau )}
1462:{\displaystyle C(0,\tau )}
822:, as the random variables
18:
3793:10.1103/RevModPhys.46.597
3773:Reviews of Modern Physics
3641:10.1107/S1600576714012424
3449:Sethna, James P. (2006).
3427:Ornstein–Zernike equation
816:autocorrelation functions
3523:Harvard University Press
2984:, but above criticality
1184:, and reference radius,
3749:Yeomans, J. M. (1992).
3689:10.1073/pnas.0905337106
3537:, Springer (Heidelberg)
3166:which is seen often in
1423:{\displaystyle t+\tau }
927:{\displaystyle \tau =0}
275:{\displaystyle t+\tau }
3504:10.1103/PhysRev.37.405
3478:Onsager, Lars (1931).
3407:
3133:
3110:
3037:
3017:and below criticality
3011:
2978:
2945:
2921:
2898:
2824:
2791:
2763:
2740:
2653:
2630:
2481:
2463:
2456:
2436:
2409:
2382:
2307:
2265:
2238:
2218:
2198:
2111:
2070:
2043:
2016:
1956:
1887:
1828:
1678:
1489:
1463:
1424:
1398:
1378:
1341:
1198:
1178:
1156:
967:
966:{\displaystyle C(r,0)}
928:
902:
870:
843:
804:
777:
750:
658:
563:
536:
505:
477:
276:
250:
230:
204:
184:
157:
105:
98:
71:
51:
3835:Statistical mechanics
3408:
3168:statistical mechanics
3134:
3111:
3038:
3012:
2979:
2946:
2922:
2920:{\displaystyle \eta }
2899:
2825:
2792:
2790:{\displaystyle T_{c}}
2764:
2741:
2654:
2631:
2482:
2457:
2437:
2435:{\displaystyle T_{C}}
2410:
2408:{\displaystyle T_{C}}
2383:
2308:
2266:
2264:{\displaystyle T_{C}}
2239:
2219:
2199:
2112:
2071:
2069:{\displaystyle T_{C}}
2044:
2042:{\displaystyle T_{C}}
2017:
1980:
1957:
1955:{\displaystyle \tau }
1888:
1829:
1679:
1490:
1464:
1425:
1399:
1379:
1342:
1199:
1179:
1157:
968:
929:
903:
901:{\displaystyle \tau }
871:
869:{\displaystyle s_{2}}
844:
842:{\displaystyle s_{1}}
805:
803:{\displaystyle s_{2}}
778:
776:{\displaystyle s_{1}}
751:
659:
564:
562:{\displaystyle s_{2}}
537:
535:{\displaystyle s_{1}}
506:
478:
277:
251:
231:
205:
185:
183:{\displaystyle s_{2}}
158:
156:{\displaystyle s_{1}}
110:statistical mechanics
99:
72:
52:
29:
3198:
3132:{\displaystyle \nu }
3123:
3057:
3021:
2988:
2955:
2935:
2911:
2837:
2801:
2774:
2753:
2666:
2652:{\displaystyle \xi }
2643:
2509:
2480:{\displaystyle \xi }
2471:
2455:{\displaystyle \xi }
2446:
2419:
2392:
2317:
2275:
2248:
2237:{\displaystyle \xi }
2228:
2217:{\displaystyle \xi }
2208:
2121:
2080:
2053:
2026:
1985:
1946:
1862:
1691:
1499:
1473:
1438:
1408:
1388:
1368:
1208:
1188:
1168:
977:
973:. It is written as:
942:
912:
892:
853:
826:
787:
760:
668:
573:
546:
519:
489:
286:
260:
240:
214:
194:
167:
140:
118:correlation function
114:correlation function
81:
70:{\displaystyle \xi }
61:
34:
19:For other uses, see
3785:1974RvMP...46..597F
3680:2009PNAS..10611511W
3588:2010PhRvB..82j4207A
3495:1931PhRv...37..405O
1488:{\displaystyle r=0}
934:. This defines the
485:Here the brackets,
229:{\displaystyle R+r}
3489:(405): 2265–2279.
3403:
3129:
3106:
3033:
3007:
2974:
2941:
2917:
2894:
2820:
2787:
2759:
2736:
2649:
2626:
2477:
2464:
2452:
2432:
2405:
2378:
2303:
2261:
2234:
2214:
2194:
2107:
2066:
2039:
2012:
1952:
1921:neutron scattering
1903:2018-12-25 at the
1883:
1824:
1674:
1485:
1459:
1420:
1394:
1374:
1337:
1194:
1174:
1152:
963:
924:
898:
866:
839:
800:
773:
746:
654:
559:
532:
501:
473:
272:
246:
226:
200:
180:
153:
134:canonical ensemble
106:
94:
67:
47:
3840:Conceptual models
3760:978-0-19-851730-6
3566:Physical Review B
3533:M. Henkel (1999)
3005:
2972:
2929:critical exponent
2888:
2724:
2699:
2572:
1762:
1594:
1397:{\displaystyle t}
1377:{\displaystyle R}
1279:
1197:{\displaystyle R}
1177:{\displaystyle t}
1072:
387:
249:{\displaystyle t}
203:{\displaystyle R}
91:
44:
3847:
3796:
3764:
3740:
3712:
3711:
3701:
3691:
3659:
3653:
3652:
3634:
3614:
3608:
3607:
3581:
3561:
3555:
3544:
3538:
3531:
3525:
3515:
3509:
3508:
3506:
3475:
3469:
3468:
3446:
3412:
3410:
3409:
3404:
3393:
3392:
3380:
3379:
3378:
3377:
3357:
3356:
3344:
3343:
3342:
3341:
3324:
3323:
3311:
3310:
3309:
3308:
3285:
3284:
3266:
3265:
3253:
3252:
3240:
3239:
3238:
3237:
3225:
3224:
3215:
3214:
3138:
3136:
3135:
3130:
3115:
3113:
3112:
3107:
3101:
3100:
3092:
3086:
3085:
3070:
3048:phase transition
3042:
3040:
3039:
3034:
3016:
3014:
3013:
3008:
3006:
2998:
2983:
2981:
2980:
2975:
2973:
2965:
2950:
2948:
2947:
2942:
2926:
2924:
2923:
2918:
2903:
2901:
2900:
2895:
2889:
2887:
2886:
2856:
2829:
2827:
2826:
2821:
2816:
2815:
2796:
2794:
2793:
2788:
2786:
2785:
2768:
2766:
2765:
2760:
2745:
2743:
2742:
2737:
2731:
2730:
2726:
2725:
2717:
2700:
2698:
2697:
2685:
2658:
2656:
2655:
2650:
2635:
2633:
2632:
2627:
2603:
2583:
2570:
2551:
2534:
2486:
2484:
2483:
2478:
2461:
2459:
2458:
2453:
2441:
2439:
2438:
2433:
2431:
2430:
2414:
2412:
2411:
2406:
2404:
2403:
2387:
2385:
2384:
2379:
2377:
2376:
2312:
2310:
2309:
2304:
2293:
2292:
2270:
2268:
2267:
2262:
2260:
2259:
2243:
2241:
2240:
2235:
2223:
2221:
2220:
2215:
2203:
2201:
2200:
2195:
2193:
2192:
2179:
2163:
2162:
2116:
2114:
2113:
2108:
2075:
2073:
2072:
2067:
2065:
2064:
2048:
2046:
2045:
2040:
2038:
2037:
2021:
2019:
2018:
2013:
1961:
1959:
1958:
1953:
1892:
1890:
1889:
1884:
1833:
1831:
1830:
1825:
1807:
1806:
1805:
1780:
1779:
1778:
1760:
1747:
1746:
1745:
1723:
1722:
1721:
1683:
1681:
1680:
1675:
1645:
1644:
1643:
1612:
1611:
1610:
1592:
1567:
1566:
1565:
1537:
1536:
1535:
1494:
1492:
1491:
1486:
1468:
1466:
1465:
1460:
1429:
1427:
1426:
1421:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1346:
1344:
1343:
1338:
1324:
1323:
1322:
1297:
1296:
1295:
1277:
1264:
1263:
1262:
1240:
1239:
1238:
1203:
1201:
1200:
1195:
1183:
1181:
1180:
1175:
1161:
1159:
1158:
1153:
1123:
1122:
1121:
1090:
1089:
1088:
1070:
1045:
1044:
1043:
1015:
1014:
1013:
972:
970:
969:
964:
933:
931:
930:
925:
907:
905:
904:
899:
875:
873:
872:
867:
865:
864:
848:
846:
845:
840:
838:
837:
809:
807:
806:
801:
799:
798:
782:
780:
779:
774:
772:
771:
755:
753:
752:
747:
715:
714:
713:
685:
684:
683:
663:
661:
660:
655:
623:
622:
621:
590:
589:
588:
568:
566:
565:
560:
558:
557:
541:
539:
538:
533:
531:
530:
510:
508:
507:
502:
482:
480:
479:
474:
438:
437:
436:
405:
404:
403:
385:
354:
353:
352:
324:
323:
322:
281:
279:
278:
273:
255:
253:
252:
247:
235:
233:
232:
227:
209:
207:
206:
201:
189:
187:
186:
181:
179:
178:
162:
160:
159:
154:
152:
151:
103:
101:
100:
95:
93:
92:
89:
76:
74:
73:
68:
56:
54:
53:
48:
46:
45:
42:
3855:
3854:
3850:
3849:
3848:
3846:
3845:
3844:
3820:
3819:
3767:
3761:
3748:
3737:
3724:
3721:
3719:Further reading
3716:
3715:
3674:(28): 11511–4.
3661:
3660:
3656:
3616:
3615:
3611:
3563:
3562:
3558:
3545:
3541:
3532:
3528:
3516:
3512:
3483:Physical Review
3477:
3476:
3472:
3465:
3448:
3447:
3440:
3435:
3423:
3384:
3369:
3364:
3348:
3333:
3328:
3315:
3300:
3295:
3276:
3257:
3244:
3229:
3216:
3206:
3201:
3196:
3195:
3188:
3172:fluid mechanics
3160:
3121:
3120:
3087:
3077:
3055:
3054:
3019:
3018:
2986:
2985:
2953:
2952:
2933:
2932:
2909:
2908:
2860:
2835:
2834:
2807:
2799:
2798:
2777:
2772:
2771:
2751:
2750:
2712:
2708:
2689:
2664:
2663:
2641:
2640:
2507:
2506:
2498:
2493:
2469:
2468:
2444:
2443:
2422:
2417:
2416:
2395:
2390:
2389:
2347:
2315:
2314:
2284:
2273:
2272:
2251:
2246:
2245:
2226:
2225:
2206:
2205:
2164:
2151:
2119:
2118:
2078:
2077:
2056:
2051:
2050:
2029:
2024:
2023:
1983:
1982:
1975:
1944:
1943:
1930:
1912:
1905:Wayback Machine
1860:
1859:
1855:
1843:microscopically
1797:
1770:
1737:
1713:
1689:
1688:
1635:
1602:
1557:
1527:
1497:
1496:
1471:
1470:
1436:
1435:
1406:
1405:
1386:
1385:
1366:
1365:
1358:
1314:
1287:
1254:
1230:
1206:
1205:
1186:
1185:
1166:
1165:
1113:
1080:
1035:
1005:
975:
974:
940:
939:
910:
909:
890:
889:
882:
856:
851:
850:
829:
824:
823:
790:
785:
784:
763:
758:
757:
705:
675:
666:
665:
613:
580:
571:
570:
549:
544:
543:
522:
517:
516:
487:
486:
428:
395:
344:
314:
284:
283:
258:
257:
238:
237:
212:
211:
192:
191:
190:, at positions
170:
165:
164:
143:
138:
137:
130:
84:
79:
78:
59:
58:
37:
32:
31:
24:
17:
12:
11:
5:
3853:
3851:
3843:
3842:
3837:
3832:
3822:
3821:
3818:
3817:
3797:
3779:(4): 597–616.
3765:
3759:
3746:
3741:
3736:978-0198566779
3735:
3720:
3717:
3714:
3713:
3654:
3609:
3572:(10): 104207.
3556:
3539:
3526:
3510:
3470:
3464:978-0198566779
3463:
3437:
3436:
3434:
3431:
3430:
3429:
3422:
3419:
3414:
3413:
3402:
3399:
3396:
3391:
3387:
3383:
3376:
3372:
3367:
3363:
3360:
3355:
3351:
3347:
3340:
3336:
3331:
3327:
3322:
3318:
3314:
3307:
3303:
3298:
3294:
3291:
3288:
3283:
3279:
3275:
3272:
3269:
3264:
3260:
3256:
3251:
3247:
3243:
3236:
3232:
3228:
3223:
3219:
3213:
3209:
3204:
3187:
3184:
3159:
3156:
3128:
3117:
3116:
3105:
3099:
3096:
3091:
3084:
3080:
3076:
3073:
3069:
3065:
3062:
3032:
3029:
3026:
3004:
3001:
2996:
2993:
2971:
2968:
2963:
2960:
2940:
2916:
2905:
2904:
2893:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2863:
2859:
2854:
2851:
2848:
2845:
2842:
2819:
2814:
2810:
2806:
2784:
2780:
2758:
2747:
2746:
2735:
2729:
2723:
2720:
2715:
2711:
2706:
2703:
2696:
2692:
2688:
2683:
2680:
2677:
2674:
2671:
2648:
2625:
2621:
2618:
2615:
2612:
2609:
2606:
2602:
2598:
2595:
2592:
2589:
2586:
2582:
2578:
2575:
2569:
2566:
2563:
2560:
2557:
2554:
2550:
2546:
2543:
2540:
2537:
2533:
2529:
2526:
2523:
2520:
2517:
2514:
2497:
2494:
2492:
2489:
2476:
2451:
2429:
2425:
2402:
2398:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2350:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2302:
2299:
2296:
2291:
2287:
2283:
2280:
2258:
2254:
2233:
2213:
2191:
2188:
2185:
2182:
2178:
2174:
2171:
2167:
2161:
2158:
2154:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2063:
2059:
2036:
2032:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1974:
1971:
1951:
1929:
1926:
1911:
1908:
1882:
1879:
1876:
1873:
1870:
1867:
1854:
1851:
1823:
1819:
1816:
1813:
1810:
1804:
1800:
1795:
1792:
1789:
1786:
1783:
1777:
1773:
1768:
1765:
1759:
1756:
1753:
1750:
1744:
1740:
1735:
1732:
1729:
1726:
1720:
1716:
1711:
1708:
1705:
1702:
1699:
1696:
1673:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1642:
1638:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1609:
1605:
1600:
1597:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1564:
1560:
1555:
1552:
1549:
1546:
1543:
1540:
1534:
1530:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1484:
1481:
1478:
1458:
1455:
1452:
1449:
1446:
1443:
1419:
1416:
1413:
1393:
1373:
1357:
1354:
1336:
1333:
1330:
1327:
1321:
1317:
1312:
1309:
1306:
1303:
1300:
1294:
1290:
1285:
1282:
1276:
1273:
1270:
1267:
1261:
1257:
1252:
1249:
1246:
1243:
1237:
1233:
1228:
1225:
1222:
1219:
1216:
1213:
1193:
1173:
1151:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1120:
1116:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1087:
1083:
1078:
1075:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1042:
1038:
1033:
1030:
1027:
1024:
1021:
1018:
1012:
1008:
1003:
1000:
997:
994:
991:
988:
985:
982:
962:
959:
956:
953:
950:
947:
923:
920:
917:
897:
881:
878:
863:
859:
836:
832:
797:
793:
770:
766:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
712:
708:
703:
700:
697:
694:
691:
688:
682:
678:
673:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
620:
616:
611:
608:
605:
602:
599:
596:
593:
587:
583:
578:
556:
552:
529:
525:
513:expected value
500:
497:
494:
472:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
435:
431:
426:
423:
420:
417:
414:
411:
408:
402:
398:
393:
390:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
351:
347:
342:
339:
336:
333:
330:
327:
321:
317:
312:
309:
306:
303:
300:
297:
294:
291:
271:
268:
265:
245:
225:
222:
219:
199:
177:
173:
150:
146:
129:
126:
87:
66:
40:
15:
13:
10:
9:
6:
4:
3:
2:
3852:
3841:
3838:
3836:
3833:
3831:
3828:
3827:
3825:
3815:
3814:
3809:
3808:J.L. Lebowitz
3805:
3801:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3770:
3769:Fisher, M. E.
3766:
3762:
3756:
3752:
3747:
3745:
3742:
3738:
3732:
3728:
3723:
3722:
3718:
3709:
3705:
3700:
3695:
3690:
3685:
3681:
3677:
3673:
3669:
3665:
3658:
3655:
3650:
3646:
3642:
3638:
3633:
3628:
3624:
3620:
3613:
3610:
3605:
3601:
3597:
3593:
3589:
3585:
3580:
3575:
3571:
3567:
3560:
3557:
3553:
3549:
3543:
3540:
3536:
3530:
3527:
3524:
3520:
3514:
3511:
3505:
3500:
3496:
3492:
3488:
3484:
3481:
3474:
3471:
3466:
3460:
3456:
3452:
3445:
3443:
3439:
3432:
3428:
3425:
3424:
3420:
3418:
3400:
3389:
3385:
3374:
3370:
3365:
3361:
3353:
3349:
3338:
3334:
3329:
3320:
3316:
3305:
3301:
3296:
3289:
3281:
3277:
3273:
3270:
3267:
3262:
3258:
3254:
3249:
3245:
3234:
3230:
3226:
3221:
3217:
3211:
3207:
3202:
3194:
3193:
3192:
3185:
3183:
3181:
3177:
3173:
3169:
3165:
3157:
3155:
3153:
3149:
3145:
3140:
3126:
3103:
3097:
3094:
3082:
3078:
3074:
3071:
3063:
3060:
3053:
3052:
3051:
3049:
3044:
3030:
3027:
3024:
3002:
2999:
2994:
2991:
2969:
2966:
2961:
2958:
2938:
2930:
2914:
2891:
2880:
2877:
2874:
2871:
2868:
2861:
2857:
2852:
2846:
2840:
2833:
2832:
2831:
2812:
2808:
2782:
2778:
2756:
2733:
2727:
2721:
2718:
2713:
2709:
2704:
2701:
2694:
2690:
2686:
2681:
2675:
2669:
2662:
2661:
2660:
2646:
2637:
2623:
2613:
2610:
2607:
2587:
2573:
2561:
2558:
2555:
2544:
2538:
2524:
2518:
2512:
2503:
2495:
2490:
2488:
2474:
2449:
2427:
2423:
2400:
2396:
2370:
2367:
2364:
2361:
2358:
2352:
2348:
2344:
2338:
2335:
2332:
2329:
2326:
2320:
2297:
2289:
2285:
2278:
2256:
2252:
2231:
2211:
2186:
2180:
2176:
2172:
2169:
2165:
2159:
2156:
2152:
2148:
2142:
2139:
2136:
2133:
2130:
2124:
2101:
2098:
2095:
2092:
2089:
2083:
2061:
2057:
2034:
2030:
2006:
2003:
2000:
1997:
1994:
1988:
1979:
1972:
1970:
1968:
1963:
1949:
1941:
1940:
1935:
1927:
1925:
1922:
1918:
1909:
1907:
1906:
1902:
1899:
1894:
1877:
1874:
1871:
1865:
1852:
1850:
1848:
1844:
1840:
1834:
1821:
1811:
1784:
1763:
1751:
1733:
1727:
1706:
1700:
1694:
1684:
1671:
1661:
1658:
1655:
1652:
1649:
1622:
1619:
1616:
1595:
1583:
1580:
1577:
1574:
1571:
1553:
1547:
1544:
1541:
1520:
1514:
1511:
1508:
1502:
1482:
1479:
1476:
1453:
1450:
1447:
1441:
1433:
1417:
1414:
1411:
1391:
1371:
1363:
1355:
1353:
1350:
1328:
1301:
1280:
1268:
1250:
1244:
1223:
1217:
1211:
1191:
1171:
1162:
1149:
1139:
1136:
1133:
1130:
1127:
1100:
1097:
1094:
1073:
1061:
1058:
1055:
1052:
1049:
1031:
1025:
1022:
1019:
998:
992:
989:
986:
980:
957:
954:
951:
945:
937:
921:
918:
915:
895:
887:
879:
877:
861:
857:
834:
830:
821:
817:
813:
795:
791:
768:
764:
737:
734:
731:
728:
725:
722:
719:
701:
695:
692:
689:
645:
642:
639:
636:
633:
630:
627:
600:
597:
594:
554:
550:
527:
523:
514:
495:
483:
470:
460:
457:
454:
451:
448:
445:
442:
415:
412:
409:
388:
376:
373:
370:
367:
364:
361:
358:
340:
334:
331:
328:
307:
301:
298:
295:
289:
269:
266:
263:
243:
223:
220:
217:
197:
175:
171:
148:
144:
135:
127:
125:
121:
119:
115:
111:
85:
64:
38:
28:
22:
3811:
3776:
3772:
3750:
3726:
3671:
3667:
3657:
3622:
3618:
3612:
3569:
3565:
3559:
3542:
3534:
3529:
3518:
3513:
3486:
3482:
3473:
3454:
3415:
3189:
3180:Bethe ansatz
3161:
3141:
3118:
3045:
2906:
2748:
2638:
2499:
2491:Applications
2465:
1964:
1937:
1934:Lars Onsager
1931:
1913:
1895:
1856:
1842:
1838:
1835:
1685:
1495:, yielding:
1431:
1361:
1359:
1163:
935:
885:
883:
484:
131:
122:
113:
107:
3625:(4): 1315.
1839:macroscopic
128:Definitions
3824:Categories
3804:M.S. Green
3433:References
236:and times
3810:editors,
3632:1402.1432
3604:119243898
3579:1006.5382
3552:arxiv.org
3398:⟩
3362:⋯
3293:⟨
3271:⋯
3227:⋯
3152:universal
3144:power law
3127:ν
3098:ν
3095:−
3075:−
3064:∝
3061:ξ
3025:ϑ
2992:ϑ
2959:η
2939:ϑ
2915:η
2881:η
2872:−
2853:≈
2818:⟩
2805:⟨
2757:ϑ
2714:−
2705:
2695:ϑ
2682:≈
2647:ξ
2620:⟩
2597:⟨
2594:⟩
2577:⟨
2574:−
2568:⟩
2545:⋅
2528:⟨
2496:Magnetism
2475:ξ
2450:ξ
2371:η
2362:−
2353:−
2345:∝
2333:τ
2301:∞
2279:ξ
2232:ξ
2212:ξ
2181:ξ
2170:−
2160:ϑ
2157:−
2149:∝
2137:τ
2096:τ
2001:τ
1950:τ
1932:In 1931,
1878:τ
1818:⟩
1812:τ
1794:⟨
1791:⟩
1767:⟨
1764:−
1758:⟩
1752:τ
1734:⋅
1710:⟨
1701:τ
1668:⟩
1662:τ
1632:⟨
1629:⟩
1599:⟨
1596:−
1590:⟩
1584:τ
1554:⋅
1524:⟨
1515:τ
1454:τ
1418:τ
1335:⟩
1311:⟨
1308:⟩
1284:⟨
1281:−
1275:⟩
1251:⋅
1227:⟨
1146:⟩
1110:⟨
1107:⟩
1077:⟨
1074:−
1068:⟩
1032:⋅
1002:⟨
916:τ
896:τ
744:⟩
738:τ
702:⋅
672:⟨
652:⟩
646:τ
610:⟨
607:⟩
577:⟨
499:⟩
496:⋅
493:⟨
467:⟩
461:τ
425:⟨
422:⟩
392:⟨
389:−
383:⟩
377:τ
341:⋅
311:⟨
302:τ
270:τ
65:ξ
3708:20716512
3649:97097937
3421:See also
2049:. Above
1901:Archived
1362:temporal
3800:C. Domb
3781:Bibcode
3699:2703671
3676:Bibcode
3584:Bibcode
3491:Bibcode
3148:scaling
886:spatial
3757:
3733:
3706:
3696:
3647:
3602:
3461:
2907:where
2571:
1761:
1593:
1278:
1071:
386:
112:, the
3645:S2CID
3627:arXiv
3600:S2CID
3574:arXiv
3550:from
3142:This
2927:is a
2500:In a
2244:. At
90:Curie
43:Curie
3755:ISBN
3731:ISBN
3704:PMID
3459:ISBN
3178:and
3170:and
2502:spin
1384:and
849:and
783:and
542:and
256:and
210:and
163:and
3789:doi
3694:PMC
3684:doi
3672:106
3637:doi
3592:doi
3499:doi
2702:exp
108:In
3826::
3806:,
3802:,
3787:.
3777:46
3775:.
3702:.
3692:.
3682:.
3670:.
3666:.
3643:.
3635:.
3623:47
3621:.
3598:.
3590:.
3582:.
3570:82
3568:.
3521:,
3497:.
3487:38
3485:.
3453:.
3441:^
3139:.
3043:.
2388:.
2076:,
1969:.
1849:.
1434:,
938:,
569:,
282::
3795:.
3791::
3783::
3763:.
3739:.
3710:.
3686::
3678::
3651:.
3639::
3629::
3606:.
3594::
3586::
3576::
3554:.
3507:.
3501::
3493::
3467:.
3401:.
3395:)
3390:n
3386:s
3382:(
3375:n
3371:i
3366:X
3359:)
3354:2
3350:s
3346:(
3339:2
3335:i
3330:X
3326:)
3321:1
3317:s
3313:(
3306:1
3302:i
3297:X
3290:=
3287:)
3282:n
3278:s
3274:,
3268:,
3263:2
3259:s
3255:,
3250:1
3246:s
3242:(
3235:n
3231:i
3222:2
3218:i
3212:1
3208:i
3203:C
3104:,
3090:|
3083:c
3079:T
3072:T
3068:|
3031:2
3028:=
3003:2
3000:1
2995:=
2970:4
2967:1
2962:=
2892:,
2884:)
2878:+
2875:2
2869:d
2866:(
2862:r
2858:1
2850:)
2847:r
2844:(
2841:C
2813:2
2809:M
2783:c
2779:T
2734:,
2728:)
2722:d
2719:r
2710:(
2691:r
2687:1
2679:)
2676:r
2673:(
2670:C
2624:.
2617:)
2614:r
2611:+
2608:R
2605:(
2601:s
2591:)
2588:R
2585:(
2581:s
2565:)
2562:r
2559:+
2556:R
2553:(
2549:s
2542:)
2539:R
2536:(
2532:s
2525:=
2522:)
2519:r
2516:(
2513:C
2428:C
2424:T
2401:C
2397:T
2374:)
2368:+
2365:2
2359:d
2356:(
2349:r
2342:)
2339:0
2336:=
2330:,
2327:r
2324:(
2321:C
2298:=
2295:)
2290:C
2286:T
2282:(
2257:C
2253:T
2190:)
2187:T
2184:(
2177:/
2173:r
2166:e
2153:r
2146:)
2143:0
2140:=
2134:,
2131:r
2128:(
2125:C
2105:)
2102:0
2099:=
2093:,
2090:r
2087:(
2084:C
2062:C
2058:T
2035:C
2031:T
2010:)
2007:0
2004:=
1998:,
1995:r
1992:(
1989:C
1881:)
1875:,
1872:r
1869:(
1866:C
1822:.
1815:)
1809:(
1803:2
1799:s
1788:)
1785:0
1782:(
1776:1
1772:s
1755:)
1749:(
1743:2
1739:s
1731:)
1728:0
1725:(
1719:1
1715:s
1707:=
1704:)
1698:(
1695:C
1672:.
1665:)
1659:+
1656:t
1653:,
1650:R
1647:(
1641:2
1637:s
1626:)
1623:t
1620:,
1617:R
1614:(
1608:1
1604:s
1587:)
1581:+
1578:t
1575:,
1572:R
1569:(
1563:2
1559:s
1551:)
1548:t
1545:,
1542:R
1539:(
1533:1
1529:s
1521:=
1518:)
1512:,
1509:0
1506:(
1503:C
1483:0
1480:=
1477:r
1457:)
1451:,
1448:0
1445:(
1442:C
1415:+
1412:t
1392:t
1372:R
1332:)
1329:r
1326:(
1320:2
1316:s
1305:)
1302:0
1299:(
1293:1
1289:s
1272:)
1269:r
1266:(
1260:2
1256:s
1248:)
1245:0
1242:(
1236:1
1232:s
1224:=
1221:)
1218:r
1215:(
1212:C
1192:R
1172:t
1150:.
1143:)
1140:t
1137:,
1134:r
1131:+
1128:R
1125:(
1119:2
1115:s
1104:)
1101:t
1098:,
1095:R
1092:(
1086:1
1082:s
1065:)
1062:t
1059:,
1056:r
1053:+
1050:R
1047:(
1041:2
1037:s
1029:)
1026:t
1023:,
1020:R
1017:(
1011:1
1007:s
999:=
996:)
993:0
990:,
987:r
984:(
981:C
961:)
958:0
955:,
952:r
949:(
946:C
922:0
919:=
862:2
858:s
835:1
831:s
796:2
792:s
769:1
765:s
741:)
735:+
732:t
729:,
726:r
723:+
720:R
717:(
711:2
707:s
699:)
696:t
693:,
690:R
687:(
681:1
677:s
649:)
643:+
640:t
637:,
634:r
631:+
628:R
625:(
619:2
615:s
604:)
601:t
598:,
595:R
592:(
586:1
582:s
555:2
551:s
528:1
524:s
471:.
464:)
458:+
455:t
452:,
449:r
446:+
443:R
440:(
434:2
430:s
419:)
416:t
413:,
410:R
407:(
401:1
397:s
380:)
374:+
371:t
368:,
365:r
362:+
359:R
356:(
350:2
346:s
338:)
335:t
332:,
329:R
326:(
320:1
316:s
308:=
305:)
299:,
296:r
293:(
290:C
267:+
264:t
244:t
224:r
221:+
218:R
198:R
176:2
172:s
149:1
145:s
86:T
39:T
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
