1367:
1909:
36:
2291:
980:
1532:
287:, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations. Physically, this means that an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean-field”.
1951:
1362:{\displaystyle {\begin{aligned}F_{0}&=\operatorname {Tr} _{1,2,\ldots ,N}{\mathcal {H}}(\xi _{1},\xi _{2},\ldots ,\xi _{N})P_{0}^{(N)}(\xi _{1},\xi _{2},\ldots ,\xi _{N})\\&+kT\,\operatorname {Tr} _{1,2,\ldots ,N}P_{0}^{(N)}(\xi _{1},\xi _{2},\ldots ,\xi _{N})\log P_{0}^{(N)}(\xi _{1},\xi _{2},\ldots ,\xi _{N}),\end{aligned}}}
1904:{\displaystyle {\begin{aligned}P_{0}^{(N)}(\xi _{1},\xi _{2},\ldots ,\xi _{N})&={\frac {1}{Z_{0}^{(N)}}}e^{-\beta {\mathcal {H}}_{0}(\xi _{1},\xi _{2},\ldots ,\xi _{N})}\\&=\prod _{i=1}^{N}{\frac {1}{Z_{0}}}e^{-\beta h_{i}(\xi _{i})}\ {\stackrel {\mathrm {def} }{=}}\ \prod _{i=1}^{N}P_{0}^{(i)}(\xi _{i}),\end{aligned}}}
3594:
If we expand the right side, we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean
760:
of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimising the right side of the inequality. The minimising reference system is then the "best" approximation to the true system using non-correlated degrees of freedom
325:
Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high
2693:
3770:
2286:{\displaystyle {\begin{aligned}F_{0}&=\sum _{(i,j)\in {\mathcal {P}}}\operatorname {Tr} _{i,j}V_{i,j}(\xi _{i},\xi _{j})P_{0}^{(i)}(\xi _{i})P_{0}^{(j)}(\xi _{j})\\&+kT\sum _{i=1}^{N}\operatorname {Tr} _{i}P_{0}^{(i)}(\xi _{i})\log P_{0}^{(i)}(\xi _{i}).\end{aligned}}}
2932:
2494:
3921:
4225:
3073:
Equating the effective field felt by all spins to a mean spin value relates the variational approach to the suppression of fluctuations. The physical interpretation of the magnetisation function is then a field of mean values for individual spins.
3526:
517:
876:
4014:
2505:
4330:
induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension
3207:
409:
3604:
724:
1456:
4521:
4978:
In mean field theory, the mean field appearing in the single-site problem is a time-independent scalar or vector quantity. However, this isn't always the case: in a variant of mean field theory called
2784:
3003:
1956:
1537:
1521:
985:
952:
280:
of a system difficult. MFT is an approximation method that often makes the original problem to be solvable and open to calculation, and in some cases MFT may give very accurate approximations.
3379:
4876:
3585:
3068:
3249:
5647:
4304:
3778:
Again, the summand can be re-expanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields
2344:
3784:
3281:
4102:
637:
4758:
3330:
2772:
268:
with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain
Gaussian
4645:
4595:
3387:
420:
903:
4824:
4696:
4794:
4546:
is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system and calculate
2332:
754:
4915:
4046:
1939:
598:
547:
4378:
4722:
4094:
4665:
4618:
4568:
4544:
4349:
4324:
4248:
4066:
3104:
972:
571:
774:
5162:
Baccelli, F.; Karpelevich, F. I.; Kelbert, M. Y.; Puhalskii, A. A.; Rybko, A. N.; Suhov, Y. M. (1992). "A mean-field limit for a class of queueing networks".
5678:
2688:{\displaystyle h_{i}^{\text{MF}}(\xi _{i})=\sum _{\{j\mid (i,j)\in {\mathcal {P}}\}}\operatorname {Tr} _{j}V_{i,j}(\xi _{i},\xi _{j})P_{0}^{(j)}(\xi _{j}).}
3929:
974:
over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by
318:
In general, dimensionality plays an active role in determining whether a mean-field approach will work for any particular problem. There is sometimes a
5435:
3112:
2728:. The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian,
360:
3765:{\displaystyle H\approx H^{\text{MF}}\equiv -J\sum _{\langle i,j\rangle }(m_{i}m_{j}+m_{i}\delta s_{j}+m_{j}\delta s_{i})-h\sum _{i}s_{i}.}
183:. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost.
605:
5683:
5673:
645:
1942:
757:
291:
5106:
Boudec, J. Y. L.; McDonald, D.; Mundinger, J. (2007). "A Generic Mean Field
Convergence Result for Systems of Interacting Objects".
1375:
3009:
17:
5627:
4389:
5046:
2927:{\displaystyle F_{V}=F_{0}+\left\langle \left(-J\sum s_{i}s_{j}-h\sum s_{i}\right)-\left(-m\sum s_{i}\right)\right\rangle _{0}.}
100:
72:
5768:
5727:
5428:
5138:
5029:
290:
Quite often, MFT provides a convenient launch point for studying higher-order fluctuations. For example, when computing the
4947:
5637:
2947:
5719:
1461:
79:
5696:
908:
4096:
prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression
3338:
53:
4829:
2720:
The
Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensional
5642:
5617:
5350:
255:
119:
3534:
639:. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as
5421:
3018:
347:
277:
86:
5537:
3595:
value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.
338:
render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.
3215:
157:
5758:
57:
5632:
5210:
2489:{\displaystyle P_{0}^{(i)}(\xi _{i})={\frac {1}{Z_{0}}}e^{-\beta h_{i}^{MF}(\xi _{i})},\quad i=1,2,\ldots ,N,}
68:
3916:{\displaystyle H^{\text{MF}}=-J\sum _{\langle i,j\rangle }{\big (}m^{2}+2m(s_{i}-m){\big )}-h\sum _{i}s_{i}.}
5763:
4962:
4943:
4264:
4220:{\displaystyle H^{\text{MF}}={\frac {Jm^{2}Nz}{2}}-\underbrace {(h+mJz)} _{h^{\text{eff.}}}\sum _{i}s_{i},}
219:
4383:
Substituting this
Hamiltonian into the partition function and solving the effective 1D problem, we obtain
3775:
These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.
326:
dimensionality, when the
Hamiltonian includes long-range forces, or when the particles are extended (e.g.
5567:
4995:
4979:
4973:
3254:
259:
3521:{\displaystyle H=-J\sum _{\langle i,j\rangle }(m_{i}+\delta s_{i})(m_{j}+\delta s_{j})-h\sum _{i}s_{i},}
2941:
the variational free energy yields the best approximation to the actual magnetisation. The minimiser is
2937:
By the
Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function that
2296:
In order to minimise, we take the derivative with respect to the single-degree-of-freedom probabilities
611:
512:{\displaystyle F\leq F_{0}\ {\stackrel {\mathrm {def} }{=}}\ \langle {\mathcal {H}}\rangle _{0}-TS_{0},}
5585:
4730:
3286:
164:
that are free to vary). Such models consider many individual components that interact with each other.
2734:
5520:
4623:
4573:
2938:
351:
5121:
884:
4982:(DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to the
4802:
4674:
2703:
Mean field theory can be applied to a number of physical systems so as to study phenomena such as
5711:
5602:
4763:
2299:
199:
168:
46:
905:
is the set of pairs that interact, the minimising procedure can be carried out formally. Define
768:
For the most common case that the target
Hamiltonian contains only pairwise interactions, i.e.,
5116:
5000:
284:
215:
93:
732:
4900:
2725:
601:
299:
231:
187:
5555:
5689:
5514:
5273:
5171:
4954:
4019:
1917:
576:
525:
335:
4354:
8:
5703:
5468:
4701:
4251:
4071:
2335:
871:{\displaystyle {\mathcal {H}}=\sum _{(i,j)\in {\mathcal {P}}}V_{i,j}(\xi _{i},\xi _{j}),}
303:
211:
5277:
5175:
3598:
The mean field approximation consists of neglecting this second-order fluctuation term:
2338:
to ensure proper normalization. The end result is the set of self-consistency equations
1458:
is the probability to find the reference system in the state specified by the variables
5735:
5579:
5399:
5379:
5327:
5289:
5263:
5233:
5206:
5187:
5144:
5076:
4939:
4889:
Similarly, MFT can be applied to other types of
Hamiltonian as in the following cases:
4650:
4603:
4553:
4529:
4334:
4309:
4233:
4051:
3089:
957:
556:
331:
319:
137:
5107:
3335:
Let us transform our spin variable by introducing the fluctuation from its mean value
156:) models by studying a simpler model that approximates the original by averaging over
5622:
5493:
5473:
5403:
5394:
5346:
5191:
5134:
5109:
Fourth
International Conference on the Quantitative Evaluation of Systems (QEST 2007)
5081:
5025:
4897:
transition. In this case, the analog of the magnetization is the superconducting gap
4547:
4009:{\displaystyle \sum _{\langle i,j\rangle }={\frac {1}{2}}\sum _{i}\sum _{j\in nn(i)}}
2704:
243:
176:
5148:
5483:
5389:
5323:
5293:
5281:
5237:
5225:
5179:
5126:
5071:
5061:
1524:
265:
191:
186:
MFT has since been applied to a wide range of fields outside of physics, including
180:
5311:
307:
207:
5463:
4921:
4894:
203:
5367:
5285:
5229:
5752:
5573:
4983:
2721:
295:
251:
247:
171:
to any one body with an average or effective interaction, sometimes called a
2724:. A magnetisation function can be calculated from the resultant approximate
5652:
5607:
5547:
5448:
5307:
5251:
5085:
269:
239:
235:
195:
5561:
5341:
Stanley, H. E. (1971). "Mean Field Theory of
Magnetic Phase Transitions".
3202:{\displaystyle H=-J\sum _{\langle i,j\rangle }s_{i}s_{j}-h\sum _{i}s_{i},}
276:). Often combinatorial problems arise that make things like computing the
5444:
5413:
4961:
Variationally minimisation like mean field theory can be also be used in
3083:
273:
5130:
4881:
This shows that MFT can account for the ferromagnetic phase transition.
5657:
5254:(2009). "More is the Same; Phase Transitions and Mean Field Theories".
5183:
4935:
4932:
4667:
as a function of temperature. This leads to the following observation:
404:{\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}+\Delta {\mathcal {H}}}
153:
5066:
4925:
161:
35:
5488:
5384:
5161:
246:. MFT has been used in the Bragg–Williams approximation, models on
5312:"L'hypothèse du champ moléculaire et la propriété ferromagnétique"
5268:
719:{\displaystyle {\mathcal {H}}_{0}=\sum _{i=1}^{N}h_{i}(\xi _{i}),}
550:
327:
133:
5612:
5498:
5648:
Maria Skłodowska-Curie National Research Institute of Oncology
1451:{\displaystyle P_{0}^{(N)}(\xi _{1},\xi _{2},\dots ,\xi _{N})}
5024:(4th print ed.). Cambridge: Cambridge University Press.
4516:{\displaystyle Z=e^{-{\frac {\beta Jm^{2}Nz}{2}}}\left^{N},}
4967:
3926:
The summation over neighboring spins can be rewritten as
5368:"Magnetisation and Mean Field Theory in the Ising Model"
5343:
Introduction to Phase Transitions and Critical Phenomena
5105:
3251:
indicates summation over the pair of nearest neighbors
3077:
4903:
4832:
4805:
4766:
4733:
4704:
4677:
4653:
4626:
4606:
4576:
4556:
4532:
4392:
4357:
4337:
4312:
4267:
4236:
4105:
4074:
4054:
4022:
3932:
3787:
3607:
3537:
3390:
3341:
3289:
3257:
3218:
3115:
3092:
3021:
2950:
2787:
2737:
2508:
2347:
2302:
1954:
1920:
1535:
1464:
1378:
983:
960:
911:
887:
777:
735:
648:
614:
579:
559:
528:
423:
363:
322:
above which MFT is valid and below which it is not.
5679:
Maria Skłodowska-Curie Monument in Warsaw (Downtown)
2998:{\displaystyle m=J\sum \langle s_{j}\rangle _{0}+h,}
160:(the number of values in the final calculation of a
1516:{\displaystyle (\xi _{1},\xi _{2},\dots ,\xi _{N})}
60:. Unsourced material may be challenged and removed.
5684:Maria Skłodowska-Curie Monument in Warsaw (Ochota)
4909:
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4008:
3915:
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3243:
3201:
3106:-dimensional lattice. The Hamiltonian is given by
3098:
3062:
2997:
2926:
2766:
2687:
2488:
2326:
2285:
1933:
1903:
1515:
1450:
1361:
966:
947:{\displaystyle \operatorname {Tr} _{i}f(\xi _{i})}
946:
897:
870:
748:
718:
631:
592:
565:
541:
511:
403:
5045:Parr, Thomas; Sajid, Noor; Friston, Karl (2020).
4550:. In particular, we can obtain the magnetization
3374:{\displaystyle m_{i}\equiv \langle s_{i}\rangle }
152:studies the behavior of high-dimensional random (
5750:
5044:
4871:{\displaystyle T_{\text{c}}={\frac {Jz}{k_{B}}}}
4254:. At this point, the Ising Hamiltonian has been
341:
5019:
4884:
4986:to study the metal–Mott-insulator transition.
4671:For temperatures greater than a certain value
3580:{\displaystyle \delta s_{i}\equiv s_{i}-m_{i}}
1523:. This probability is given by the normalized
346:The formal basis for mean-field theory is the
5429:
5365:
3879:
3831:
3063:{\displaystyle m={\text{tanh}}(zJ\beta m)+h.}
4258:into a sum of one-body Hamiltonians with an
3950:
3938:
3824:
3812:
3650:
3638:
3420:
3408:
3368:
3355:
3270:
3258:
3236:
3224:
3145:
3133:
2977:
2963:
2582:
2548:
604:. The average is taken over the equilibrium
481:
470:
3244:{\displaystyle \sum _{\langle i,j\rangle }}
310:that correct the mean-field approximation.
5443:
5436:
5422:
5204:
5674:Maria Skłodowska-Curie Monument in Lublin
5393:
5383:
5267:
5120:
5075:
5065:
4306:, which is the sum of the external field
1176:
954:as the generalized sum of the observable
608:of the reference system with Hamiltonian
120:Learn how and when to remove this message
5250:
5020:Chaikin, P. M.; Lubensky, T. C. (2007).
5340:
4968:Extension to time-dependent mean fields
167:The main idea of MFT is to replace all
14:
5751:
5366:Sakthivadivel, Dalton A R (Jan 2022).
5334:
5022:Principles of condensed matter physics
5728:Marie Curie: The Courage of Knowledge
5417:
5306:
4299:{\displaystyle h^{\text{eff.}}=h+Jzm}
4948:Self-consistent mean field (biology)
4826:is given by the following relation:
4760:, there are two non-zero solutions:
3381:. We may rewrite the Hamiltonian as
2715:
230:The idea first appeared in physics (
58:adding citations to reliable sources
29:
5720:Marie Curie, une femme sur le front
4600:We thus have two equations between
3276:{\displaystyle \langle i,j\rangle }
3078:Non-interacting spins approximation
24:
4928:of the director field is non-zero.
4904:
2577:
2001:
1825:
1822:
1819:
1658:
1038:
890:
813:
780:
652:
632:{\displaystyle {\mathcal {H}}_{0}}
618:
475:
459:
456:
453:
396:
391:
377:
366:
350:. This inequality states that the
27:Approximation of physical behavior
25:
5780:
5643:Pierre and Marie Curie University
5638:Maria Curie-Skłodowska University
5618:IEEE Marie Sklodowska-Curie Award
4753:{\displaystyle T<T_{\text{c}}}
3325:{\displaystyle s_{i},s_{j}=\pm 1}
2499:where the mean field is given by
5697:Maria Skłodowska-Curie Medallion
5395:10.21468/SciPostPhysLectNotes.35
5328:10.1051/jphystap:019070060066100
2767:{\displaystyle -m\sum _{i}s_{i}}
334:is the formal expression of how
298:of the interaction terms in the
34:
5218:Japanese Journal of Mathematics
4640:{\displaystyle h^{\text{eff.}}}
4590:{\displaystyle h^{\text{eff.}}}
2778:the variational free energy is
2698:
2455:
414:has the following upper bound:
45:needs additional citations for
5359:
5300:
5256:Journal of Statistical Physics
5244:
5198:
5164:Journal of Statistical Physics
5155:
5099:
5038:
5013:
4796:. The system is ferromagnetic.
4172:
4154:
4035:
4029:
4001:
3995:
3874:
3855:
3730:
3655:
3486:
3457:
3454:
3425:
3048:
3033:
2710:
2679:
2666:
2661:
2655:
2642:
2616:
2569:
2557:
2537:
2524:
2447:
2434:
2382:
2369:
2364:
2358:
2319:
2313:
2273:
2260:
2255:
2249:
2230:
2217:
2212:
2206:
2143:
2130:
2125:
2119:
2106:
2093:
2088:
2082:
2069:
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1993:
1981:
1891:
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1873:
1867:
1801:
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1669:
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1631:
1606:
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1229:
1224:
1218:
1157:
1112:
1107:
1101:
1088:
1043:
941:
928:
898:{\displaystyle {\mathcal {P}}}
862:
836:
805:
793:
710:
697:
302:can sometimes at best produce
254:, Pierre–Weiss approximation,
13:
1:
5628:Maria Skłodowska-Curie Bridge
5538:Maria Skłodowska-Curie Museum
5372:SciPost Physics Lecture Notes
5006:
4724:. The system is paramagnetic.
3332:are neighboring Ising spins.
354:of a system with Hamiltonian
342:Formal approach (Hamiltonian)
256:Flory–Huggins solution theory
5769:Electronic structure methods
4944:protein structure prediction
4885:Application to other systems
4819:{\displaystyle T_{\text{c}}}
4691:{\displaystyle T_{\text{c}}}
3012:of spin. This simplifies to
220:quantal response equilibrium
212:computer-network performance
150:Self-consistent field theory
7:
5633:Maria Skłodowska-Curie Park
5345:. Oxford University Press.
4996:Dynamical mean field theory
4989:
4980:dynamical mean field theory
4974:Dynamical mean field theory
4789:{\displaystyle m=\pm m_{0}}
4647:, allowing us to determine
4048:means "nearest neighbor of
2327:{\displaystyle P_{0}^{(i)}}
313:
10:
5785:
4971:
225:
5666:
5595:
5546:
5530:
5521:Treatise on Radioactivity
5507:
5456:
5286:10.1007/s10955-009-9814-1
5230:10.1007/s11537-007-0657-8
5047:"Modules or Mean-Fields?"
4931:To determine the optimal
4920:The molecular field of a
4957:of a composite material.
763:mean field approximation
749:{\displaystyle \xi _{i}}
18:Mean field approximation
5712:Les Palmes de M. Schutz
4910:{\displaystyle \Delta }
4698:, the only solution is
602:Helmholtz free energies
260:Scheutjens–Fleer theory
200:artificial intelligence
5568:Hélène Langevin-Joliot
5001:Mean field game theory
4963:statistical inference.
4938:packing given a fixed
4924:that emerges when the
4911:
4872:
4820:
4790:
4754:
4718:
4692:
4661:
4641:
4614:
4591:
4564:
4540:
4517:
4374:
4345:
4320:
4300:
4244:
4221:
4090:
4062:
4042:
4010:
3917:
3766:
3581:
3522:
3375:
3326:
3277:
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3203:
3100:
3064:
2999:
2928:
2768:
2689:
2490:
2328:
2287:
2182:
1935:
1905:
1856:
1749:
1517:
1452:
1363:
968:
948:
899:
872:
750:
720:
686:
633:
594:
567:
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513:
405:
5759:Statistical mechanics
5586:Frédéric Joliot-Curie
4912:
4873:
4821:
4791:
4755:
4719:
4693:
4662:
4642:
4615:
4592:
4565:
4541:
4518:
4375:
4346:
4321:
4301:
4245:
4222:
4091:
4063:
4043:
4041:{\displaystyle nn(i)}
4011:
3918:
3767:
3582:
3523:
3376:
3327:
3278:
3246:
3204:
3101:
3065:
3000:
2929:
2769:
2690:
2491:
2329:
2288:
2162:
1936:
1934:{\displaystyle Z_{0}}
1906:
1836:
1729:
1518:
1453:
1364:
969:
949:
900:
873:
751:
721:
666:
634:
595:
593:{\displaystyle F_{0}}
568:
544:
542:{\displaystyle S_{0}}
514:
406:
348:Bogoliubov inequality
232:statistical mechanics
188:statistical inference
5690:Marie Curie Gargoyle
5316:J. Phys. Theor. Appl
4901:
4830:
4803:
4764:
4731:
4702:
4675:
4651:
4624:
4604:
4574:
4554:
4530:
4390:
4373:{\displaystyle z=2d}
4355:
4335:
4310:
4265:
4260:effective mean field
4234:
4103:
4072:
4052:
4020:
3930:
3785:
3605:
3535:
3388:
3339:
3287:
3255:
3216:
3113:
3090:
3019:
2948:
2785:
2735:
2506:
2345:
2300:
1952:
1918:
1533:
1462:
1376:
981:
958:
909:
885:
775:
761:and is known as the
733:
646:
612:
577:
557:
526:
421:
361:
54:improve this article
5764:Concepts in physics
5278:2009JSP...137..777K
5176:1992JSP....66..803B
5131:10.1109/QEST.2007.8
4893:To study the metal–
4717:{\displaystyle m=0}
4252:coordination number
4089:{\displaystyle 1/2}
2665:
2523:
2433:
2368:
2336:Lagrange multiplier
2323:
2259:
2216:
2129:
2092:
1877:
1641:
1560:
1399:
1303:
1228:
1111:
175:. This reduces any
69:"Mean-field theory"
5582:(Pierre's brother)
5580:Paul-Jacques Curie
5556:Irène Joliot-Curie
5211:"Mean field games"
5184:10.1007/BF01055703
4955:elastic properties
4907:
4868:
4816:
4786:
4750:
4714:
4688:
4657:
4637:
4610:
4587:
4560:
4548:critical exponents
4536:
4513:
4370:
4341:
4316:
4296:
4240:
4217:
4203:
4193:
4179:
4086:
4058:
4038:
4006:
4005:
3977:
3954:
3913:
3899:
3828:
3762:
3748:
3654:
3577:
3518:
3504:
3424:
3371:
3322:
3273:
3241:
3240:
3199:
3185:
3149:
3096:
3060:
2995:
2924:
2764:
2753:
2685:
2645:
2586:
2509:
2486:
2416:
2348:
2324:
2303:
2283:
2281:
2239:
2196:
2109:
2072:
2007:
1943:partition function
1931:
1901:
1899:
1857:
1621:
1540:
1513:
1448:
1379:
1359:
1357:
1283:
1208:
1091:
964:
944:
895:
868:
819:
758:degrees of freedom
746:
716:
629:
590:
563:
539:
509:
401:
332:Ginzburg criterion
320:critical dimension
292:partition function
278:partition function
179:into an effective
158:degrees of freedom
138:probability theory
5746:
5745:
5623:Marie Curie Medal
5515:Curie's principle
5479:Mean-field theory
5474:Curie temperature
5140:978-0-7695-2883-0
5067:10.3390/e22050552
5031:978-0-521-79450-3
4953:To determine the
4866:
4840:
4813:
4747:
4685:
4660:{\displaystyle m}
4634:
4613:{\displaystyle m}
4584:
4570:as a function of
4563:{\displaystyle m}
4539:{\displaystyle N}
4493:
4486:
4436:
4344:{\displaystyle d}
4319:{\displaystyle h}
4275:
4243:{\displaystyle z}
4194:
4189:
4152:
4150:
4145:
4113:
4061:{\displaystyle i}
3978:
3968:
3966:
3933:
3890:
3807:
3795:
3739:
3633:
3621:
3495:
3403:
3219:
3176:
3128:
3099:{\displaystyle d}
3031:
2744:
2716:Formal derivation
2705:phase transitions
2543:
2521:
2403:
1976:
1835:
1830:
1808:
1765:
1642:
967:{\displaystyle f}
788:
566:{\displaystyle F}
469:
464:
442:
272:theories, the 1D
244:phase transitions
234:) in the work of
177:many-body problem
142:Mean-field theory
130:
129:
122:
104:
16:(Redirected from
5776:
5484:Piezoelectricity
5438:
5431:
5424:
5415:
5414:
5408:
5407:
5397:
5387:
5363:
5357:
5356:
5338:
5332:
5331:
5304:
5298:
5297:
5271:
5262:(5–6): 777–797.
5248:
5242:
5241:
5215:
5202:
5196:
5195:
5159:
5153:
5152:
5124:
5114:
5103:
5097:
5096:
5094:
5092:
5079:
5069:
5051:
5042:
5036:
5035:
5017:
4940:protein backbone
4916:
4914:
4913:
4908:
4877:
4875:
4874:
4869:
4867:
4865:
4864:
4855:
4847:
4842:
4841:
4838:
4825:
4823:
4822:
4817:
4815:
4814:
4811:
4795:
4793:
4792:
4787:
4785:
4784:
4759:
4757:
4756:
4751:
4749:
4748:
4745:
4723:
4721:
4720:
4715:
4697:
4695:
4694:
4689:
4687:
4686:
4683:
4666:
4664:
4663:
4658:
4646:
4644:
4643:
4638:
4636:
4635:
4632:
4619:
4617:
4616:
4611:
4596:
4594:
4593:
4588:
4586:
4585:
4582:
4569:
4567:
4566:
4561:
4545:
4543:
4542:
4537:
4522:
4520:
4519:
4514:
4509:
4508:
4503:
4499:
4498:
4494:
4492:
4488:
4487:
4484:
4477:
4460:
4439:
4438:
4437:
4432:
4425:
4424:
4408:
4379:
4377:
4376:
4371:
4350:
4348:
4347:
4342:
4325:
4323:
4322:
4317:
4305:
4303:
4302:
4297:
4277:
4276:
4273:
4249:
4247:
4246:
4241:
4226:
4224:
4223:
4218:
4213:
4212:
4202:
4192:
4191:
4190:
4187:
4180:
4175:
4146:
4141:
4134:
4133:
4120:
4115:
4114:
4111:
4095:
4093:
4092:
4087:
4082:
4067:
4065:
4064:
4059:
4047:
4045:
4044:
4039:
4015:
4013:
4012:
4007:
4004:
3976:
3967:
3959:
3953:
3922:
3920:
3919:
3914:
3909:
3908:
3898:
3883:
3882:
3867:
3866:
3845:
3844:
3835:
3834:
3827:
3797:
3796:
3793:
3771:
3769:
3768:
3763:
3758:
3757:
3747:
3729:
3728:
3716:
3715:
3703:
3702:
3690:
3689:
3677:
3676:
3667:
3666:
3653:
3623:
3622:
3619:
3586:
3584:
3583:
3578:
3576:
3575:
3563:
3562:
3550:
3549:
3531:where we define
3527:
3525:
3524:
3519:
3514:
3513:
3503:
3485:
3484:
3469:
3468:
3453:
3452:
3437:
3436:
3423:
3380:
3378:
3377:
3372:
3367:
3366:
3351:
3350:
3331:
3329:
3328:
3323:
3312:
3311:
3299:
3298:
3282:
3280:
3279:
3274:
3250:
3248:
3247:
3242:
3239:
3208:
3206:
3205:
3200:
3195:
3194:
3184:
3169:
3168:
3159:
3158:
3148:
3105:
3103:
3102:
3097:
3069:
3067:
3066:
3061:
3032:
3029:
3010:ensemble average
3004:
3002:
3001:
2996:
2985:
2984:
2975:
2974:
2933:
2931:
2930:
2925:
2920:
2919:
2914:
2910:
2909:
2905:
2904:
2903:
2877:
2873:
2872:
2871:
2853:
2852:
2843:
2842:
2810:
2809:
2797:
2796:
2773:
2771:
2770:
2765:
2763:
2762:
2752:
2694:
2692:
2691:
2686:
2678:
2677:
2664:
2653:
2641:
2640:
2628:
2627:
2615:
2614:
2596:
2595:
2585:
2581:
2580:
2536:
2535:
2522:
2519:
2517:
2495:
2493:
2492:
2487:
2451:
2450:
2446:
2445:
2432:
2424:
2404:
2402:
2401:
2389:
2381:
2380:
2367:
2356:
2333:
2331:
2330:
2325:
2322:
2311:
2292:
2290:
2289:
2284:
2282:
2272:
2271:
2258:
2247:
2229:
2228:
2215:
2204:
2192:
2191:
2181:
2176:
2149:
2142:
2141:
2128:
2117:
2105:
2104:
2091:
2080:
2068:
2067:
2055:
2054:
2042:
2041:
2023:
2022:
2006:
2005:
2004:
1968:
1967:
1940:
1938:
1937:
1932:
1930:
1929:
1910:
1908:
1907:
1902:
1900:
1890:
1889:
1876:
1865:
1855:
1850:
1833:
1832:
1831:
1829:
1828:
1816:
1811:
1806:
1805:
1804:
1800:
1799:
1787:
1786:
1766:
1764:
1763:
1751:
1748:
1743:
1722:
1718:
1717:
1713:
1712:
1694:
1693:
1681:
1680:
1668:
1667:
1662:
1661:
1643:
1640:
1629:
1617:
1605:
1604:
1586:
1585:
1573:
1572:
1559:
1548:
1525:Boltzmann factor
1522:
1520:
1519:
1514:
1509:
1508:
1490:
1489:
1477:
1476:
1457:
1455:
1454:
1449:
1444:
1443:
1425:
1424:
1412:
1411:
1398:
1387:
1368:
1366:
1365:
1360:
1358:
1348:
1347:
1329:
1328:
1316:
1315:
1302:
1291:
1273:
1272:
1254:
1253:
1241:
1240:
1227:
1216:
1204:
1203:
1163:
1156:
1155:
1137:
1136:
1124:
1123:
1110:
1099:
1087:
1086:
1068:
1067:
1055:
1054:
1042:
1041:
1032:
1031:
997:
996:
973:
971:
970:
965:
953:
951:
950:
945:
940:
939:
921:
920:
904:
902:
901:
896:
894:
893:
877:
875:
874:
869:
861:
860:
848:
847:
835:
834:
818:
817:
816:
784:
783:
755:
753:
752:
747:
745:
744:
725:
723:
722:
717:
709:
708:
696:
695:
685:
680:
662:
661:
656:
655:
638:
636:
635:
630:
628:
627:
622:
621:
599:
597:
596:
591:
589:
588:
572:
570:
569:
564:
548:
546:
545:
540:
538:
537:
518:
516:
515:
510:
505:
504:
489:
488:
479:
478:
467:
466:
465:
463:
462:
450:
445:
440:
439:
438:
410:
408:
407:
402:
400:
399:
387:
386:
381:
380:
370:
369:
308:Feynman diagrams
192:graphical models
181:one-body problem
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
5784:
5783:
5779:
5778:
5777:
5775:
5774:
5773:
5749:
5748:
5747:
5742:
5662:
5603:Curie Institute
5591:
5570:(granddaughter)
5542:
5526:
5503:
5469:Curie–Weiss law
5452:
5442:
5412:
5411:
5364:
5360:
5353:
5339:
5335:
5305:
5301:
5252:Kadanoff, L. P.
5249:
5245:
5213:
5203:
5199:
5160:
5156:
5141:
5122:10.1.1.110.2612
5112:
5104:
5100:
5090:
5088:
5049:
5043:
5039:
5032:
5018:
5014:
5009:
4992:
4976:
4970:
4902:
4899:
4898:
4887:
4860:
4856:
4848:
4846:
4837:
4833:
4831:
4828:
4827:
4810:
4806:
4804:
4801:
4800:
4780:
4776:
4765:
4762:
4761:
4744:
4740:
4732:
4729:
4728:
4703:
4700:
4699:
4682:
4678:
4676:
4673:
4672:
4652:
4649:
4648:
4631:
4627:
4625:
4622:
4621:
4605:
4602:
4601:
4581:
4577:
4575:
4572:
4571:
4555:
4552:
4551:
4531:
4528:
4527:
4504:
4483:
4479:
4478:
4461:
4459:
4455:
4445:
4441:
4440:
4420:
4416:
4409:
4407:
4403:
4399:
4391:
4388:
4387:
4356:
4353:
4352:
4336:
4333:
4332:
4311:
4308:
4307:
4272:
4268:
4266:
4263:
4262:
4235:
4232:
4231:
4208:
4204:
4198:
4186:
4182:
4181:
4153:
4151:
4129:
4125:
4121:
4119:
4110:
4106:
4104:
4101:
4100:
4078:
4073:
4070:
4069:
4053:
4050:
4049:
4021:
4018:
4017:
3982:
3972:
3958:
3937:
3931:
3928:
3927:
3904:
3900:
3894:
3878:
3877:
3862:
3858:
3840:
3836:
3830:
3829:
3811:
3792:
3788:
3786:
3783:
3782:
3753:
3749:
3743:
3724:
3720:
3711:
3707:
3698:
3694:
3685:
3681:
3672:
3668:
3662:
3658:
3637:
3618:
3614:
3606:
3603:
3602:
3571:
3567:
3558:
3554:
3545:
3541:
3536:
3533:
3532:
3509:
3505:
3499:
3480:
3476:
3464:
3460:
3448:
3444:
3432:
3428:
3407:
3389:
3386:
3385:
3362:
3358:
3346:
3342:
3340:
3337:
3336:
3307:
3303:
3294:
3290:
3288:
3285:
3284:
3256:
3253:
3252:
3223:
3217:
3214:
3213:
3190:
3186:
3180:
3164:
3160:
3154:
3150:
3132:
3114:
3111:
3110:
3091:
3088:
3087:
3080:
3028:
3020:
3017:
3016:
2980:
2976:
2970:
2966:
2949:
2946:
2945:
2915:
2899:
2895:
2885:
2881:
2867:
2863:
2848:
2844:
2838:
2834:
2824:
2820:
2819:
2815:
2814:
2805:
2801:
2792:
2788:
2786:
2783:
2782:
2758:
2754:
2748:
2736:
2733:
2732:
2718:
2713:
2701:
2673:
2669:
2654:
2649:
2636:
2632:
2623:
2619:
2604:
2600:
2591:
2587:
2576:
2575:
2547:
2531:
2527:
2518:
2513:
2507:
2504:
2503:
2441:
2437:
2425:
2420:
2409:
2405:
2397:
2393:
2388:
2376:
2372:
2357:
2352:
2346:
2343:
2342:
2312:
2307:
2301:
2298:
2297:
2280:
2279:
2267:
2263:
2248:
2243:
2224:
2220:
2205:
2200:
2187:
2183:
2177:
2166:
2147:
2146:
2137:
2133:
2118:
2113:
2100:
2096:
2081:
2076:
2063:
2059:
2050:
2046:
2031:
2027:
2012:
2008:
2000:
1999:
1980:
1969:
1963:
1959:
1955:
1953:
1950:
1949:
1925:
1921:
1919:
1916:
1915:
1898:
1897:
1885:
1881:
1866:
1861:
1851:
1840:
1818:
1817:
1812:
1810:
1809:
1795:
1791:
1782:
1778:
1771:
1767:
1759:
1755:
1750:
1744:
1733:
1720:
1719:
1708:
1704:
1689:
1685:
1676:
1672:
1663:
1657:
1656:
1655:
1648:
1644:
1630:
1625:
1616:
1609:
1600:
1596:
1581:
1577:
1568:
1564:
1549:
1544:
1536:
1534:
1531:
1530:
1504:
1500:
1485:
1481:
1472:
1468:
1463:
1460:
1459:
1439:
1435:
1420:
1416:
1407:
1403:
1388:
1383:
1377:
1374:
1373:
1356:
1355:
1343:
1339:
1324:
1320:
1311:
1307:
1292:
1287:
1268:
1264:
1249:
1245:
1236:
1232:
1217:
1212:
1181:
1177:
1161:
1160:
1151:
1147:
1132:
1128:
1119:
1115:
1100:
1095:
1082:
1078:
1063:
1059:
1050:
1046:
1037:
1036:
1009:
1005:
998:
992:
988:
984:
982:
979:
978:
959:
956:
955:
935:
931:
916:
912:
910:
907:
906:
889:
888:
886:
883:
882:
856:
852:
843:
839:
824:
820:
812:
811:
792:
779:
778:
776:
773:
772:
740:
736:
734:
731:
730:
704:
700:
691:
687:
681:
670:
657:
651:
650:
649:
647:
644:
643:
623:
617:
616:
615:
613:
610:
609:
584:
580:
578:
575:
574:
558:
555:
554:
533:
529:
527:
524:
523:
500:
496:
484:
480:
474:
473:
452:
451:
446:
444:
443:
434:
430:
422:
419:
418:
395:
394:
382:
376:
375:
374:
365:
364:
362:
359:
358:
344:
316:
294:, studying the
228:
208:queueing theory
204:epidemic models
173:molecular field
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
5782:
5772:
5771:
5766:
5761:
5744:
5743:
5741:
5740:
5732:
5724:
5716:
5708:
5700:
5693:
5686:
5681:
5676:
5670:
5668:
5664:
5663:
5661:
5660:
5655:
5650:
5645:
5640:
5635:
5630:
5625:
5620:
5615:
5610:
5605:
5599:
5597:
5593:
5592:
5590:
5589:
5583:
5577:
5571:
5565:
5559:
5552:
5550:
5544:
5543:
5541:
5540:
5534:
5532:
5528:
5527:
5525:
5524:
5517:
5511:
5509:
5505:
5504:
5502:
5501:
5496:
5491:
5486:
5481:
5476:
5471:
5466:
5460:
5458:
5454:
5453:
5441:
5440:
5433:
5426:
5418:
5410:
5409:
5358:
5351:
5333:
5322:(1): 661–690.
5299:
5243:
5205:Lasry, J. M.;
5197:
5154:
5139:
5098:
5037:
5030:
5011:
5010:
5008:
5005:
5004:
5003:
4998:
4991:
4988:
4972:Main article:
4969:
4966:
4959:
4958:
4951:
4929:
4922:liquid crystal
4918:
4906:
4895:superconductor
4886:
4883:
4863:
4859:
4854:
4851:
4845:
4836:
4809:
4798:
4797:
4783:
4779:
4775:
4772:
4769:
4743:
4739:
4736:
4725:
4713:
4710:
4707:
4681:
4656:
4630:
4609:
4580:
4559:
4535:
4524:
4523:
4512:
4507:
4502:
4497:
4491:
4482:
4476:
4473:
4470:
4467:
4464:
4458:
4454:
4451:
4448:
4444:
4435:
4431:
4428:
4423:
4419:
4415:
4412:
4406:
4402:
4398:
4395:
4369:
4366:
4363:
4360:
4340:
4315:
4295:
4292:
4289:
4286:
4283:
4280:
4271:
4239:
4228:
4227:
4216:
4211:
4207:
4201:
4197:
4185:
4178:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4149:
4144:
4140:
4137:
4132:
4128:
4124:
4118:
4109:
4085:
4081:
4077:
4057:
4037:
4034:
4031:
4028:
4025:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3981:
3975:
3971:
3965:
3962:
3957:
3952:
3949:
3946:
3943:
3940:
3936:
3924:
3923:
3912:
3907:
3903:
3897:
3893:
3889:
3886:
3881:
3876:
3873:
3870:
3865:
3861:
3857:
3854:
3851:
3848:
3843:
3839:
3833:
3826:
3823:
3820:
3817:
3814:
3810:
3806:
3803:
3800:
3791:
3773:
3772:
3761:
3756:
3752:
3746:
3742:
3738:
3735:
3732:
3727:
3723:
3719:
3714:
3710:
3706:
3701:
3697:
3693:
3688:
3684:
3680:
3675:
3671:
3665:
3661:
3657:
3652:
3649:
3646:
3643:
3640:
3636:
3632:
3629:
3626:
3617:
3613:
3610:
3591:of the spin.
3587:; this is the
3574:
3570:
3566:
3561:
3557:
3553:
3548:
3544:
3540:
3529:
3528:
3517:
3512:
3508:
3502:
3498:
3494:
3491:
3488:
3483:
3479:
3475:
3472:
3467:
3463:
3459:
3456:
3451:
3447:
3443:
3440:
3435:
3431:
3427:
3422:
3419:
3416:
3413:
3410:
3406:
3402:
3399:
3396:
3393:
3370:
3365:
3361:
3357:
3354:
3349:
3345:
3321:
3318:
3315:
3310:
3306:
3302:
3297:
3293:
3272:
3269:
3266:
3263:
3260:
3238:
3235:
3232:
3229:
3226:
3222:
3210:
3209:
3198:
3193:
3189:
3183:
3179:
3175:
3172:
3167:
3163:
3157:
3153:
3147:
3144:
3141:
3138:
3135:
3131:
3127:
3124:
3121:
3118:
3095:
3079:
3076:
3071:
3070:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3027:
3024:
3006:
3005:
2994:
2991:
2988:
2983:
2979:
2973:
2969:
2965:
2962:
2959:
2956:
2953:
2935:
2934:
2923:
2918:
2913:
2908:
2902:
2898:
2894:
2891:
2888:
2884:
2880:
2876:
2870:
2866:
2862:
2859:
2856:
2851:
2847:
2841:
2837:
2833:
2830:
2827:
2823:
2818:
2813:
2808:
2804:
2800:
2795:
2791:
2776:
2775:
2761:
2757:
2751:
2747:
2743:
2740:
2717:
2714:
2712:
2709:
2700:
2697:
2696:
2695:
2684:
2681:
2676:
2672:
2668:
2663:
2660:
2657:
2652:
2648:
2644:
2639:
2635:
2631:
2626:
2622:
2618:
2613:
2610:
2607:
2603:
2599:
2594:
2590:
2584:
2579:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2546:
2542:
2539:
2534:
2530:
2526:
2516:
2512:
2497:
2496:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2454:
2449:
2444:
2440:
2436:
2431:
2428:
2423:
2419:
2415:
2412:
2408:
2400:
2396:
2392:
2387:
2384:
2379:
2375:
2371:
2366:
2363:
2360:
2355:
2351:
2321:
2318:
2315:
2310:
2306:
2294:
2293:
2278:
2275:
2270:
2266:
2262:
2257:
2254:
2251:
2246:
2242:
2238:
2235:
2232:
2227:
2223:
2219:
2214:
2211:
2208:
2203:
2199:
2195:
2190:
2186:
2180:
2175:
2172:
2169:
2165:
2161:
2158:
2155:
2152:
2150:
2148:
2145:
2140:
2136:
2132:
2127:
2124:
2121:
2116:
2112:
2108:
2103:
2099:
2095:
2090:
2087:
2084:
2079:
2075:
2071:
2066:
2062:
2058:
2053:
2049:
2045:
2040:
2037:
2034:
2030:
2026:
2021:
2018:
2015:
2011:
2003:
1998:
1995:
1992:
1989:
1986:
1983:
1979:
1975:
1972:
1970:
1966:
1962:
1958:
1957:
1928:
1924:
1912:
1911:
1896:
1893:
1888:
1884:
1880:
1875:
1872:
1869:
1864:
1860:
1854:
1849:
1846:
1843:
1839:
1827:
1824:
1821:
1815:
1803:
1798:
1794:
1790:
1785:
1781:
1777:
1774:
1770:
1762:
1758:
1754:
1747:
1742:
1739:
1736:
1732:
1728:
1725:
1723:
1721:
1716:
1711:
1707:
1703:
1700:
1697:
1692:
1688:
1684:
1679:
1675:
1671:
1666:
1660:
1654:
1651:
1647:
1639:
1636:
1633:
1628:
1624:
1620:
1615:
1612:
1610:
1608:
1603:
1599:
1595:
1592:
1589:
1584:
1580:
1576:
1571:
1567:
1563:
1558:
1555:
1552:
1547:
1543:
1539:
1538:
1512:
1507:
1503:
1499:
1496:
1493:
1488:
1484:
1480:
1475:
1471:
1467:
1447:
1442:
1438:
1434:
1431:
1428:
1423:
1419:
1415:
1410:
1406:
1402:
1397:
1394:
1391:
1386:
1382:
1370:
1369:
1354:
1351:
1346:
1342:
1338:
1335:
1332:
1327:
1323:
1319:
1314:
1310:
1306:
1301:
1298:
1295:
1290:
1286:
1282:
1279:
1276:
1271:
1267:
1263:
1260:
1257:
1252:
1248:
1244:
1239:
1235:
1231:
1226:
1223:
1220:
1215:
1211:
1207:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1180:
1175:
1172:
1169:
1166:
1164:
1162:
1159:
1154:
1150:
1146:
1143:
1140:
1135:
1131:
1127:
1122:
1118:
1114:
1109:
1106:
1103:
1098:
1094:
1090:
1085:
1081:
1077:
1074:
1071:
1066:
1062:
1058:
1053:
1049:
1045:
1040:
1035:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1008:
1004:
1001:
999:
995:
991:
987:
986:
963:
943:
938:
934:
930:
927:
924:
919:
915:
892:
879:
878:
867:
864:
859:
855:
851:
846:
842:
838:
833:
830:
827:
823:
815:
810:
807:
804:
801:
798:
795:
791:
787:
782:
743:
739:
727:
726:
715:
712:
707:
703:
699:
694:
690:
684:
679:
676:
673:
669:
665:
660:
654:
626:
620:
587:
583:
562:
536:
532:
520:
519:
508:
503:
499:
495:
492:
487:
483:
477:
472:
461:
458:
455:
449:
437:
433:
429:
426:
412:
411:
398:
393:
390:
385:
379:
373:
368:
343:
340:
315:
312:
227:
224:
128:
127:
110:September 2022
42:
40:
33:
26:
9:
6:
4:
3:
2:
5781:
5770:
5767:
5765:
5762:
5760:
5757:
5756:
5754:
5738:
5737:
5733:
5730:
5729:
5725:
5722:
5721:
5717:
5714:
5713:
5709:
5706:
5705:
5701:
5699:
5698:
5694:
5692:
5691:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5671:
5669:
5665:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5600:
5598:
5594:
5587:
5584:
5581:
5578:
5575:
5574:Pierre Joliot
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5553:
5551:
5549:
5545:
5539:
5536:
5535:
5533:
5529:
5523:
5522:
5518:
5516:
5513:
5512:
5510:
5506:
5500:
5497:
5495:
5494:Radioactivity
5492:
5490:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5461:
5459:
5455:
5450:
5446:
5439:
5434:
5432:
5427:
5425:
5420:
5419:
5416:
5405:
5401:
5396:
5391:
5386:
5381:
5377:
5373:
5369:
5362:
5354:
5352:0-19-505316-8
5348:
5344:
5337:
5329:
5325:
5321:
5317:
5313:
5309:
5308:Weiss, Pierre
5303:
5295:
5291:
5287:
5283:
5279:
5275:
5270:
5265:
5261:
5257:
5253:
5247:
5239:
5235:
5231:
5227:
5223:
5219:
5212:
5208:
5201:
5193:
5189:
5185:
5181:
5177:
5173:
5169:
5165:
5158:
5150:
5146:
5142:
5136:
5132:
5128:
5123:
5118:
5115:. p. 3.
5111:
5110:
5102:
5087:
5083:
5078:
5073:
5068:
5063:
5059:
5055:
5048:
5041:
5033:
5027:
5023:
5016:
5012:
5002:
4999:
4997:
4994:
4993:
4987:
4985:
4984:Hubbard model
4981:
4975:
4965:
4964:
4956:
4952:
4949:
4945:
4941:
4937:
4934:
4930:
4927:
4923:
4919:
4896:
4892:
4891:
4890:
4882:
4879:
4861:
4857:
4852:
4849:
4843:
4834:
4807:
4781:
4777:
4773:
4770:
4767:
4741:
4737:
4734:
4726:
4711:
4708:
4705:
4679:
4670:
4669:
4668:
4654:
4628:
4607:
4598:
4578:
4557:
4549:
4533:
4510:
4505:
4500:
4495:
4489:
4480:
4474:
4471:
4468:
4465:
4462:
4456:
4452:
4449:
4446:
4442:
4433:
4429:
4426:
4421:
4417:
4413:
4410:
4404:
4400:
4396:
4393:
4386:
4385:
4384:
4381:
4367:
4364:
4361:
4358:
4338:
4329:
4313:
4293:
4290:
4287:
4284:
4281:
4278:
4269:
4261:
4257:
4253:
4237:
4214:
4209:
4205:
4199:
4195:
4183:
4176:
4169:
4166:
4163:
4160:
4157:
4147:
4142:
4138:
4135:
4130:
4126:
4122:
4116:
4107:
4099:
4098:
4097:
4083:
4079:
4075:
4055:
4032:
4026:
4023:
3998:
3992:
3989:
3986:
3983:
3979:
3973:
3969:
3963:
3960:
3955:
3947:
3944:
3941:
3934:
3910:
3905:
3901:
3895:
3891:
3887:
3884:
3871:
3868:
3863:
3859:
3852:
3849:
3846:
3841:
3837:
3821:
3818:
3815:
3808:
3804:
3801:
3798:
3789:
3781:
3780:
3779:
3776:
3759:
3754:
3750:
3744:
3740:
3736:
3733:
3725:
3721:
3717:
3712:
3708:
3704:
3699:
3695:
3691:
3686:
3682:
3678:
3673:
3669:
3663:
3659:
3647:
3644:
3641:
3634:
3630:
3627:
3624:
3615:
3611:
3608:
3601:
3600:
3599:
3596:
3592:
3590:
3572:
3568:
3564:
3559:
3555:
3551:
3546:
3542:
3538:
3515:
3510:
3506:
3500:
3496:
3492:
3489:
3481:
3477:
3473:
3470:
3465:
3461:
3449:
3445:
3441:
3438:
3433:
3429:
3417:
3414:
3411:
3404:
3400:
3397:
3394:
3391:
3384:
3383:
3382:
3363:
3359:
3352:
3347:
3343:
3333:
3319:
3316:
3313:
3308:
3304:
3300:
3295:
3291:
3267:
3264:
3261:
3233:
3230:
3227:
3220:
3196:
3191:
3187:
3181:
3177:
3173:
3170:
3165:
3161:
3155:
3151:
3142:
3139:
3136:
3129:
3125:
3122:
3119:
3116:
3109:
3108:
3107:
3093:
3085:
3082:Consider the
3075:
3057:
3054:
3051:
3045:
3042:
3039:
3036:
3025:
3022:
3015:
3014:
3013:
3011:
3008:which is the
2992:
2989:
2986:
2981:
2971:
2967:
2960:
2957:
2954:
2951:
2944:
2943:
2942:
2940:
2921:
2916:
2911:
2906:
2900:
2896:
2892:
2889:
2886:
2882:
2878:
2874:
2868:
2864:
2860:
2857:
2854:
2849:
2845:
2839:
2835:
2831:
2828:
2825:
2821:
2816:
2811:
2806:
2802:
2798:
2793:
2789:
2781:
2780:
2779:
2759:
2755:
2749:
2745:
2741:
2738:
2731:
2730:
2729:
2727:
2723:
2722:Ising lattice
2708:
2706:
2682:
2674:
2670:
2658:
2650:
2646:
2637:
2633:
2629:
2624:
2620:
2611:
2608:
2605:
2601:
2597:
2592:
2588:
2572:
2566:
2563:
2560:
2554:
2551:
2544:
2540:
2532:
2528:
2514:
2510:
2502:
2501:
2500:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2452:
2442:
2438:
2429:
2426:
2421:
2417:
2413:
2410:
2406:
2398:
2394:
2390:
2385:
2377:
2373:
2361:
2353:
2349:
2341:
2340:
2339:
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71: –
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65:Find sources:
59:
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43:This article
41:
37:
32:
31:
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5734:
5726:
5718:
5710:
5704:Madame Curie
5702:
5695:
5688:
5653:Curie Island
5588:(son-in-law)
5519:
5508:Publications
5478:
5375:
5371:
5361:
5342:
5336:
5319:
5315:
5302:
5259:
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5217:
5207:Lions, P. L.
5200:
5170:(3–4): 803.
5167:
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5101:
5089:. Retrieved
5060:(552): 552.
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2699:Applications
2498:
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1913:
1371:
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767:
762:
728:
521:
413:
345:
336:fluctuations
324:
317:
304:perturbation
289:
285:field theory
282:
270:random-field
264:
242:to describe
240:Pierre Weiss
236:Pierre Curie
229:
218:, as in the
196:neuroscience
185:
172:
169:interactions
166:
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145:
141:
131:
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107:
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64:
52:Please help
47:verification
44:
5739:(2019 film)
5736:Radioactive
5731:(2016 film)
5723:(2014 film)
5715:(1997 film)
5707:(1943 film)
5464:Curie's law
5457:Discoveries
5224:: 229–260.
4326:and of the
4068:", and the
3589:fluctuation
3084:Ising model
2726:free energy
2711:Ising model
352:free energy
306:results or
300:Hamiltonian
274:Ising model
216:game theory
5753:Categories
5667:Depictions
5658:7000 Curie
5576:(grandson)
5564:(daughter)
5558:(daughter)
5385:2102.00960
5007:References
4936:side chain
4933:amino acid
4328:mean field
3212:where the
154:stochastic
80:newspapers
5596:Namesakes
5562:Ève Curie
5404:237623181
5269:0906.0653
5192:120840517
5117:CiteSeerX
4926:Laplacian
4905:Δ
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756:are the
606:ensemble
328:polymers
314:Validity
5531:Museums
5294:9074428
5274:Bibcode
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5054:Entropy
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551:entropy
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226:Origins
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800:,
797:i
794:(
786:=
781:H
742:i
714:,
711:)
706:i
698:(
693:i
689:h
683:N
678:1
675:=
672:i
664:=
659:0
653:H
625:0
619:H
586:0
582:F
561:F
535:0
531:S
507:,
502:0
498:S
494:T
486:0
476:H
460:f
457:e
454:d
448:=
436:0
432:F
425:F
397:H
389:+
384:0
378:H
372:=
367:H
144:(
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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