656:
2650:: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? The problem here is that the ThomasāFermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so particles simply 'disappear' from the continuum of states. It turns out, however, that the macroscopic equation gives an accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term to accept the particles that fall out of the continuum:
1296:
33:
669:
3056:
2994:
3551:
This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number,
3046:
Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change
1957:
The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the BoseāEinstein condensate and will be dealt with in the next sections. As will be seen,
3004:
The above standard treatment of a macroscopic Bose gas is straightforward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic
4632:
It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that
3690:
All thermodynamic properties can be computed from this potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an
4625:
3047:
the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite). See the article BoseāEinstein condensate.
2099:
3923:
2842:
3685:
1935:
2424:
3500:
5010:
Tarasov, S. V.; Kocharovsky, Vl. V.; Kocharovsky, V. V. (2015-09-07). "Grand
Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of BoseāEinstein Condensation of an Ideal Gas in Arbitrary Trap".
2774:
2598:
1467:
1087:
1572:
4266:
1801:
4925:
1303:, Bose gas) in three dimensions. The Bose gas pressure is lower than an equivalent classical gas, especially below the critical temperature (marked with ā
) where particles begin moving en masse into the zero-pressure condensed phase.
2240:
4069:
1286:
is from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0).
4782:
4413:
1683:
4338:
2319:, below which the ThomasāFermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature:
1315:, which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral. This replacement gives the macroscopic grand potential function
4689:
4525:
1976:
4183:
3191:
2989:{\displaystyle {\frac {N_{0}}{N}}={\begin{cases}1-\left({\frac {T}{T_{\rm {c}}}}\right)^{\alpha }&{\mbox{if }}\alpha >1{\mbox{ and }}T<T_{\rm {c}},\\0&{\mbox{otherwise}}.\end{cases}}}
3827:
1239:
3566:
3364:
1837:
1182:
2325:
3386:
3994:
2656:
4101:
2476:
1368:
3716:
3552:
which is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.
973:
3796:
3761:
1482:
4833:
2461:
1340:
4457:
4494:
3377:
in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.
1360:
3117:, systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy
1263:. All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variables
4189:
4519:
4126:
3948:
1720:
4844:
3819:
2150:
4000:
3522:
4700:
745:, who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a
4344:
1609:
4272:
700:
1958:
even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the uncondensed portion of the gas.
4639:
3087:. Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential
4620:{\displaystyle \approx \ln \left({\frac {\zeta (\alpha )}{\tau ^{\alpha }}}\right)-{\frac {\zeta (\alpha )}{2^{\alpha }\tau ^{\alpha }}}}
3022:
and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the
2094:{\displaystyle N_{\rm {m}}=-z{\frac {\partial \Omega _{m}}{\partial z}}={\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{c})^{\alpha }}}.}
5123:
Mullin, W. J.; FernĆ”ndez, J. P. (2003). "BoseāEinstein condensation, fluctuations, and recurrence relations in statistical mechanics".
4136:
917:
693:
5306:
4994:
3918:{\displaystyle \approx {\frac {\zeta (\alpha )}{\tau ^{\alpha }}}-{\frac {\zeta ^{2}(\alpha )}{2^{\alpha }\tau ^{2\alpha }}}}
3239:
This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for
3127:
5185:
4950:. In one dimensional case correlation functions also were evaluated. In one dimension Bose gas is equivalent to quantum
1279:. All partial derivatives are taken with respect to one of these three variables while the other two are held constant.
3680:{\displaystyle \Omega =g_{0}\ln(1-z)-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\rm {c}}\right)^{\alpha }}}}
4951:
1930:{\displaystyle \Omega _{\rm {m}}=-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\text{c}}\right)^{\alpha }}},}
1197:
5195:
686:
673:
3327:
2419:{\displaystyle T_{\rm {c}}=\left({\frac {N}{\zeta (\alpha )}}\right)^{1/\alpha }{\frac {E_{\rm {c}}}{k_{\rm {B}}}}}
1136:
3495:{\displaystyle N={\frac {g_{0}\,z}{1-z}}+N~{\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}~\tau ^{\alpha }}
655:
4984:
438:
2769:{\displaystyle N=N_{0}+N_{\rm {m}}=N_{0}+{\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{\rm {c}})^{\alpha }}}}
3958:
5326:
2593:{\displaystyle T_{\rm {c}}=\left({\frac {N}{Vf\zeta (3/2)}}\right)^{2/3}{\frac {h^{2}}{2\pi mk_{\rm {B}}}}}
890:
765:
746:
730:
613:
93:
4935:
1462:{\displaystyle \Omega _{\rm {m}}=\int _{0}^{\infty }\ln \left(1-ze^{-\beta E}\right)\,dg\approx \Omega .}
4075:
2120:(i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).
1082:{\displaystyle \Omega =-\ln({\mathcal {Z}})=\sum _{i}g_{i}\ln \left(1-ze^{-\beta \epsilon _{i}}\right).}
839:. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The
5316:
5062:
Yukalov, V I (2005-03-01). "Number-of-particle fluctuations in systems with Bose-Einstein condensate".
932:
618:
243:
4939:
3694:
1567:{\displaystyle dg={\frac {1}{\Gamma (\alpha )}}\,{\frac {E^{\,\alpha -1}}{E_{\rm {c}}^{\alpha }}}~dE}
508:
183:
2871:
3380:
The equation for the number of particles can be written in terms of the normalized temperature as:
960:
503:
498:
24:
3767:
3732:
588:
5321:
4963:
4793:
2432:
1318:
193:
4423:
598:
4463:
3006:
1704:
583:
523:
493:
443:
163:
53:
5253:
4261:{\displaystyle ={\frac {{\textrm {Li}}_{\alpha \!+\!1}(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
2254:
1345:
854:
of vibration of the crystal lattice of a metal, can be treated as effective massless bosons.
623:
238:
223:
5268:
5142:
5081:
5020:
1796:{\displaystyle {\frac {1}{(\beta E_{c})^{\alpha }}}={\frac {f}{(\hbar \omega \beta )^{3}}}}
836:
213:
103:
4920:{\displaystyle TS=(\alpha +1)+\ln \left({\frac {\tau ^{\alpha }}{\zeta (\alpha )}}\right)}
4500:
4107:
3929:
1961:
8:
2235:{\displaystyle N_{\rm {m,max}}={\frac {\zeta (\alpha )}{(\beta E_{\rm {c}})^{\alpha }}},}
904:
The theory of Bose-Einstein condensates and Bose gases can also explain some features of
878:
828:
734:
453:
263:
113:
5272:
5146:
5085:
5024:
4064:{\displaystyle ={\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
5311:
5284:
5166:
5132:
5105:
5071:
5044:
3804:
3114:
3040:
1689:
1252:
1124:
817:
633:
593:
568:
316:
307:
4777:{\displaystyle C_{V}={\frac {\partial U}{\partial T}}=k_{\rm {B}}(\alpha +1)\,U\beta }
1123:
is the absolute activity (or "fugacity"), which may also be expressed in terms of the
5288:
5280:
5191:
5158:
5109:
5097:
5048:
5036:
4990:
4408:{\displaystyle \approx 1-{\frac {\zeta (\alpha )}{2^{\alpha \!+\!1}\tau ^{\alpha }}}}
1678:{\displaystyle {\frac {1}{(\beta E_{\rm {c}})^{\alpha }}}={\frac {Vf}{\Lambda ^{3}}}}
905:
898:
761:
563:
408:
298:
218:
3373:
in the limit of low temperature and, except for the chemical potential, linear in 1/
2836:. This approach gives the fraction of condensed particles in the macroscopic limit:
1092:
where each term in the sum corresponds to a particular single-particle energy level
5276:
5150:
5089:
5028:
3043:, which fixes the total particle number, however the calculations are not as easy.
886:
268:
233:
228:
188:
158:
128:
88:
48:
5170:
4333:{\displaystyle ={\frac {\zeta (\alpha \!+\!1)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
947:
and high temperature, both the Fermi gas and the Bose gas behave like a classical
3023:
1967:
964:
832:
742:
718:
578:
528:
398:
153:
65:
2621:
approaches 1), and thus for example for a gas in a one- or two-dimensional box (
1295:
4947:
4938:. In one dimension bosons with delta interaction behave as fermions, they obey
1593:
944:
909:
785:
769:
660:
628:
608:
603:
558:
478:
413:
311:
198:
43:
5032:
3059:
Figure 1: Various Bose gas parameters as a function of normalized temperature
5300:
5162:
5101:
5040:
3005:
fluctuation catastrophe: the number of particles in any given state follow a
2130:, the number of particles only increases up to a finite maximum value, i.e.,
1951:
916:) and behave like bosons. As a result, superconductors behave like having no
870:
859:
809:
339:
320:
302:
203:
123:
4942:. In one dimension Bose gas with delta interaction can be solved exactly by
3691:
approximation symbol indicates that only the first few terms of a series in
533:
5093:
4943:
3548:
approaches infinity, which can be easily determined from these expansions.
3261:
3232:. Now, the behaviour is smooth when crossing the critical temperature, and
1312:
1308:
894:
553:
543:
513:
473:
468:
448:
293:
273:
133:
3720:
5137:
5076:
1260:
913:
855:
851:
847:
773:
638:
573:
548:
518:
463:
458:
390:
32:
885:. The Bose gas is the most simple quantitative model that explains this
4946:. The bulk free energy and thermodynamic potentials were calculated by
4684:{\displaystyle U={\frac {\partial \Omega }{\partial \beta }}=\alpha PV}
1962:
Limit on number of particles in uncondensed phase, critical temperature
1476:
may be expressed for many different situations by the general formula:
893:, a state where a large number of bosons occupy the lowest energy, the
824:
738:
483:
325:
118:
5154:
3544:
approaches infinity. In particular, we are interested in the limit as
3309:. The red lines plot values of the negative of the chemical potential
3055:
1300:
1299:
Pressure vs temperature curves of classical and quantum ideal gases (
1290:
959:
The thermodynamics of an ideal Bose gas is best calculated using the
948:
936:
882:
801:
722:
538:
488:
361:
208:
108:
869:
atoms. When a system of He atoms is cooled down to temperature near
4694:
A similar situation holds for the specific heat at constant volume
928:
924:
866:
793:
789:
3533:. From these expansions, we can find the behavior of the gas near
940:
813:
772:. These particles can be classified as elementary: these are the
418:
403:
366:
357:
352:
5184:
Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1997-03-06).
2999:
823:
The first model that treated a gas with several bosons, was the
1831:
This integral expression for the grand potential evaluates to:
874:
873:, many quantum mechanical effects are present. Below 2.17
841:
812:
in more complex systems can also be considered bosons like the
797:
777:
726:
371:
347:
78:
3369:
It can be seen that each of these parameters become linear in
5009:
4178:{\displaystyle {\frac {PV\beta }{N}}=-{\frac {\Omega }{N}}\,}
805:
781:
757:
376:
73:
5187:
Quantum
Inverse Scattering Method and Correlation Functions
2982:
2786:
is the number of particles in the ground state condensate.
2429:
For example, for the three-dimensional Bose gas in a box (
3298:. The blue lines are the fraction of condensed particles
3264:. The solid black line is the fraction of excited states
889:. Mainly when a gas of bosons is cooled down, it forms a
83:
5183:
733:. The statistical mechanics of bosons were developed by
5019:(4). Springer Science and Business Media LLC: 942ā964.
3186:{\displaystyle \Omega =g_{0}\ln(1-z)+\Omega _{\rm {m}}}
3050:
897:, and quantum effects are macroscopically visible like
865:
An interesting example of a Bose gas is an ensemble of
2970:
2933:
2917:
2610:, there is no upper limit on the number of particles (
923:
The equivalent model for half-integer particles (like
858:
used the phonon gas model to explain the behaviour of
4847:
4796:
4703:
4642:
4528:
4503:
4466:
4426:
4347:
4275:
4192:
4139:
4110:
4078:
4003:
3961:
3932:
3830:
3807:
3770:
3735:
3697:
3569:
3389:
3330:
3313:
and the green lines plot the corresponding values of
3130:
2845:
2659:
2479:
2435:
2328:
2153:
1979:
1840:
1723:
1612:
1485:
1371:
1348:
1321:
1200:
1139:
976:
5233:
4838:
Note that in the limit of high temperature, we have
3317:. The horizontal axis is the normalized temperature
3529:or as an asymptotic expansion in inverse powers of
3091:is shown in red, and green lines are the values of
2116:= 1 is however crucially dependent on the value of
729:, which have an integer value of spin and abide by
5116:
5055:
4919:
4827:
4776:
4683:
4619:
4513:
4488:
4451:
4407:
4332:
4260:
4177:
4120:
4095:
4063:
3988:
3942:
3917:
3813:
3790:
3755:
3710:
3679:
3494:
3358:
3185:
2988:
2768:
2592:
2455:
2418:
2234:
2093:
1929:
1795:
1677:
1566:
1461:
1354:
1334:
1291:Macroscopic limit, result for uncondensed fraction
1233:
1176:
1081:
5254:"General Thermal Wavelength and its Applications"
4633:for a classical ideal gas over all temperatures:
4386:
4382:
4295:
4291:
4215:
4211:
3236:approaches 1 very closely but does not reach it.
5298:
2638:
2635:respectively) there is no critical temperature.
1703:for simple spinless bosons). For a massive Bose
1596:. For example, for a massive Bose gas in a box,
1234:{\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}
831:. This model leads to a better understanding of
5242:
5122:
3039:. Calculations can instead be performed in the
3359:{\displaystyle \tau ={\frac {T}{T_{\rm {c}}}}}
3291:and the dotted black line is the solution for
1821:/2 is the harmonic potential. It is seen that
3000:Limitations of the macroscopic Bose gas model
2297:. This corresponds to a critical temperature
1177:{\displaystyle z(\beta ,\mu )=e^{\beta \mu }}
694:
3009:, meaning that when condensation happens at
752:
5245:BoseāEinstein Condensation in Dilute Gases
5003:
2643:The above problem raises the question for
701:
687:
31:
5136:
5075:
4989:. Springer Science & Business Media.
4824:
4767:
4510:
4485:
4448:
4319:
4247:
4174:
4117:
4092:
4050:
3985:
3939:
3787:
3752:
3409:
2813:takes up the remainder of particles. For
2279:). Thus, for a fixed number of particles
1524:
1516:
1443:
1307:Following the procedure described in the
5247:. Cambridge: Cambridge University Press.
3054:
1294:
768:, or equivalently, that possess integer
5236:Statistical Physics, 3rd Edition Part 1
5224:
5061:
4982:
5299:
5234:Landau, L. D.; E. M. Lifshitz (1996).
3989:{\displaystyle 1-{\frac {N_{0}}{N}}\,}
3560:Expanded out, the grand potential is:
2112:= +1). The behaviour when approaching
5215:
5208:
4934:= 3/2 is simply a restatement of the
1970:is found from the grand potential by
1714:and the critical energy is given by:
1603:and the critical energy is given by:
877:, the ensemble starts to behave as a
741:and extended to massive particles by
3051:Approximate behaviour in small gases
2826:there is the normal behaviour, with
2789:Thus in the macroscopic limit, when
1110:is the number of states with energy
954:
5251:
2463:and using the above noted value of
13:
4746:
4728:
4720:
4660:
4655:
4652:
4166:
4096:{\displaystyle =\tau ^{\alpha }\,}
3658:
3570:
3513:, this equation can be solved for
3348:
3177:
3172:
3131:
2951:
2896:
2747:
2685:
2581:
2486:
2408:
2396:
2335:
2286:, the largest possible value that
2210:
2172:
2169:
2166:
2160:
2104:This increases monotonically with
2019:
2008:
2004:
1986:
1847:
1842:
1664:
1631:
1542:
1501:
1453:
1397:
1378:
1373:
1349:
1323:
1219:
997:
977:
14:
5338:
5243:Pethick, C. J.; H. Smith (2004).
3555:
1771:
827:, a gas of photons, developed by
5238:. Oxford: Butterworth-Heinemann.
5220:. New York: John Wiley and Sons.
3540:and in the MaxwellāBoltzmann as
939:(an ensemble of non-interacting
792:; or composite like the atom of
668:
667:
654:
4952:non-linear Schrƶdinger equation
3711:{\displaystyle \tau ^{\alpha }}
3517:and then a series solution for
5190:. Cambridge University Press.
5177:
5013:Journal of Statistical Physics
4976:
4907:
4901:
4869:
4857:
4764:
4752:
4589:
4583:
4554:
4548:
4482:
4476:
4445:
4439:
4369:
4363:
4313:
4307:
4299:
4285:
4241:
4235:
4227:
4221:
4044:
4038:
4030:
4024:
3884:
3878:
3846:
3840:
3639:
3633:
3604:
3592:
3521:can be found by the method of
3473:
3467:
3459:
3453:
3165:
3153:
2754:
2735:
2730:
2724:
2529:
2515:
2364:
2358:
2217:
2198:
2193:
2187:
2076:
2059:
2054:
2048:
1888:
1882:
1828:is a function of volume only.
1781:
1768:
1747:
1730:
1638:
1619:
1510:
1504:
1155:
1143:
1002:
992:
862:of metals at low temperature.
16:State of matter of many bosons
1:
4983:Schwabl, Franz (2013-03-09).
4969:
2639:Inclusion of the ground state
2290:can have is a critical value
4522:
4497:
4460:
4418:
4341:
4269:
4186:
4131:
4104:
4072:
3997:
3953:
3926:
3824:
3822:
3801:
3791:{\displaystyle T\gg T_{c}\,}
3764:
3756:{\displaystyle T\ll T_{c}\,}
3729:
3726:
3723:
3067:is 3/2. Solid lines are for
967:for a Bose gas is given by:
7:
5229:. New York: Academic Press.
5125:American Journal of Physics
4957:
4828:{\displaystyle TS=U+PV-G\,}
3121:=0 in the grand potential:
3095:. It has been assumed that
2456:{\displaystyle \alpha =3/2}
1335:{\displaystyle \Omega _{m}}
881:, a fluid with almost zero
721:, analogous to a classical
10:
5343:
5281:10.1088/0143-0807/21/6/314
4452:{\displaystyle G=\ln(z)\,}
1313:ThomasāFermi approximation
1311:article, we can apply the
943:). At low enough particle
850:, is an example where the
244:Spin gapless semiconductor
5033:10.1007/s10955-015-1361-3
4940:Pauli exclusion principle
4787:The entropy is given by:
4489:{\displaystyle =\ln(z)\,}
3260:, which corresponds to a
1282:The permissible range of
753:Introduction and examples
184:Electronic band structure
5307:BoseāEinstein statistics
1696:is a degeneracy factor (
961:grand canonical ensemble
891:BoseāEinstein condensate
808:etc. Additionally, some
766:BoseāEinstein statistics
747:BoseāEinstein condensate
731:BoseāEinstein statistics
717:is a quantum-mechanical
94:BoseāEinstein condensate
25:Condensed matter physics
4936:SackurāTetrode equation
3080:, dotted lines are for
1355:{\displaystyle \Omega }
5216:Huang, Kerson (1967).
5094:10.1002/lapl.200410157
4921:
4829:
4778:
4685:
4621:
4515:
4490:
4453:
4409:
4334:
4262:
4179:
4122:
4097:
4065:
3990:
3944:
3919:
3815:
3792:
3757:
3712:
3681:
3525:, either in powers of
3496:
3360:
3262:gas of bosons in a box
3187:
3110:
3026:becomes unbounded for
3007:geometric distribution
2990:
2770:
2594:
2457:
2420:
2236:
2095:
1931:
1797:
1705:gas in a harmonic trap
1679:
1568:
1463:
1356:
1336:
1304:
1235:
1178:
1083:
933:FermiāDirac statistics
918:electrical resistivity
764:particles that follow
5218:Statistical Mechanics
5064:Laser Physics Letters
4986:Statistical Mechanics
4922:
4830:
4779:
4686:
4622:
4516:
4491:
4454:
4410:
4335:
4263:
4180:
4123:
4098:
4066:
3991:
3945:
3920:
3816:
3793:
3758:
3713:
3682:
3497:
3361:
3188:
3058:
2991:
2771:
2595:
2458:
2421:
2255:Riemann zeta function
2237:
2096:
1932:
1798:
1680:
1592:energy, and Ī is the
1569:
1464:
1357:
1337:
1298:
1236:
1179:
1084:
920:at low temperatures.
788:and the hypothetical
239:Topological insulator
5225:Isihara, A. (1971).
4845:
4794:
4701:
4640:
4526:
4514:{\displaystyle =0\,}
4501:
4464:
4424:
4345:
4273:
4190:
4137:
4121:{\displaystyle =1\,}
4108:
4076:
4001:
3959:
3943:{\displaystyle =1\,}
3930:
3828:
3805:
3768:
3733:
3695:
3567:
3387:
3328:
3196:which gives instead
3128:
2843:
2657:
2477:
2433:
2326:
2151:
1977:
1838:
1721:
1610:
1483:
1369:
1346:
1342:, which is close to
1319:
1198:
1137:
974:
931:atoms), that follow
837:black-body radiation
818:charge density waves
725:. It is composed of
257:Electronic phenomena
104:Fermionic condensate
5327:Satyendra Nath Bose
5273:2000EJPh...21..625Y
5252:Yan, Zijun (2000).
5227:Statistical Physics
5147:2003AmJPh..71..661M
5086:2005LaPhL...2..156Y
5025:2015JSP...161..942T
4964:TonksāGirardeau gas
3523:inversion of series
2806:is pinned to 1 and
2108:(up to the maximum
1968:number of particles
1552:
1401:
735:Satyendra Nath Bose
264:Quantum Hall effect
5209:General references
4917:
4825:
4774:
4681:
4617:
4511:
4486:
4449:
4405:
4330:
4258:
4175:
4118:
4093:
4061:
3986:
3940:
3915:
3811:
3788:
3753:
3708:
3677:
3492:
3356:
3183:
3111:
3041:canonical ensemble
2986:
2981:
2974:
2937:
2921:
2766:
2590:
2453:
2416:
2232:
2091:
1927:
1793:
1690:thermal wavelength
1675:
1564:
1536:
1459:
1387:
1352:
1332:
1305:
1253:Boltzmann constant
1231:
1174:
1125:chemical potential
1079:
1017:
762:quantum mechanical
661:Physics portal
5317:Quantum mechanics
5155:10.1119/1.1544520
4996:978-3-662-04702-6
4911:
4735:
4667:
4630:
4629:
4615:
4568:
4419:Gibbs Free Energy
4403:
4317:
4245:
4204:
4172:
4156:
4132:Equation of state
4048:
4015:
3983:
3913:
3860:
3814:{\displaystyle z}
3675:
3618:
3481:
3477:
3444:
3435:
3425:
3354:
2973:
2936:
2920:
2902:
2861:
2764:
2715:
2588:
2533:
2414:
2368:
2227:
2086:
2039:
2026:
1922:
1908:
1867:
1791:
1757:
1673:
1648:
1557:
1553:
1514:
1229:
1008:
955:Macroscopic limit
912:couple in pairs (
906:superconductivity
899:wave interference
800:, the nucleus of
711:
710:
409:Granular material
177:Electronic phases
5334:
5292:
5258:
5248:
5239:
5230:
5221:
5202:
5201:
5181:
5175:
5174:
5140:
5138:cond-mat/0211115
5120:
5114:
5113:
5079:
5077:cond-mat/0504473
5059:
5053:
5052:
5007:
5001:
5000:
4980:
4926:
4924:
4923:
4918:
4916:
4912:
4910:
4896:
4895:
4886:
4834:
4832:
4831:
4826:
4783:
4781:
4780:
4775:
4751:
4750:
4749:
4736:
4734:
4726:
4718:
4713:
4712:
4690:
4688:
4687:
4682:
4668:
4666:
4658:
4650:
4626:
4624:
4623:
4618:
4616:
4614:
4613:
4612:
4603:
4602:
4592:
4578:
4573:
4569:
4567:
4566:
4557:
4543:
4520:
4518:
4517:
4512:
4495:
4493:
4492:
4487:
4458:
4456:
4455:
4450:
4414:
4412:
4411:
4406:
4404:
4402:
4401:
4400:
4391:
4390:
4372:
4358:
4339:
4337:
4336:
4331:
4329:
4328:
4318:
4316:
4302:
4280:
4267:
4265:
4264:
4259:
4257:
4256:
4246:
4244:
4230:
4220:
4219:
4206:
4205:
4202:
4197:
4184:
4182:
4181:
4176:
4173:
4165:
4157:
4152:
4141:
4127:
4125:
4124:
4119:
4102:
4100:
4099:
4094:
4091:
4090:
4070:
4068:
4067:
4062:
4060:
4059:
4049:
4047:
4033:
4023:
4022:
4017:
4016:
4013:
4008:
3995:
3993:
3992:
3987:
3984:
3979:
3978:
3969:
3949:
3947:
3946:
3941:
3924:
3922:
3921:
3916:
3914:
3912:
3911:
3910:
3898:
3897:
3887:
3877:
3876:
3866:
3861:
3859:
3858:
3849:
3835:
3820:
3818:
3817:
3812:
3797:
3795:
3794:
3789:
3786:
3785:
3762:
3760:
3759:
3754:
3751:
3750:
3721:
3717:
3715:
3714:
3709:
3707:
3706:
3686:
3684:
3683:
3678:
3676:
3674:
3673:
3668:
3664:
3663:
3662:
3661:
3642:
3632:
3631:
3620:
3619:
3616:
3611:
3585:
3584:
3539:
3501:
3499:
3498:
3493:
3491:
3490:
3479:
3478:
3476:
3462:
3452:
3451:
3446:
3445:
3442:
3437:
3433:
3426:
3424:
3413:
3408:
3407:
3397:
3365:
3363:
3362:
3357:
3355:
3353:
3352:
3351:
3338:
3297:
3290:
3289:
3288:
3277:
3259:
3245:
3231:
3230:
3228:
3227:
3221:
3218:
3192:
3190:
3189:
3184:
3182:
3181:
3180:
3146:
3145:
3108:
3086:
3079:
3078:
3077:
3038:
3021:
2995:
2993:
2992:
2987:
2985:
2984:
2975:
2971:
2956:
2955:
2954:
2938:
2934:
2922:
2918:
2913:
2912:
2907:
2903:
2901:
2900:
2899:
2886:
2862:
2857:
2856:
2847:
2835:
2825:
2801:
2775:
2773:
2772:
2767:
2765:
2763:
2762:
2761:
2752:
2751:
2750:
2733:
2723:
2722:
2717:
2716:
2713:
2708:
2703:
2702:
2690:
2689:
2688:
2675:
2674:
2649:
2634:
2627:
2609:
2599:
2597:
2596:
2591:
2589:
2587:
2586:
2585:
2584:
2564:
2563:
2554:
2552:
2551:
2547:
2538:
2534:
2532:
2525:
2501:
2491:
2490:
2489:
2462:
2460:
2459:
2454:
2449:
2425:
2423:
2422:
2417:
2415:
2413:
2412:
2411:
2401:
2400:
2399:
2389:
2387:
2386:
2382:
2373:
2369:
2367:
2350:
2340:
2339:
2338:
2318:
2278:
2241:
2239:
2238:
2233:
2228:
2226:
2225:
2224:
2215:
2214:
2213:
2196:
2182:
2177:
2176:
2175:
2143:
2129:
2100:
2098:
2097:
2092:
2087:
2085:
2084:
2083:
2074:
2073:
2057:
2047:
2046:
2041:
2040:
2037:
2032:
2027:
2025:
2017:
2016:
2015:
2002:
1991:
1990:
1989:
1936:
1934:
1933:
1928:
1923:
1921:
1920:
1915:
1911:
1910:
1909:
1906:
1891:
1881:
1880:
1869:
1868:
1865:
1860:
1852:
1851:
1850:
1802:
1800:
1799:
1794:
1792:
1790:
1789:
1788:
1763:
1758:
1756:
1755:
1754:
1745:
1744:
1725:
1713:
1702:
1684:
1682:
1681:
1676:
1674:
1672:
1671:
1662:
1654:
1649:
1647:
1646:
1645:
1636:
1635:
1634:
1614:
1602:
1573:
1571:
1570:
1565:
1555:
1554:
1551:
1546:
1545:
1535:
1534:
1518:
1515:
1513:
1496:
1468:
1466:
1465:
1460:
1442:
1438:
1437:
1436:
1400:
1395:
1383:
1382:
1381:
1361:
1359:
1358:
1353:
1341:
1339:
1338:
1333:
1331:
1330:
1240:
1238:
1237:
1232:
1230:
1228:
1224:
1223:
1222:
1208:
1183:
1181:
1180:
1175:
1173:
1172:
1088:
1086:
1085:
1080:
1075:
1071:
1070:
1069:
1068:
1067:
1027:
1026:
1016:
1001:
1000:
935:, is called the
887:phase transition
846:, also known as
703:
696:
689:
676:
671:
670:
663:
659:
658:
269:Spin Hall effect
159:Phase transition
129:Luttinger liquid
66:States of matter
49:Phase transition
35:
21:
20:
5342:
5341:
5337:
5336:
5335:
5333:
5332:
5331:
5297:
5296:
5295:
5256:
5211:
5206:
5205:
5198:
5182:
5178:
5121:
5117:
5060:
5056:
5008:
5004:
4997:
4981:
4977:
4972:
4960:
4897:
4891:
4887:
4885:
4881:
4846:
4843:
4842:
4795:
4792:
4791:
4745:
4744:
4740:
4727:
4719:
4717:
4708:
4704:
4702:
4699:
4698:
4659:
4651:
4649:
4641:
4638:
4637:
4608:
4604:
4598:
4594:
4593:
4579:
4577:
4562:
4558:
4544:
4542:
4538:
4527:
4524:
4523:
4502:
4499:
4498:
4465:
4462:
4461:
4425:
4422:
4421:
4420:
4396:
4392:
4378:
4374:
4373:
4359:
4357:
4346:
4343:
4342:
4324:
4320:
4303:
4281:
4279:
4274:
4271:
4270:
4252:
4248:
4231:
4207:
4201:
4200:
4199:
4198:
4196:
4191:
4188:
4187:
4164:
4142:
4140:
4138:
4135:
4134:
4133:
4109:
4106:
4105:
4086:
4082:
4077:
4074:
4073:
4055:
4051:
4034:
4018:
4012:
4011:
4010:
4009:
4007:
4002:
3999:
3998:
3974:
3970:
3968:
3960:
3957:
3956:
3955:
3931:
3928:
3927:
3903:
3899:
3893:
3889:
3888:
3872:
3868:
3867:
3865:
3854:
3850:
3836:
3834:
3829:
3826:
3825:
3806:
3803:
3802:
3781:
3777:
3769:
3766:
3765:
3746:
3742:
3734:
3731:
3730:
3702:
3698:
3696:
3693:
3692:
3669:
3657:
3656:
3652:
3648:
3644:
3643:
3621:
3615:
3614:
3613:
3612:
3610:
3580:
3576:
3568:
3565:
3564:
3558:
3534:
3486:
3482:
3463:
3447:
3441:
3440:
3439:
3438:
3436:
3414:
3403:
3399:
3398:
3396:
3388:
3385:
3384:
3347:
3346:
3342:
3337:
3329:
3326:
3325:
3304:
3292:
3286:
3284:
3279:
3272:
3265:
3257:
3247:
3240:
3222:
3219:
3214:
3208:
3207:
3205:
3203:
3197:
3176:
3175:
3171:
3141:
3137:
3129:
3126:
3125:
3106:
3096:
3081:
3075:
3073:
3068:
3063:. The value of
3053:
3037:
3027:
3024:compressibility
3020:
3010:
3002:
2980:
2979:
2969:
2967:
2961:
2960:
2950:
2949:
2945:
2935: and
2932:
2916:
2914:
2908:
2895:
2894:
2890:
2885:
2881:
2880:
2867:
2866:
2852:
2848:
2846:
2844:
2841:
2840:
2833:
2827:
2824:
2814:
2812:
2802:, the value of
2800:
2790:
2785:
2757:
2753:
2746:
2745:
2741:
2734:
2718:
2712:
2711:
2710:
2709:
2707:
2698:
2694:
2684:
2683:
2679:
2670:
2666:
2658:
2655:
2654:
2644:
2641:
2629:
2622:
2616:
2604:
2580:
2579:
2575:
2565:
2559:
2555:
2553:
2543:
2539:
2521:
2505:
2500:
2496:
2495:
2485:
2484:
2480:
2478:
2475:
2474:
2469:
2445:
2434:
2431:
2430:
2407:
2406:
2402:
2395:
2394:
2390:
2388:
2378:
2374:
2354:
2349:
2345:
2344:
2334:
2333:
2329:
2327:
2324:
2323:
2317:
2311:
2304:
2298:
2296:
2285:
2264:
2258:
2220:
2216:
2209:
2208:
2204:
2197:
2183:
2181:
2159:
2158:
2154:
2152:
2149:
2148:
2138:
2136:
2124:
2079:
2075:
2069:
2065:
2058:
2042:
2036:
2035:
2034:
2033:
2031:
2018:
2011:
2007:
2003:
2001:
1985:
1984:
1980:
1978:
1975:
1974:
1964:
1945:
1916:
1905:
1901:
1897:
1893:
1892:
1870:
1864:
1863:
1862:
1861:
1859:
1846:
1845:
1841:
1839:
1836:
1835:
1827:
1784:
1780:
1767:
1762:
1750:
1746:
1740:
1736:
1729:
1724:
1722:
1719:
1718:
1708:
1697:
1688:where Ī is the
1667:
1663:
1655:
1653:
1641:
1637:
1630:
1629:
1625:
1618:
1613:
1611:
1608:
1607:
1597:
1587:
1581:is a constant,
1547:
1541:
1540:
1523:
1519:
1517:
1500:
1495:
1484:
1481:
1480:
1472:The degeneracy
1426:
1422:
1412:
1408:
1396:
1391:
1377:
1376:
1372:
1370:
1367:
1366:
1347:
1344:
1343:
1326:
1322:
1320:
1317:
1316:
1293:
1250:
1218:
1217:
1213:
1212:
1207:
1199:
1196:
1195:
1165:
1161:
1138:
1135:
1134:
1118:
1109:
1100:
1063:
1059:
1052:
1048:
1038:
1034:
1022:
1018:
1012:
996:
995:
975:
972:
971:
965:grand potential
957:
910:charge carriers
755:
743:Albert Einstein
719:phase of matter
707:
666:
653:
652:
645:
644:
643:
433:
425:
424:
423:
399:Amorphous solid
393:
383:
382:
381:
360:
342:
332:
331:
330:
319:
317:Antiferromagnet
310:
308:Superparamagnet
301:
288:
287:Magnetic phases
280:
279:
278:
258:
250:
249:
248:
178:
170:
169:
168:
154:Order parameter
148:
147:Phase phenomena
140:
139:
138:
68:
58:
17:
12:
11:
5:
5340:
5330:
5329:
5324:
5322:Thermodynamics
5319:
5314:
5309:
5294:
5293:
5267:(6): 625ā631.
5249:
5240:
5231:
5222:
5212:
5210:
5207:
5204:
5203:
5196:
5176:
5131:(7): 661ā669.
5115:
5070:(3): 156ā161.
5054:
5002:
4995:
4974:
4973:
4971:
4968:
4967:
4966:
4959:
4956:
4948:Chen-Ning Yang
4928:
4927:
4915:
4909:
4906:
4903:
4900:
4894:
4890:
4884:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4836:
4835:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4785:
4784:
4773:
4770:
4766:
4763:
4760:
4757:
4754:
4748:
4743:
4739:
4733:
4730:
4725:
4722:
4716:
4711:
4707:
4692:
4691:
4680:
4677:
4674:
4671:
4665:
4662:
4657:
4654:
4648:
4645:
4628:
4627:
4611:
4607:
4601:
4597:
4591:
4588:
4585:
4582:
4576:
4572:
4565:
4561:
4556:
4553:
4550:
4547:
4541:
4537:
4534:
4531:
4521:
4509:
4506:
4496:
4484:
4481:
4478:
4475:
4472:
4469:
4459:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4416:
4415:
4399:
4395:
4389:
4385:
4381:
4377:
4371:
4368:
4365:
4362:
4356:
4353:
4350:
4340:
4327:
4323:
4315:
4312:
4309:
4306:
4301:
4298:
4294:
4290:
4287:
4284:
4278:
4268:
4255:
4251:
4243:
4240:
4237:
4234:
4229:
4226:
4223:
4218:
4214:
4210:
4195:
4185:
4171:
4168:
4163:
4160:
4155:
4151:
4148:
4145:
4129:
4128:
4116:
4113:
4103:
4089:
4085:
4081:
4071:
4058:
4054:
4046:
4043:
4040:
4037:
4032:
4029:
4026:
4021:
4006:
3996:
3982:
3977:
3973:
3967:
3964:
3954:Vapor fraction
3951:
3950:
3938:
3935:
3925:
3909:
3906:
3902:
3896:
3892:
3886:
3883:
3880:
3875:
3871:
3864:
3857:
3853:
3848:
3845:
3842:
3839:
3833:
3823:
3821:
3810:
3799:
3798:
3784:
3780:
3776:
3773:
3763:
3749:
3745:
3741:
3738:
3728:
3725:
3705:
3701:
3688:
3687:
3672:
3667:
3660:
3655:
3651:
3647:
3641:
3638:
3635:
3630:
3627:
3624:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3583:
3579:
3575:
3572:
3557:
3556:Thermodynamics
3554:
3503:
3502:
3489:
3485:
3475:
3472:
3469:
3466:
3461:
3458:
3455:
3450:
3432:
3429:
3423:
3420:
3417:
3412:
3406:
3402:
3395:
3392:
3367:
3366:
3350:
3345:
3341:
3336:
3333:
3302:
3270:
3255:
3212:
3201:
3194:
3193:
3179:
3174:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3144:
3140:
3136:
3133:
3104:
3052:
3049:
3035:
3018:
3001:
2998:
2997:
2996:
2983:
2978:
2968:
2966:
2963:
2962:
2959:
2953:
2948:
2944:
2941:
2931:
2928:
2925:
2915:
2911:
2906:
2898:
2893:
2889:
2884:
2879:
2876:
2873:
2872:
2870:
2865:
2860:
2855:
2851:
2831:
2822:
2810:
2798:
2783:
2777:
2776:
2760:
2756:
2749:
2744:
2740:
2737:
2732:
2729:
2726:
2721:
2706:
2701:
2697:
2693:
2687:
2682:
2678:
2673:
2669:
2665:
2662:
2640:
2637:
2614:
2601:
2600:
2583:
2578:
2574:
2571:
2568:
2562:
2558:
2550:
2546:
2542:
2537:
2531:
2528:
2524:
2520:
2517:
2514:
2511:
2508:
2504:
2499:
2494:
2488:
2483:
2467:
2452:
2448:
2444:
2441:
2438:
2427:
2426:
2410:
2405:
2398:
2393:
2385:
2381:
2377:
2372:
2366:
2363:
2360:
2357:
2353:
2348:
2343:
2337:
2332:
2315:
2309:
2302:
2294:
2283:
2260:
2243:
2242:
2231:
2223:
2219:
2212:
2207:
2203:
2200:
2195:
2192:
2189:
2186:
2180:
2174:
2171:
2168:
2165:
2162:
2157:
2134:
2102:
2101:
2090:
2082:
2078:
2072:
2068:
2064:
2061:
2056:
2053:
2050:
2045:
2030:
2024:
2021:
2014:
2010:
2006:
2000:
1997:
1994:
1988:
1983:
1963:
1960:
1941:
1938:
1937:
1926:
1919:
1914:
1904:
1900:
1896:
1890:
1887:
1884:
1879:
1876:
1873:
1858:
1855:
1849:
1844:
1825:
1804:
1803:
1787:
1783:
1779:
1776:
1773:
1770:
1766:
1761:
1753:
1749:
1743:
1739:
1735:
1732:
1728:
1686:
1685:
1670:
1666:
1661:
1658:
1652:
1644:
1640:
1633:
1628:
1624:
1621:
1617:
1594:gamma function
1585:
1575:
1574:
1563:
1560:
1550:
1544:
1539:
1533:
1530:
1527:
1522:
1512:
1509:
1506:
1503:
1499:
1494:
1491:
1488:
1470:
1469:
1458:
1455:
1452:
1449:
1446:
1441:
1435:
1432:
1429:
1425:
1421:
1418:
1415:
1411:
1407:
1404:
1399:
1394:
1390:
1386:
1380:
1375:
1351:
1329:
1325:
1292:
1289:
1248:
1242:
1241:
1227:
1221:
1216:
1211:
1206:
1203:
1185:
1184:
1171:
1168:
1164:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1114:
1105:
1096:
1090:
1089:
1078:
1074:
1066:
1062:
1058:
1055:
1051:
1047:
1044:
1041:
1037:
1033:
1030:
1025:
1021:
1015:
1011:
1007:
1004:
999:
994:
991:
988:
985:
982:
979:
956:
953:
945:number density
810:quasiparticles
796:, the atom of
754:
751:
709:
708:
706:
705:
698:
691:
683:
680:
679:
678:
677:
664:
647:
646:
642:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
526:
521:
516:
511:
506:
501:
496:
491:
486:
481:
476:
471:
466:
461:
456:
451:
446:
441:
435:
434:
431:
430:
427:
426:
422:
421:
416:
414:Liquid crystal
411:
406:
401:
395:
394:
389:
388:
385:
384:
380:
379:
374:
369:
364:
355:
350:
344:
343:
340:Quasiparticles
338:
337:
334:
333:
329:
328:
323:
314:
305:
299:Superdiamagnet
296:
290:
289:
286:
285:
282:
281:
277:
276:
271:
266:
260:
259:
256:
255:
252:
251:
247:
246:
241:
236:
231:
226:
224:Thermoelectric
221:
219:Superconductor
216:
211:
206:
201:
199:Mott insulator
196:
191:
186:
180:
179:
176:
175:
172:
171:
167:
166:
161:
156:
150:
149:
146:
145:
142:
141:
137:
136:
131:
126:
121:
116:
111:
106:
101:
96:
91:
86:
81:
76:
70:
69:
64:
63:
60:
59:
57:
56:
51:
46:
40:
37:
36:
28:
27:
15:
9:
6:
4:
3:
2:
5339:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5304:
5302:
5290:
5286:
5282:
5278:
5274:
5270:
5266:
5262:
5255:
5250:
5246:
5241:
5237:
5232:
5228:
5223:
5219:
5214:
5213:
5199:
5197:9780521586467
5193:
5189:
5188:
5180:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5144:
5139:
5134:
5130:
5126:
5119:
5111:
5107:
5103:
5099:
5095:
5091:
5087:
5083:
5078:
5073:
5069:
5065:
5058:
5050:
5046:
5042:
5038:
5034:
5030:
5026:
5022:
5018:
5014:
5006:
4998:
4992:
4988:
4987:
4979:
4975:
4965:
4962:
4961:
4955:
4953:
4949:
4945:
4941:
4937:
4933:
4913:
4904:
4898:
4892:
4888:
4882:
4878:
4875:
4872:
4866:
4863:
4860:
4854:
4851:
4848:
4841:
4840:
4839:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4790:
4789:
4788:
4771:
4768:
4761:
4758:
4755:
4741:
4737:
4731:
4723:
4714:
4709:
4705:
4697:
4696:
4695:
4678:
4675:
4672:
4669:
4663:
4646:
4643:
4636:
4635:
4634:
4609:
4605:
4599:
4595:
4586:
4580:
4574:
4570:
4563:
4559:
4551:
4545:
4539:
4535:
4532:
4529:
4507:
4504:
4479:
4473:
4470:
4467:
4442:
4436:
4433:
4430:
4427:
4417:
4397:
4393:
4387:
4383:
4379:
4375:
4366:
4360:
4354:
4351:
4348:
4325:
4321:
4310:
4304:
4296:
4292:
4288:
4282:
4276:
4253:
4249:
4238:
4232:
4224:
4216:
4212:
4208:
4193:
4169:
4161:
4158:
4153:
4149:
4146:
4143:
4130:
4114:
4111:
4087:
4083:
4079:
4056:
4052:
4041:
4035:
4027:
4019:
4004:
3980:
3975:
3971:
3965:
3962:
3952:
3936:
3933:
3907:
3904:
3900:
3894:
3890:
3881:
3873:
3869:
3862:
3855:
3851:
3843:
3837:
3831:
3808:
3800:
3782:
3778:
3774:
3771:
3747:
3743:
3739:
3736:
3722:
3719:
3703:
3699:
3670:
3665:
3653:
3649:
3645:
3636:
3628:
3625:
3622:
3607:
3601:
3598:
3595:
3589:
3586:
3581:
3577:
3573:
3563:
3562:
3561:
3553:
3549:
3547:
3543:
3537:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3487:
3483:
3470:
3464:
3456:
3448:
3430:
3427:
3421:
3418:
3415:
3410:
3404:
3400:
3393:
3390:
3383:
3382:
3381:
3378:
3376:
3372:
3343:
3339:
3334:
3331:
3324:
3323:
3322:
3320:
3316:
3312:
3308:
3301:
3295:
3282:
3276:
3269:
3263:
3254:
3250:
3243:
3237:
3235:
3226:
3217:
3211:
3200:
3168:
3162:
3159:
3156:
3150:
3147:
3142:
3138:
3134:
3124:
3123:
3122:
3120:
3116:
3113:For smaller,
3103:
3099:
3094:
3090:
3084:
3071:
3066:
3062:
3057:
3048:
3044:
3042:
3034:
3030:
3025:
3017:
3013:
3008:
2976:
2964:
2957:
2946:
2942:
2939:
2929:
2926:
2923:
2909:
2904:
2891:
2887:
2882:
2877:
2874:
2868:
2863:
2858:
2853:
2849:
2839:
2838:
2837:
2830:
2821:
2817:
2809:
2805:
2797:
2793:
2787:
2782:
2758:
2742:
2738:
2727:
2719:
2704:
2699:
2695:
2691:
2680:
2676:
2671:
2667:
2663:
2660:
2653:
2652:
2651:
2647:
2636:
2632:
2625:
2620:
2613:
2607:
2576:
2572:
2569:
2566:
2560:
2556:
2548:
2544:
2540:
2535:
2526:
2522:
2518:
2512:
2509:
2506:
2502:
2497:
2492:
2481:
2473:
2472:
2471:
2466:
2450:
2446:
2442:
2439:
2436:
2403:
2391:
2383:
2379:
2375:
2370:
2361:
2355:
2351:
2346:
2341:
2330:
2322:
2321:
2320:
2314:
2308:
2301:
2293:
2289:
2282:
2276:
2272:
2268:
2263:
2256:
2252:
2248:
2229:
2221:
2205:
2201:
2190:
2184:
2178:
2163:
2155:
2147:
2146:
2145:
2141:
2137:is finite at
2133:
2127:
2121:
2119:
2115:
2111:
2107:
2088:
2080:
2070:
2066:
2062:
2051:
2043:
2028:
2022:
2012:
1998:
1995:
1992:
1981:
1973:
1972:
1971:
1969:
1959:
1955:
1953:
1952:polylogarithm
1949:
1944:
1924:
1917:
1912:
1902:
1898:
1894:
1885:
1877:
1874:
1871:
1856:
1853:
1834:
1833:
1832:
1829:
1824:
1820:
1817:
1813:
1809:
1785:
1777:
1774:
1764:
1759:
1751:
1741:
1737:
1733:
1726:
1717:
1716:
1715:
1711:
1707:we will have
1706:
1700:
1695:
1691:
1668:
1659:
1656:
1650:
1642:
1626:
1622:
1615:
1606:
1605:
1604:
1600:
1595:
1591:
1584:
1580:
1561:
1558:
1548:
1537:
1531:
1528:
1525:
1520:
1507:
1497:
1492:
1489:
1486:
1479:
1478:
1477:
1475:
1456:
1450:
1447:
1444:
1439:
1433:
1430:
1427:
1423:
1419:
1416:
1413:
1409:
1405:
1402:
1392:
1388:
1384:
1365:
1364:
1363:
1327:
1314:
1310:
1302:
1297:
1288:
1285:
1280:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1247:
1225:
1214:
1209:
1204:
1201:
1194:
1193:
1192:
1190:
1169:
1166:
1162:
1158:
1152:
1149:
1146:
1140:
1133:
1132:
1131:
1130:by defining:
1129:
1126:
1122:
1117:
1113:
1108:
1104:
1099:
1095:
1076:
1072:
1064:
1060:
1056:
1053:
1049:
1045:
1042:
1039:
1035:
1031:
1028:
1023:
1019:
1013:
1009:
1005:
989:
986:
983:
980:
970:
969:
968:
966:
962:
952:
950:
946:
942:
938:
934:
930:
926:
921:
919:
915:
911:
907:
902:
900:
896:
892:
888:
884:
880:
876:
872:
871:absolute zero
868:
863:
861:
860:heat capacity
857:
853:
849:
845:
843:
838:
834:
830:
826:
821:
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
750:
748:
744:
740:
736:
732:
728:
724:
720:
716:
704:
699:
697:
692:
690:
685:
684:
682:
681:
675:
665:
662:
657:
651:
650:
649:
648:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
445:
442:
440:
439:Van der Waals
437:
436:
429:
428:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
396:
392:
387:
386:
378:
375:
373:
370:
368:
365:
363:
359:
356:
354:
351:
349:
346:
345:
341:
336:
335:
327:
324:
322:
318:
315:
313:
309:
306:
304:
300:
297:
295:
292:
291:
284:
283:
275:
272:
270:
267:
265:
262:
261:
254:
253:
245:
242:
240:
237:
235:
234:Ferroelectric
232:
230:
229:Piezoelectric
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
204:Semiconductor
202:
200:
197:
195:
192:
190:
187:
185:
182:
181:
174:
173:
165:
162:
160:
157:
155:
152:
151:
144:
143:
135:
132:
130:
127:
125:
124:Superfluidity
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
75:
72:
71:
67:
62:
61:
55:
52:
50:
47:
45:
42:
41:
39:
38:
34:
30:
29:
26:
23:
22:
19:
5264:
5261:Eur. J. Phys
5260:
5244:
5235:
5226:
5217:
5186:
5179:
5128:
5124:
5118:
5067:
5063:
5057:
5016:
5012:
5005:
4985:
4978:
4944:Bethe ansatz
4931:
4929:
4837:
4786:
4693:
4631:
3689:
3559:
3550:
3545:
3541:
3535:
3530:
3526:
3518:
3514:
3510:
3506:
3505:For a given
3504:
3379:
3374:
3370:
3368:
3318:
3314:
3310:
3306:
3299:
3293:
3280:
3274:
3267:
3252:
3248:
3241:
3238:
3233:
3224:
3215:
3209:
3198:
3195:
3118:
3112:
3101:
3097:
3092:
3088:
3082:
3069:
3064:
3060:
3045:
3032:
3028:
3015:
3011:
3003:
2828:
2819:
2815:
2807:
2803:
2795:
2791:
2788:
2780:
2778:
2645:
2642:
2630:
2623:
2618:
2617:diverges as
2611:
2605:
2602:
2464:
2428:
2312:
2306:
2299:
2291:
2287:
2280:
2274:
2270:
2266:
2261:
2250:
2246:
2244:
2139:
2131:
2125:
2122:
2117:
2113:
2109:
2105:
2103:
1965:
1956:
1947:
1942:
1939:
1830:
1822:
1818:
1815:
1811:
1807:
1805:
1709:
1698:
1693:
1687:
1598:
1589:
1582:
1578:
1576:
1473:
1471:
1309:gas in a box
1306:
1283:
1281:
1276:
1272:
1268:
1264:
1256:
1245:
1243:
1191:defined as:
1188:
1186:
1127:
1120:
1115:
1111:
1106:
1102:
1097:
1093:
1091:
958:
922:
914:Cooper pairs
903:
895:ground state
864:
852:normal modes
840:
833:Planck's law
822:
756:
714:
712:
569:von Klitzing
274:Kondo effect
134:Time crystal
114:Fermi liquid
98:
18:
4930:which, for
3321:defined by
1261:temperature
856:Peter Debye
848:Debye model
816:(quanta of
774:Higgs boson
391:Soft matter
312:Ferromagnet
5301:Categories
4970:References
3718:is shown.
3115:mesoscopic
2470:) we get:
1966:The total
1954:function.
879:superfluid
825:photon gas
739:photon gas
534:Louis NĆ©el
524:Schrieffer
432:Scientists
326:Spin glass
321:Metamagnet
303:Paramagnet
119:Supersolid
5312:Ideal gas
5289:250870934
5163:0002-9505
5110:119073938
5102:1612-2011
5049:118614846
5041:0022-4715
4905:α
4899:ζ
4893:α
4889:τ
4879:
4861:α
4819:−
4772:β
4756:α
4729:∂
4721:∂
4673:α
4664:β
4661:∂
4656:Ω
4653:∂
4610:α
4606:τ
4600:α
4587:α
4581:ζ
4575:−
4564:α
4560:τ
4552:α
4546:ζ
4536:
4530:≈
4474:
4437:
4398:α
4394:τ
4380:α
4367:α
4361:ζ
4355:−
4349:≈
4326:α
4322:τ
4311:α
4305:ζ
4289:α
4283:ζ
4254:α
4250:τ
4239:α
4233:ζ
4209:α
4167:Ω
4162:−
4150:β
4088:α
4084:τ
4057:α
4053:τ
4042:α
4036:ζ
4020:α
3966:−
3908:α
3901:τ
3895:α
3882:α
3870:ζ
3863:−
3856:α
3852:τ
3844:α
3838:ζ
3832:≈
3775:≫
3740:≪
3724:Quantity
3704:α
3700:τ
3671:α
3650:β
3623:α
3608:−
3599:−
3590:
3571:Ω
3488:α
3484:τ
3471:α
3465:ζ
3449:α
3419:−
3332:τ
3173:Ω
3160:−
3151:
3132:Ω
2972:otherwise
2924:α
2910:α
2878:−
2759:α
2739:β
2720:α
2570:π
2513:ζ
2437:α
2384:α
2362:α
2356:ζ
2253:) is the
2222:α
2202:β
2191:α
2185:ζ
2081:α
2063:β
2044:α
2020:∂
2009:Ω
2005:∂
1996:−
1950:) is the
1918:α
1899:β
1872:α
1857:−
1843:Ω
1778:β
1775:ω
1772:ℏ
1752:α
1734:β
1665:Λ
1643:α
1623:β
1549:α
1529:−
1526:α
1508:α
1502:Γ
1454:Ω
1451:≈
1431:β
1428:−
1417:−
1406:
1398:∞
1389:∫
1374:Ω
1350:Ω
1324:Ω
1301:Fermi gas
1202:β
1170:μ
1167:β
1153:μ
1147:β
1061:ϵ
1057:β
1054:−
1043:−
1032:
1010:∑
990:
984:−
978:Ω
949:ideal gas
937:Fermi gas
925:electrons
883:viscosity
802:deuterium
723:ideal gas
713:An ideal
634:Wetterich
614:Abrikosov
529:Josephson
499:Van Vleck
489:Luttinger
362:Polariton
294:Diamagnet
214:Conductor
209:Semimetal
194:Insulator
109:Fermi gas
4958:See also
3727:General
2919:if
1940:where Li
1590:critical
941:fermions
929:helium-3
867:helium-4
835:and the
814:plasmons
794:hydrogen
790:graviton
715:Bose gas
674:Category
619:Ginzburg
594:Laughlin
554:Kadanoff
509:Shockley
494:Anderson
449:von Laue
99:Bose gas
5269:Bibcode
5143:Bibcode
5082:Bibcode
5021:Bibcode
3246:, with
3229:
3206:
2257:(using
1275:), and
1259:is the
1251:is the
624:Leggett
599:Stƶrmer
584:Bednorz
544:Giaever
514:Bardeen
504:Hubbard
479:Peierls
469:Onsager
419:Polymer
404:Colloid
367:Polaron
358:Plasmon
353:Exciton
5287:
5194:
5171:949741
5169:
5161:
5108:
5100:
5047:
5039:
4993:
3480:
3434:
3296:= 1000
3102:ε
3089:μ
3085:= 1000
3065:α
3061:τ
2779:where
2648:> 1
2245:where
2128:> 1
1806:where
1692:, and
1577:where
1556:
1244:where
963:. The
908:where
842:phonon
806:mesons
784:, the
780:, the
778:photon
776:, the
758:Bosons
737:for a
727:bosons
672:
639:Perdew
629:Parisi
589:MĆ¼ller
579:Rohrer
574:Binnig
564:Wilson
559:Fisher
519:Cooper
484:Landau
372:Magnon
348:Phonon
189:Plasma
89:Plasma
79:Liquid
44:Phases
5285:S2CID
5257:(PDF)
5167:S2CID
5133:arXiv
5106:S2CID
5072:arXiv
5045:S2CID
3244:= 3/2
3031:<
3014:<
2818:>
2794:<
2626:= 1/2
1601:= 3/2
1588:is a
782:gluon
539:Esaki
464:Bloch
459:Debye
454:Bragg
444:Onnes
377:Roton
74:Solid
5192:ISBN
5159:ISSN
5098:ISSN
5037:ISSN
4991:ISBN
3509:and
3278:for
3266:1 ā
3223:1 ā
2943:<
2927:>
2628:and
2603:For
2305:= 1/
2269:) =
2123:For
1814:) =
1271:(or
1255:and
1187:and
829:Bose
770:spin
760:are
609:Tsui
604:Yang
549:Kohn
474:Mott
5277:doi
5151:doi
5090:doi
5029:doi
5017:161
3538:= 0
3287:000
3258:= 1
3107:= 1
3076:000
2834:= 0
2633:= 1
2608:ā¤ 1
2142:= 1
1712:= 3
1701:= 1
927:or
844:gas
820:).
786:W/Z
164:QCP
84:Gas
54:QCP
5303::
5283:.
5275:.
5265:21
5263:.
5259:.
5165:.
5157:.
5149:.
5141:.
5129:71
5127:.
5104:.
5096:.
5088:.
5080:.
5066:.
5043:.
5035:.
5027:.
5015:.
4954:.
4876:ln
4533:ln
4471:ln
4434:ln
4203:Li
4014:Li
3617:Li
3587:ln
3443:Li
3285:10
3283:=
3251:=
3204:=
3148:ln
3100:=
3074:10
3072:=
2714:Li
2259:Li
2144::
2038:Li
1866:Li
1816:mĻ
1474:dg
1403:ln
1362::
1267:,
1119:;
1101:;
1029:ln
987:ln
951:.
901:.
804:,
749:.
5291:.
5279::
5271::
5200:.
5173:.
5153::
5145::
5135::
5112:.
5092::
5084::
5074::
5068:2
5051:.
5031::
5023::
4999:.
4932:Ī±
4914:)
4908:)
4902:(
4883:(
4873:+
4870:)
4867:1
4864:+
4858:(
4855:=
4852:S
4849:T
4822:G
4816:V
4813:P
4810:+
4807:U
4804:=
4801:S
4798:T
4769:U
4765:)
4762:1
4759:+
4753:(
4747:B
4742:k
4738:=
4732:T
4724:U
4715:=
4710:V
4706:C
4679:V
4676:P
4670:=
4647:=
4644:U
4596:2
4590:)
4584:(
4571:)
4555:)
4549:(
4540:(
4508:0
4505:=
4483:)
4480:z
4477:(
4468:=
4446:)
4443:z
4440:(
4431:=
4428:G
4388:1
4384:+
4376:2
4370:)
4364:(
4352:1
4314:)
4308:(
4300:)
4297:1
4293:+
4286:(
4277:=
4242:)
4236:(
4228:)
4225:z
4222:(
4217:1
4213:+
4194:=
4170:N
4159:=
4154:N
4147:V
4144:P
4115:1
4112:=
4080:=
4045:)
4039:(
4031:)
4028:z
4025:(
4005:=
3981:N
3976:0
3972:N
3963:1
3937:1
3934:=
3905:2
3891:2
3885:)
3879:(
3874:2
3847:)
3841:(
3809:z
3783:c
3779:T
3772:T
3748:c
3744:T
3737:T
3666:)
3659:c
3654:E
3646:(
3640:)
3637:z
3634:(
3629:1
3626:+
3605:)
3602:z
3596:1
3593:(
3582:0
3578:g
3574:=
3546:N
3542:T
3536:T
3531:Ļ
3527:Ļ
3519:z
3515:Ļ
3511:Ļ
3507:N
3474:)
3468:(
3460:)
3457:z
3454:(
3431:N
3428:+
3422:z
3416:1
3411:z
3405:0
3401:g
3394:=
3391:N
3375:Ļ
3371:Ļ
3349:c
3344:T
3340:T
3335:=
3319:Ļ
3315:z
3311:Ī¼
3307:N
3305:/
3303:0
3300:N
3294:N
3281:N
3275:N
3273:/
3271:0
3268:N
3256:c
3253:Īµ
3249:k
3242:Ī±
3234:z
3225:z
3220:/
3216:z
3213:0
3210:g
3202:0
3199:N
3178:m
3169:+
3166:)
3163:z
3157:1
3154:(
3143:0
3139:g
3135:=
3119:Īµ
3109:.
3105:c
3098:k
3093:z
3083:N
3070:N
3036:c
3033:T
3029:T
3019:c
3016:T
3012:T
2977:.
2965:0
2958:,
2952:c
2947:T
2940:T
2930:1
2905:)
2897:c
2892:T
2888:T
2883:(
2875:1
2869:{
2864:=
2859:N
2854:0
2850:N
2832:0
2829:N
2823:c
2820:T
2816:T
2811:0
2808:N
2804:z
2799:c
2796:T
2792:T
2784:0
2781:N
2755:)
2748:c
2743:E
2736:(
2731:)
2728:z
2725:(
2705:+
2700:0
2696:N
2692:=
2686:m
2681:N
2677:+
2672:0
2668:N
2664:=
2661:N
2646:Ī±
2631:Ī±
2624:Ī±
2619:z
2615:m
2612:N
2606:Ī±
2582:B
2577:k
2573:m
2567:2
2561:2
2557:h
2549:3
2545:/
2541:2
2536:)
2530:)
2527:2
2523:/
2519:3
2516:(
2510:f
2507:V
2503:N
2498:(
2493:=
2487:c
2482:T
2468:c
2465:E
2451:2
2447:/
2443:3
2440:=
2409:B
2404:k
2397:c
2392:E
2380:/
2376:1
2371:)
2365:)
2359:(
2352:N
2347:(
2342:=
2336:c
2331:T
2316:c
2313:Ī²
2310:B
2307:k
2303:c
2300:T
2295:c
2292:Ī²
2288:Ī²
2284:m
2281:N
2277:)
2275:Ī±
2273:(
2271:Ī¶
2267:1
2265:(
2262:Ī±
2251:Ī±
2249:(
2247:Ī¶
2230:,
2218:)
2211:c
2206:E
2199:(
2194:)
2188:(
2179:=
2173:x
2170:a
2167:m
2164:,
2161:m
2156:N
2140:z
2135:m
2132:N
2126:Ī±
2118:Ī±
2114:z
2110:z
2106:z
2089:.
2077:)
2071:c
2067:E
2060:(
2055:)
2052:z
2049:(
2029:=
2023:z
2013:m
1999:z
1993:=
1987:m
1982:N
1948:x
1946:(
1943:s
1925:,
1913:)
1907:c
1903:E
1895:(
1889:)
1886:z
1883:(
1878:1
1875:+
1854:=
1848:m
1826:c
1823:E
1819:r
1812:r
1810:(
1808:V
1786:3
1782:)
1769:(
1765:f
1760:=
1748:)
1742:c
1738:E
1731:(
1727:1
1710:Ī±
1699:f
1694:f
1669:3
1660:f
1657:V
1651:=
1639:)
1632:c
1627:E
1620:(
1616:1
1599:Ī±
1586:c
1583:E
1579:Ī±
1562:E
1559:d
1543:c
1538:E
1532:1
1521:E
1511:)
1505:(
1498:1
1493:=
1490:g
1487:d
1457:.
1448:g
1445:d
1440:)
1434:E
1424:e
1420:z
1414:1
1410:(
1393:0
1385:=
1379:m
1328:m
1284:z
1277:V
1273:T
1269:Ī²
1265:z
1257:T
1249:B
1246:k
1226:T
1220:B
1215:k
1210:1
1205:=
1189:Ī²
1163:e
1159:=
1156:)
1150:,
1144:(
1141:z
1128:Ī¼
1121:z
1116:i
1112:Īµ
1107:i
1103:g
1098:i
1094:Īµ
1077:.
1073:)
1065:i
1050:e
1046:z
1040:1
1036:(
1024:i
1020:g
1014:i
1006:=
1003:)
998:Z
993:(
981:=
875:K
798:O
702:e
695:t
688:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
