646:
2640:: 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:
1286:
33:
659:
3046:
2984:
3541:
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,
3036:
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
1947:
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,
2994:
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
4622:
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
3680:
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
4615:
3037:
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.
2089:
3913:
2832:
3675:
1925:
2414:
3490:
5000:
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".
2764:
2588:
1457:
1077:
1562:
4256:
1791:
4915:
1293:, 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.
2230:
4059:
1276:
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).
4772:
4403:
1673:
4328:
2309:, 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:
1305:, 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
4679:
4515:
1966:
4173:
3181:
2979:{\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}}}
3817:
1229:
3556:
3354:
1827:
1172:
2315:
3376:
3984:
2646:
4091:
2466:
1358:
3706:
3542:
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.
963:
3786:
3751:
1472:
4823:
2451:
1330:
4447:
4484:
3367:
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.
1350:
3107:, 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
1253:. 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
4179:
4509:
4116:
3938:
1710:
4834:
3809:
2140:
3990:
3512:
4690:
735:, 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
4334:
1599:
4262:
690:
1948:
even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the uncondensed portion of the gas.
4629:
3077:. Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential
4610:{\displaystyle \approx \ln \left({\frac {\zeta (\alpha )}{\tau ^{\alpha }}}\right)-{\frac {\zeta (\alpha )}{2^{\alpha }\tau ^{\alpha }}}}
3012:
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
2084:{\displaystyle N_{\rm {m}}=-z{\frac {\partial \Omega _{m}}{\partial z}}={\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{c})^{\alpha }}}.}
5113:
Mullin, W. J.; FernĆ”ndez, J. P. (2003). "BoseāEinstein condensation, fluctuations, and recurrence relations in statistical mechanics".
4126:
907:
683:
5296:
4984:
3908:{\displaystyle \approx {\frac {\zeta (\alpha )}{\tau ^{\alpha }}}-{\frac {\zeta ^{2}(\alpha )}{2^{\alpha }\tau ^{2\alpha }}}}
3229:
This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for
3117:
5175:
4940:. In one dimensional case correlation functions also were evaluated. In one dimension Bose gas is equivalent to quantum
1269:. All partial derivatives are taken with respect to one of these three variables while the other two are held constant.
3670:{\displaystyle \Omega =g_{0}\ln(1-z)-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\rm {c}}\right)^{\alpha }}}}
4941:
1920:{\displaystyle \Omega _{\rm {m}}=-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\text{c}}\right)^{\alpha }}},}
1187:
5185:
676:
663:
3317:
2409:{\displaystyle T_{\rm {c}}=\left({\frac {N}{\zeta (\alpha )}}\right)^{1/\alpha }{\frac {E_{\rm {c}}}{k_{\rm {B}}}}}
1126:
3485:{\displaystyle N={\frac {g_{0}\,z}{1-z}}+N~{\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}~\tau ^{\alpha }}
645:
4974:
438:
2759:{\displaystyle N=N_{0}+N_{\rm {m}}=N_{0}+{\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{\rm {c}})^{\alpha }}}}
3948:
5316:
2583:{\displaystyle T_{\rm {c}}=\left({\frac {N}{Vf\zeta (3/2)}}\right)^{2/3}{\frac {h^{2}}{2\pi mk_{\rm {B}}}}}
880:
755:
736:
720:
613:
93:
4925:
1452:{\displaystyle \Omega _{\rm {m}}=\int _{0}^{\infty }\ln \left(1-ze^{-\beta E}\right)\,dg\approx \Omega .}
4065:
2110:(i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).
1072:{\displaystyle \Omega =-\ln({\mathcal {Z}})=\sum _{i}g_{i}\ln \left(1-ze^{-\beta \epsilon _{i}}\right).}
829:. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The
5306:
5052:
Yukalov, V I (2005-03-01). "Number-of-particle fluctuations in systems with Bose-Einstein condensate".
922:
618:
243:
4929:
3684:
1557:{\displaystyle dg={\frac {1}{\Gamma (\alpha )}}\,{\frac {E^{\,\alpha -1}}{E_{\rm {c}}^{\alpha }}}~dE}
508:
183:
2861:
3370:
The equation for the number of particles can be written in terms of the normalized temperature as:
950:
503:
498:
24:
3757:
3722:
588:
5311:
4953:
4783:
2422:
1308:
193:
4413:
598:
4453:
2996:
1694:
583:
523:
493:
443:
163:
53:
5243:
4251:{\displaystyle ={\frac {{\textrm {Li}}_{\alpha \!+\!1}(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
2244:
1335:
844:
of vibration of the crystal lattice of a metal, can be treated as effective massless bosons.
623:
238:
223:
5258:
5132:
5071:
5010:
1786:{\displaystyle {\frac {1}{(\beta E_{c})^{\alpha }}}={\frac {f}{(\hbar \omega \beta )^{3}}}}
826:
213:
103:
4910:{\displaystyle TS=(\alpha +1)+\ln \left({\frac {\tau ^{\alpha }}{\zeta (\alpha )}}\right)}
4490:
4097:
3919:
1951:
8:
2225:{\displaystyle N_{\rm {m,max}}={\frac {\zeta (\alpha )}{(\beta E_{\rm {c}})^{\alpha }}},}
894:
The theory of Bose-Einstein condensates and Bose gases can also explain some features of
868:
818:
724:
453:
263:
113:
5262:
5136:
5075:
5014:
4054:{\displaystyle ={\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
5301:
5274:
5156:
5122:
5095:
5061:
5034:
3794:
3104:
3030:
1679:
1242:
1114:
807:
593:
568:
316:
307:
4767:{\displaystyle C_{V}={\frac {\partial U}{\partial T}}=k_{\rm {B}}(\alpha +1)\,U\beta }
1113:
is the absolute activity (or "fugacity"), which may also be expressed in terms of the
5278:
5270:
5181:
5148:
5099:
5087:
5038:
5026:
4980:
4398:{\displaystyle \approx 1-{\frac {\zeta (\alpha )}{2^{\alpha \!+\!1}\tau ^{\alpha }}}}
1668:{\displaystyle {\frac {1}{(\beta E_{\rm {c}})^{\alpha }}}={\frac {Vf}{\Lambda ^{3}}}}
895:
888:
751:
563:
408:
298:
218:
3363:
in the limit of low temperature and, except for the chemical potential, linear in 1/
2826:. This approach gives the fraction of condensed particles in the macroscopic limit:
1082:
where each term in the sum corresponds to a particular single-particle energy level
5266:
5140:
5079:
5018:
3033:, which fixes the total particle number, however the calculations are not as easy.
876:
268:
233:
228:
188:
158:
128:
88:
48:
5160:
4323:{\displaystyle ={\frac {\zeta (\alpha \!+\!1)}{\zeta (\alpha )}}\,\tau ^{\alpha }}
937:
and high temperature, both the Fermi gas and the Bose gas behave like a classical
3013:
1957:
954:
822:
732:
708:
578:
528:
398:
153:
65:
2611:
approaches 1), and thus for example for a gas in a one- or two-dimensional box (
1285:
4937:
4928:. In one dimension bosons with delta interaction behave as fermions, they obey
1583:
934:
899:
775:
759:
650:
628:
608:
603:
558:
478:
413:
311:
198:
43:
5022:
3049:
Figure 1: Various Bose gas parameters as a function of normalized temperature
5290:
5152:
5091:
5030:
2995:
fluctuation catastrophe: the number of particles in any given state follow a
2120:, the number of particles only increases up to a finite maximum value, i.e.,
1941:
906:) and behave like bosons. As a result, superconductors behave like having no
860:
849:
799:
339:
320:
302:
203:
123:
4932:. In one dimension Bose gas with delta interaction can be solved exactly by
3681:
approximation symbol indicates that only the first few terms of a series in
533:
5083:
4933:
3538:
approaches infinity, which can be easily determined from these expansions.
3251:
3222:. Now, the behaviour is smooth when crossing the critical temperature, and
1302:
1298:
884:
553:
543:
513:
473:
468:
448:
293:
273:
133:
3710:
5127:
5066:
1250:
903:
845:
841:
837:
763:
573:
548:
518:
463:
458:
390:
32:
875:. The Bose gas is the most simple quantitative model that explains this
4936:. The bulk free energy and thermodynamic potentials were calculated by
4674:{\displaystyle U={\frac {\partial \Omega }{\partial \beta }}=\alpha PV}
1952:
Limit on number of particles in uncondensed phase, critical temperature
1466:
may be expressed for many different situations by the general formula:
883:, a state where a large number of bosons occupy the lowest energy, the
814:
728:
483:
325:
118:
5144:
3534:
approaches infinity. In particular, we are interested in the limit as
3299:. The red lines plot values of the negative of the chemical potential
3045:
1290:
1289:
Pressure vs temperature curves of classical and quantum ideal gases (
1280:
949:
The thermodynamics of an ideal Bose gas is best calculated using the
938:
926:
872:
791:
712:
538:
488:
361:
208:
108:
859:
atoms. When a system of He atoms is cooled down to temperature near
4684:
A similar situation holds for the specific heat at constant volume
918:
914:
856:
783:
779:
3523:. From these expansions, we can find the behavior of the gas near
930:
803:
762:. These particles can be classified as elementary: these are the
418:
403:
366:
357:
352:
5174:
Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1997-03-06).
2989:
813:
The first model that treated a gas with several bosons, was the
1821:
This integral expression for the grand potential evaluates to:
864:
863:, many quantum mechanical effects are present. Below 2.17
831:
802:
in more complex systems can also be considered bosons like the
787:
767:
716:
371:
347:
78:
3359:
It can be seen that each of these parameters become linear in
4999:
4168:{\displaystyle {\frac {PV\beta }{N}}=-{\frac {\Omega }{N}}\,}
795:
771:
747:
376:
73:
5177:
Quantum
Inverse Scattering Method and Correlation Functions
2972:
2776:
is the number of particles in the ground state condensate.
2419:
For example, for the three-dimensional Bose gas in a box (
3288:. The blue lines are the fraction of condensed particles
3254:. The solid black line is the fraction of excited states
879:. Mainly when a gas of bosons is cooled down, it forms a
83:
5173:
723:. The statistical mechanics of bosons were developed by
5009:(4). Springer Science and Business Media LLC: 942ā964.
3176:{\displaystyle \Omega =g_{0}\ln(1-z)+\Omega _{\rm {m}}}
3040:
887:, and quantum effects are macroscopically visible like
855:
An interesting example of a Bose gas is an ensemble of
2960:
2923:
2907:
2600:, there is no upper limit on the number of particles (
913:
The equivalent model for half-integer particles (like
848:
used the phonon gas model to explain the behaviour of
4837:
4786:
4693:
4632:
4518:
4493:
4456:
4416:
4337:
4265:
4182:
4129:
4100:
4068:
3993:
3951:
3922:
3820:
3797:
3760:
3725:
3687:
3559:
3379:
3320:
3303:
and the green lines plot the corresponding values of
3120:
2835:
2649:
2469:
2425:
2318:
2143:
1969:
1830:
1713:
1602:
1475:
1361:
1338:
1311:
1190:
1129:
966:
5223:
4828:
Note that in the limit of high temperature, we have
3307:. The horizontal axis is the normalized temperature
3519:or as an asymptotic expansion in inverse powers of
3081:is shown in red, and green lines are the values of
2106:= 1 is however crucially dependent on the value of
719:, which have an integer value of spin and abide by
5106:
5045:
4909:
4817:
4766:
4673:
4609:
4503:
4478:
4441:
4397:
4322:
4250:
4167:
4110:
4085:
4053:
3978:
3932:
3907:
3803:
3780:
3745:
3700:
3669:
3484:
3348:
3175:
2978:
2758:
2582:
2445:
2408:
2224:
2083:
1919:
1785:
1667:
1556:
1451:
1344:
1324:
1281:Macroscopic limit, result for uncondensed fraction
1223:
1166:
1071:
5244:"General Thermal Wavelength and its Applications"
4623:for a classical ideal gas over all temperatures:
4376:
4372:
4285:
4281:
4205:
4201:
3226:approaches 1 very closely but does not reach it.
5288:
2628:
2625:respectively) there is no critical temperature.
1693:for simple spinless bosons). For a massive Bose
1586:. For example, for a massive Bose gas in a box,
1224:{\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}
821:. This model leads to a better understanding of
5232:
5112:
3029:. Calculations can instead be performed in the
3349:{\displaystyle \tau ={\frac {T}{T_{\rm {c}}}}}
3281:and the dotted black line is the solution for
1811:/2 is the harmonic potential. It is seen that
2990:Limitations of the macroscopic Bose gas model
2287:. This corresponds to a critical temperature
1167:{\displaystyle z(\beta ,\mu )=e^{\beta \mu }}
684:
2999:, meaning that when condensation happens at
742:
5235:BoseāEinstein Condensation in Dilute Gases
4993:
2633:The above problem raises the question for
691:
677:
31:
5126:
5065:
4979:. Springer Science & Business Media.
4814:
4757:
4500:
4475:
4438:
4309:
4237:
4164:
4107:
4082:
4040:
3975:
3929:
3777:
3742:
3399:
2803:takes up the remainder of particles. For
2269:). Thus, for a fixed number of particles
1514:
1506:
1433:
1297:Following the procedure described in the
5237:. Cambridge: Cambridge University Press.
3044:
1284:
758:, or equivalently, that possess integer
5226:Statistical Physics, 3rd Edition Part 1
5214:
5051:
4972:
5289:
5224:Landau, L. D.; E. M. Lifshitz (1996).
3979:{\displaystyle 1-{\frac {N_{0}}{N}}\,}
3550:Expanded out, the grand potential is:
2102:= +1). The behaviour when approaching
5205:
5198:
4924:= 3/2 is simply a restatement of the
1960:is found from the grand potential by
1704:and the critical energy is given by:
1593:and the critical energy is given by:
867:, the ensemble starts to behave as a
731:and extended to massive particles by
3041:Approximate behaviour in small gases
2816:there is the normal behaviour, with
2779:Thus in the macroscopic limit, when
1100:is the number of states with energy
944:
5241:
2453:and using the above noted value of
13:
4736:
4718:
4710:
4650:
4645:
4642:
4156:
4086:{\displaystyle =\tau ^{\alpha }\,}
3648:
3560:
3503:, this equation can be solved for
3338:
3167:
3162:
3121:
2941:
2886:
2737:
2675:
2571:
2476:
2398:
2386:
2325:
2276:, the largest possible value that
2200:
2162:
2159:
2156:
2150:
2094:This increases monotonically with
2009:
1998:
1994:
1976:
1837:
1832:
1654:
1621:
1532:
1491:
1443:
1387:
1368:
1363:
1339:
1313:
1209:
987:
967:
14:
5328:
5233:Pethick, C. J.; H. Smith (2004).
3545:
1761:
817:, a gas of photons, developed by
5228:. Oxford: Butterworth-Heinemann.
5210:. New York: John Wiley and Sons.
3530:and in the MaxwellāBoltzmann as
929:(an ensemble of non-interacting
782:; or composite like the atom of
658:
657:
644:
4942:non-linear Schrƶdinger equation
3701:{\displaystyle \tau ^{\alpha }}
3507:and then a series solution for
5180:. Cambridge University Press.
5167:
5003:Journal of Statistical Physics
4966:
4897:
4891:
4859:
4847:
4754:
4742:
4579:
4573:
4544:
4538:
4472:
4466:
4435:
4429:
4359:
4353:
4303:
4297:
4289:
4275:
4231:
4225:
4217:
4211:
4034:
4028:
4020:
4014:
3874:
3868:
3836:
3830:
3629:
3623:
3594:
3582:
3511:can be found by the method of
3463:
3457:
3449:
3443:
3155:
3143:
2744:
2725:
2720:
2714:
2519:
2505:
2354:
2348:
2207:
2188:
2183:
2177:
2066:
2049:
2044:
2038:
1878:
1872:
1818:is a function of volume only.
1771:
1758:
1737:
1720:
1628:
1609:
1500:
1494:
1145:
1133:
992:
982:
852:of metals at low temperature.
16:State of matter of many bosons
1:
4973:Schwabl, Franz (2013-03-09).
4959:
2629:Inclusion of the ground state
2280:can have is a critical value
4512:
4487:
4450:
4408:
4331:
4259:
4176:
4121:
4094:
4062:
3987:
3943:
3916:
3814:
3812:
3791:
3781:{\displaystyle T\gg T_{c}\,}
3754:
3746:{\displaystyle T\ll T_{c}\,}
3719:
3716:
3713:
3057:is 3/2. Solid lines are for
957:for a Bose gas is given by:
7:
5219:. New York: Academic Press.
5115:American Journal of Physics
4947:
4818:{\displaystyle TS=U+PV-G\,}
3111:=0 in the grand potential:
3085:. It has been assumed that
2446:{\displaystyle \alpha =3/2}
1325:{\displaystyle \Omega _{m}}
871:, a fluid with almost zero
711:, analogous to a classical
10:
5333:
5271:10.1088/0143-0807/21/6/314
4442:{\displaystyle G=\ln(z)\,}
1303:ThomasāFermi approximation
1301:article, we can apply the
933:). At low enough particle
840:, is an example where the
244:Spin gapless semiconductor
5023:10.1007/s10955-015-1361-3
4930:Pauli exclusion principle
4777:The entropy is given by:
4479:{\displaystyle =\ln(z)\,}
3250:, which corresponds to a
1272:The permissible range of
743:Introduction and examples
184:Electronic band structure
5297:BoseāEinstein statistics
1686:is a degeneracy factor (
951:grand canonical ensemble
881:BoseāEinstein condensate
798:etc. Additionally, some
756:BoseāEinstein statistics
737:BoseāEinstein condensate
721:BoseāEinstein statistics
707:is a quantum-mechanical
94:BoseāEinstein condensate
25:Condensed matter physics
4926:SackurāTetrode equation
3070:, dotted lines are for
1345:{\displaystyle \Omega }
5206:Huang, Kerson (1967).
5084:10.1002/lapl.200410157
4911:
4819:
4768:
4675:
4611:
4505:
4480:
4443:
4399:
4324:
4252:
4169:
4112:
4087:
4055:
3980:
3934:
3909:
3805:
3782:
3747:
3702:
3671:
3515:, either in powers of
3486:
3350:
3252:gas of bosons in a box
3177:
3100:
3016:becomes unbounded for
2997:geometric distribution
2980:
2760:
2584:
2447:
2410:
2226:
2085:
1921:
1787:
1695:gas in a harmonic trap
1669:
1558:
1453:
1346:
1326:
1294:
1225:
1168:
1073:
923:FermiāDirac statistics
908:electrical resistivity
754:particles that follow
5208:Statistical Mechanics
5054:Laser Physics Letters
4976:Statistical Mechanics
4912:
4820:
4769:
4676:
4612:
4506:
4481:
4444:
4400:
4325:
4253:
4170:
4113:
4088:
4056:
3981:
3935:
3910:
3806:
3783:
3748:
3703:
3672:
3487:
3351:
3178:
3048:
2981:
2761:
2585:
2448:
2411:
2245:Riemann zeta function
2227:
2086:
1922:
1788:
1670:
1582:energy, and Ī is the
1559:
1454:
1347:
1327:
1288:
1226:
1169:
1074:
910:at low temperatures.
778:and the hypothetical
239:Topological insulator
5215:Isihara, A. (1971).
4835:
4784:
4691:
4630:
4516:
4504:{\displaystyle =0\,}
4491:
4454:
4414:
4335:
4263:
4180:
4127:
4111:{\displaystyle =1\,}
4098:
4066:
3991:
3949:
3933:{\displaystyle =1\,}
3920:
3818:
3795:
3758:
3723:
3685:
3557:
3377:
3318:
3186:which gives instead
3118:
2833:
2647:
2467:
2423:
2316:
2141:
1967:
1828:
1711:
1600:
1473:
1359:
1336:
1332:, which is close to
1309:
1188:
1127:
964:
921:atoms), that follow
827:black-body radiation
808:charge density waves
715:. It is composed of
257:Electronic phenomena
104:Fermionic condensate
5317:Satyendra Nath Bose
5263:2000EJPh...21..625Y
5242:Yan, Zijun (2000).
5217:Statistical Physics
5137:2003AmJPh..71..661M
5076:2005LaPhL...2..156Y
5015:2015JSP...161..942T
4954:TonksāGirardeau gas
3513:inversion of series
2796:is pinned to 1 and
2098:(up to the maximum
1958:number of particles
1542:
1391:
725:Satyendra Nath Bose
264:Quantum Hall effect
5199:General references
4907:
4815:
4764:
4671:
4607:
4501:
4476:
4439:
4395:
4320:
4248:
4165:
4108:
4083:
4051:
3976:
3930:
3905:
3801:
3778:
3743:
3698:
3667:
3482:
3346:
3173:
3101:
3031:canonical ensemble
2976:
2971:
2964:
2927:
2911:
2756:
2580:
2443:
2406:
2222:
2081:
1917:
1783:
1680:thermal wavelength
1665:
1554:
1526:
1449:
1377:
1342:
1322:
1295:
1243:Boltzmann constant
1221:
1164:
1115:chemical potential
1069:
1007:
752:quantum mechanical
651:Physics portal
5307:Quantum mechanics
5145:10.1119/1.1544520
4986:978-3-662-04702-6
4901:
4725:
4657:
4620:
4619:
4605:
4558:
4409:Gibbs Free Energy
4393:
4307:
4235:
4194:
4162:
4146:
4122:Equation of state
4038:
4005:
3973:
3903:
3850:
3804:{\displaystyle z}
3665:
3608:
3471:
3467:
3434:
3425:
3415:
3344:
2963:
2926:
2910:
2892:
2851:
2754:
2705:
2578:
2523:
2404:
2358:
2217:
2076:
2029:
2016:
1912:
1898:
1857:
1781:
1747:
1663:
1638:
1547:
1543:
1504:
1219:
998:
945:Macroscopic limit
902:couple in pairs (
896:superconductivity
889:wave interference
790:, the nucleus of
701:
700:
409:Granular material
177:Electronic phases
5324:
5282:
5248:
5238:
5229:
5220:
5211:
5192:
5191:
5171:
5165:
5164:
5130:
5128:cond-mat/0211115
5110:
5104:
5103:
5069:
5067:cond-mat/0504473
5049:
5043:
5042:
4997:
4991:
4990:
4970:
4916:
4914:
4913:
4908:
4906:
4902:
4900:
4886:
4885:
4876:
4824:
4822:
4821:
4816:
4773:
4771:
4770:
4765:
4741:
4740:
4739:
4726:
4724:
4716:
4708:
4703:
4702:
4680:
4678:
4677:
4672:
4658:
4656:
4648:
4640:
4616:
4614:
4613:
4608:
4606:
4604:
4603:
4602:
4593:
4592:
4582:
4568:
4563:
4559:
4557:
4556:
4547:
4533:
4510:
4508:
4507:
4502:
4485:
4483:
4482:
4477:
4448:
4446:
4445:
4440:
4404:
4402:
4401:
4396:
4394:
4392:
4391:
4390:
4381:
4380:
4362:
4348:
4329:
4327:
4326:
4321:
4319:
4318:
4308:
4306:
4292:
4270:
4257:
4255:
4254:
4249:
4247:
4246:
4236:
4234:
4220:
4210:
4209:
4196:
4195:
4192:
4187:
4174:
4172:
4171:
4166:
4163:
4155:
4147:
4142:
4131:
4117:
4115:
4114:
4109:
4092:
4090:
4089:
4084:
4081:
4080:
4060:
4058:
4057:
4052:
4050:
4049:
4039:
4037:
4023:
4013:
4012:
4007:
4006:
4003:
3998:
3985:
3983:
3982:
3977:
3974:
3969:
3968:
3959:
3939:
3937:
3936:
3931:
3914:
3912:
3911:
3906:
3904:
3902:
3901:
3900:
3888:
3887:
3877:
3867:
3866:
3856:
3851:
3849:
3848:
3839:
3825:
3810:
3808:
3807:
3802:
3787:
3785:
3784:
3779:
3776:
3775:
3752:
3750:
3749:
3744:
3741:
3740:
3711:
3707:
3705:
3704:
3699:
3697:
3696:
3676:
3674:
3673:
3668:
3666:
3664:
3663:
3658:
3654:
3653:
3652:
3651:
3632:
3622:
3621:
3610:
3609:
3606:
3601:
3575:
3574:
3529:
3491:
3489:
3488:
3483:
3481:
3480:
3469:
3468:
3466:
3452:
3442:
3441:
3436:
3435:
3432:
3427:
3423:
3416:
3414:
3403:
3398:
3397:
3387:
3355:
3353:
3352:
3347:
3345:
3343:
3342:
3341:
3328:
3287:
3280:
3279:
3278:
3267:
3249:
3235:
3221:
3220:
3218:
3217:
3211:
3208:
3182:
3180:
3179:
3174:
3172:
3171:
3170:
3136:
3135:
3098:
3076:
3069:
3068:
3067:
3028:
3011:
2985:
2983:
2982:
2977:
2975:
2974:
2965:
2961:
2946:
2945:
2944:
2928:
2924:
2912:
2908:
2903:
2902:
2897:
2893:
2891:
2890:
2889:
2876:
2852:
2847:
2846:
2837:
2825:
2815:
2791:
2765:
2763:
2762:
2757:
2755:
2753:
2752:
2751:
2742:
2741:
2740:
2723:
2713:
2712:
2707:
2706:
2703:
2698:
2693:
2692:
2680:
2679:
2678:
2665:
2664:
2639:
2624:
2617:
2599:
2589:
2587:
2586:
2581:
2579:
2577:
2576:
2575:
2574:
2554:
2553:
2544:
2542:
2541:
2537:
2528:
2524:
2522:
2515:
2491:
2481:
2480:
2479:
2452:
2450:
2449:
2444:
2439:
2415:
2413:
2412:
2407:
2405:
2403:
2402:
2401:
2391:
2390:
2389:
2379:
2377:
2376:
2372:
2363:
2359:
2357:
2340:
2330:
2329:
2328:
2308:
2268:
2231:
2229:
2228:
2223:
2218:
2216:
2215:
2214:
2205:
2204:
2203:
2186:
2172:
2167:
2166:
2165:
2133:
2119:
2090:
2088:
2087:
2082:
2077:
2075:
2074:
2073:
2064:
2063:
2047:
2037:
2036:
2031:
2030:
2027:
2022:
2017:
2015:
2007:
2006:
2005:
1992:
1981:
1980:
1979:
1926:
1924:
1923:
1918:
1913:
1911:
1910:
1905:
1901:
1900:
1899:
1896:
1881:
1871:
1870:
1859:
1858:
1855:
1850:
1842:
1841:
1840:
1792:
1790:
1789:
1784:
1782:
1780:
1779:
1778:
1753:
1748:
1746:
1745:
1744:
1735:
1734:
1715:
1703:
1692:
1674:
1672:
1671:
1666:
1664:
1662:
1661:
1652:
1644:
1639:
1637:
1636:
1635:
1626:
1625:
1624:
1604:
1592:
1563:
1561:
1560:
1555:
1545:
1544:
1541:
1536:
1535:
1525:
1524:
1508:
1505:
1503:
1486:
1458:
1456:
1455:
1450:
1432:
1428:
1427:
1426:
1390:
1385:
1373:
1372:
1371:
1351:
1349:
1348:
1343:
1331:
1329:
1328:
1323:
1321:
1320:
1230:
1228:
1227:
1222:
1220:
1218:
1214:
1213:
1212:
1198:
1173:
1171:
1170:
1165:
1163:
1162:
1078:
1076:
1075:
1070:
1065:
1061:
1060:
1059:
1058:
1057:
1017:
1016:
1006:
991:
990:
925:, is called the
877:phase transition
836:, also known as
693:
686:
679:
666:
661:
660:
653:
649:
648:
269:Spin Hall effect
159:Phase transition
129:Luttinger liquid
66:States of matter
49:Phase transition
35:
21:
20:
5332:
5331:
5327:
5326:
5325:
5323:
5322:
5321:
5287:
5286:
5285:
5246:
5201:
5196:
5195:
5188:
5172:
5168:
5111:
5107:
5050:
5046:
4998:
4994:
4987:
4971:
4967:
4962:
4950:
4887:
4881:
4877:
4875:
4871:
4836:
4833:
4832:
4785:
4782:
4781:
4735:
4734:
4730:
4717:
4709:
4707:
4698:
4694:
4692:
4689:
4688:
4649:
4641:
4639:
4631:
4628:
4627:
4598:
4594:
4588:
4584:
4583:
4569:
4567:
4552:
4548:
4534:
4532:
4528:
4517:
4514:
4513:
4492:
4489:
4488:
4455:
4452:
4451:
4415:
4412:
4411:
4410:
4386:
4382:
4368:
4364:
4363:
4349:
4347:
4336:
4333:
4332:
4314:
4310:
4293:
4271:
4269:
4264:
4261:
4260:
4242:
4238:
4221:
4197:
4191:
4190:
4189:
4188:
4186:
4181:
4178:
4177:
4154:
4132:
4130:
4128:
4125:
4124:
4123:
4099:
4096:
4095:
4076:
4072:
4067:
4064:
4063:
4045:
4041:
4024:
4008:
4002:
4001:
4000:
3999:
3997:
3992:
3989:
3988:
3964:
3960:
3958:
3950:
3947:
3946:
3945:
3921:
3918:
3917:
3893:
3889:
3883:
3879:
3878:
3862:
3858:
3857:
3855:
3844:
3840:
3826:
3824:
3819:
3816:
3815:
3796:
3793:
3792:
3771:
3767:
3759:
3756:
3755:
3736:
3732:
3724:
3721:
3720:
3692:
3688:
3686:
3683:
3682:
3659:
3647:
3646:
3642:
3638:
3634:
3633:
3611:
3605:
3604:
3603:
3602:
3600:
3570:
3566:
3558:
3555:
3554:
3548:
3524:
3476:
3472:
3453:
3437:
3431:
3430:
3429:
3428:
3426:
3404:
3393:
3389:
3388:
3386:
3378:
3375:
3374:
3337:
3336:
3332:
3327:
3319:
3316:
3315:
3294:
3282:
3276:
3274:
3269:
3262:
3255:
3247:
3237:
3230:
3212:
3209:
3204:
3198:
3197:
3195:
3193:
3187:
3166:
3165:
3161:
3131:
3127:
3119:
3116:
3115:
3096:
3086:
3071:
3065:
3063:
3058:
3053:. The value of
3043:
3027:
3017:
3014:compressibility
3010:
3000:
2992:
2970:
2969:
2959:
2957:
2951:
2950:
2940:
2939:
2935:
2925: and
2922:
2906:
2904:
2898:
2885:
2884:
2880:
2875:
2871:
2870:
2857:
2856:
2842:
2838:
2836:
2834:
2831:
2830:
2823:
2817:
2814:
2804:
2802:
2792:, the value of
2790:
2780:
2775:
2747:
2743:
2736:
2735:
2731:
2724:
2708:
2702:
2701:
2700:
2699:
2697:
2688:
2684:
2674:
2673:
2669:
2660:
2656:
2648:
2645:
2644:
2634:
2631:
2619:
2612:
2606:
2594:
2570:
2569:
2565:
2555:
2549:
2545:
2543:
2533:
2529:
2511:
2495:
2490:
2486:
2485:
2475:
2474:
2470:
2468:
2465:
2464:
2459:
2435:
2424:
2421:
2420:
2397:
2396:
2392:
2385:
2384:
2380:
2378:
2368:
2364:
2344:
2339:
2335:
2334:
2324:
2323:
2319:
2317:
2314:
2313:
2307:
2301:
2294:
2288:
2286:
2275:
2254:
2248:
2210:
2206:
2199:
2198:
2194:
2187:
2173:
2171:
2149:
2148:
2144:
2142:
2139:
2138:
2128:
2126:
2114:
2069:
2065:
2059:
2055:
2048:
2032:
2026:
2025:
2024:
2023:
2021:
2008:
2001:
1997:
1993:
1991:
1975:
1974:
1970:
1968:
1965:
1964:
1954:
1935:
1906:
1895:
1891:
1887:
1883:
1882:
1860:
1854:
1853:
1852:
1851:
1849:
1836:
1835:
1831:
1829:
1826:
1825:
1817:
1774:
1770:
1757:
1752:
1740:
1736:
1730:
1726:
1719:
1714:
1712:
1709:
1708:
1698:
1687:
1678:where Ī is the
1657:
1653:
1645:
1643:
1631:
1627:
1620:
1619:
1615:
1608:
1603:
1601:
1598:
1597:
1587:
1577:
1571:is a constant,
1537:
1531:
1530:
1513:
1509:
1507:
1490:
1485:
1474:
1471:
1470:
1462:The degeneracy
1416:
1412:
1402:
1398:
1386:
1381:
1367:
1366:
1362:
1360:
1357:
1356:
1337:
1334:
1333:
1316:
1312:
1310:
1307:
1306:
1283:
1240:
1208:
1207:
1203:
1202:
1197:
1189:
1186:
1185:
1155:
1151:
1128:
1125:
1124:
1108:
1099:
1090:
1053:
1049:
1042:
1038:
1028:
1024:
1012:
1008:
1002:
986:
985:
965:
962:
961:
955:grand potential
947:
900:charge carriers
745:
733:Albert Einstein
709:phase of matter
697:
656:
643:
642:
635:
634:
633:
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:
5330:
5320:
5319:
5314:
5312:Thermodynamics
5309:
5304:
5299:
5284:
5283:
5257:(6): 625ā631.
5239:
5230:
5221:
5212:
5202:
5200:
5197:
5194:
5193:
5186:
5166:
5121:(7): 661ā669.
5105:
5060:(3): 156ā161.
5044:
4992:
4985:
4964:
4963:
4961:
4958:
4957:
4956:
4949:
4946:
4938:Chen-Ning Yang
4918:
4917:
4905:
4899:
4896:
4893:
4890:
4884:
4880:
4874:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4826:
4825:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4775:
4774:
4763:
4760:
4756:
4753:
4750:
4747:
4744:
4738:
4733:
4729:
4723:
4720:
4715:
4712:
4706:
4701:
4697:
4682:
4681:
4670:
4667:
4664:
4661:
4655:
4652:
4647:
4644:
4638:
4635:
4618:
4617:
4601:
4597:
4591:
4587:
4581:
4578:
4575:
4572:
4566:
4562:
4555:
4551:
4546:
4543:
4540:
4537:
4531:
4527:
4524:
4521:
4511:
4499:
4496:
4486:
4474:
4471:
4468:
4465:
4462:
4459:
4449:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4406:
4405:
4389:
4385:
4379:
4375:
4371:
4367:
4361:
4358:
4355:
4352:
4346:
4343:
4340:
4330:
4317:
4313:
4305:
4302:
4299:
4296:
4291:
4288:
4284:
4280:
4277:
4274:
4268:
4258:
4245:
4241:
4233:
4230:
4227:
4224:
4219:
4216:
4213:
4208:
4204:
4200:
4185:
4175:
4161:
4158:
4153:
4150:
4145:
4141:
4138:
4135:
4119:
4118:
4106:
4103:
4093:
4079:
4075:
4071:
4061:
4048:
4044:
4036:
4033:
4030:
4027:
4022:
4019:
4016:
4011:
3996:
3986:
3972:
3967:
3963:
3957:
3954:
3944:Vapor fraction
3941:
3940:
3928:
3925:
3915:
3899:
3896:
3892:
3886:
3882:
3876:
3873:
3870:
3865:
3861:
3854:
3847:
3843:
3838:
3835:
3832:
3829:
3823:
3813:
3811:
3800:
3789:
3788:
3774:
3770:
3766:
3763:
3753:
3739:
3735:
3731:
3728:
3718:
3715:
3695:
3691:
3678:
3677:
3662:
3657:
3650:
3645:
3641:
3637:
3631:
3628:
3625:
3620:
3617:
3614:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3573:
3569:
3565:
3562:
3547:
3546:Thermodynamics
3544:
3493:
3492:
3479:
3475:
3465:
3462:
3459:
3456:
3451:
3448:
3445:
3440:
3422:
3419:
3413:
3410:
3407:
3402:
3396:
3392:
3385:
3382:
3357:
3356:
3340:
3335:
3331:
3326:
3323:
3292:
3260:
3245:
3202:
3191:
3184:
3183:
3169:
3164:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3134:
3130:
3126:
3123:
3094:
3042:
3039:
3025:
3008:
2991:
2988:
2987:
2986:
2973:
2968:
2958:
2956:
2953:
2952:
2949:
2943:
2938:
2934:
2931:
2921:
2918:
2915:
2905:
2901:
2896:
2888:
2883:
2879:
2874:
2869:
2866:
2863:
2862:
2860:
2855:
2850:
2845:
2841:
2821:
2812:
2800:
2788:
2773:
2767:
2766:
2750:
2746:
2739:
2734:
2730:
2727:
2722:
2719:
2716:
2711:
2696:
2691:
2687:
2683:
2677:
2672:
2668:
2663:
2659:
2655:
2652:
2630:
2627:
2604:
2591:
2590:
2573:
2568:
2564:
2561:
2558:
2552:
2548:
2540:
2536:
2532:
2527:
2521:
2518:
2514:
2510:
2507:
2504:
2501:
2498:
2494:
2489:
2484:
2478:
2473:
2457:
2442:
2438:
2434:
2431:
2428:
2417:
2416:
2400:
2395:
2388:
2383:
2375:
2371:
2367:
2362:
2356:
2353:
2350:
2347:
2343:
2338:
2333:
2327:
2322:
2305:
2299:
2292:
2284:
2273:
2250:
2233:
2232:
2221:
2213:
2209:
2202:
2197:
2193:
2190:
2185:
2182:
2179:
2176:
2170:
2164:
2161:
2158:
2155:
2152:
2147:
2124:
2092:
2091:
2080:
2072:
2068:
2062:
2058:
2054:
2051:
2046:
2043:
2040:
2035:
2020:
2014:
2011:
2004:
2000:
1996:
1990:
1987:
1984:
1978:
1973:
1953:
1950:
1931:
1928:
1927:
1916:
1909:
1904:
1894:
1890:
1886:
1880:
1877:
1874:
1869:
1866:
1863:
1848:
1845:
1839:
1834:
1815:
1794:
1793:
1777:
1773:
1769:
1766:
1763:
1760:
1756:
1751:
1743:
1739:
1733:
1729:
1725:
1722:
1718:
1676:
1675:
1660:
1656:
1651:
1648:
1642:
1634:
1630:
1623:
1618:
1614:
1611:
1607:
1584:gamma function
1575:
1565:
1564:
1553:
1550:
1540:
1534:
1529:
1523:
1520:
1517:
1512:
1502:
1499:
1496:
1493:
1489:
1484:
1481:
1478:
1460:
1459:
1448:
1445:
1442:
1439:
1436:
1431:
1425:
1422:
1419:
1415:
1411:
1408:
1405:
1401:
1397:
1394:
1389:
1384:
1380:
1376:
1370:
1365:
1341:
1319:
1315:
1282:
1279:
1238:
1232:
1231:
1217:
1211:
1206:
1201:
1196:
1193:
1175:
1174:
1161:
1158:
1154:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1104:
1095:
1086:
1080:
1079:
1068:
1064:
1056:
1052:
1048:
1045:
1041:
1037:
1034:
1031:
1027:
1023:
1020:
1015:
1011:
1005:
1001:
997:
994:
989:
984:
981:
978:
975:
972:
969:
946:
943:
935:number density
800:quasiparticles
786:, the atom of
744:
741:
699:
698:
696:
695:
688:
681:
673:
670:
669:
668:
667:
654:
637:
636:
632:
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:
5329:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5294:
5292:
5280:
5276:
5272:
5268:
5264:
5260:
5256:
5252:
5245:
5240:
5236:
5231:
5227:
5222:
5218:
5213:
5209:
5204:
5203:
5189:
5187:9780521586467
5183:
5179:
5178:
5170:
5162:
5158:
5154:
5150:
5146:
5142:
5138:
5134:
5129:
5124:
5120:
5116:
5109:
5101:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5068:
5063:
5059:
5055:
5048:
5040:
5036:
5032:
5028:
5024:
5020:
5016:
5012:
5008:
5004:
4996:
4988:
4982:
4978:
4977:
4969:
4965:
4955:
4952:
4951:
4945:
4943:
4939:
4935:
4931:
4927:
4923:
4903:
4894:
4888:
4882:
4878:
4872:
4868:
4865:
4862:
4856:
4853:
4850:
4844:
4841:
4838:
4831:
4830:
4829:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4780:
4779:
4778:
4761:
4758:
4751:
4748:
4745:
4731:
4727:
4721:
4713:
4704:
4699:
4695:
4687:
4686:
4685:
4668:
4665:
4662:
4659:
4653:
4636:
4633:
4626:
4625:
4624:
4599:
4595:
4589:
4585:
4576:
4570:
4564:
4560:
4553:
4549:
4541:
4535:
4529:
4525:
4522:
4519:
4497:
4494:
4469:
4463:
4460:
4457:
4432:
4426:
4423:
4420:
4417:
4407:
4387:
4383:
4377:
4373:
4369:
4365:
4356:
4350:
4344:
4341:
4338:
4315:
4311:
4300:
4294:
4286:
4282:
4278:
4272:
4266:
4243:
4239:
4228:
4222:
4214:
4206:
4202:
4198:
4183:
4159:
4151:
4148:
4143:
4139:
4136:
4133:
4120:
4104:
4101:
4077:
4073:
4069:
4046:
4042:
4031:
4025:
4017:
4009:
3994:
3970:
3965:
3961:
3955:
3952:
3942:
3926:
3923:
3897:
3894:
3890:
3884:
3880:
3871:
3863:
3859:
3852:
3845:
3841:
3833:
3827:
3821:
3798:
3790:
3772:
3768:
3764:
3761:
3737:
3733:
3729:
3726:
3712:
3709:
3693:
3689:
3660:
3655:
3643:
3639:
3635:
3626:
3618:
3615:
3612:
3597:
3591:
3588:
3585:
3579:
3576:
3571:
3567:
3563:
3553:
3552:
3551:
3543:
3539:
3537:
3533:
3527:
3522:
3518:
3514:
3510:
3506:
3502:
3498:
3477:
3473:
3460:
3454:
3446:
3438:
3420:
3417:
3411:
3408:
3405:
3400:
3394:
3390:
3383:
3380:
3373:
3372:
3371:
3368:
3366:
3362:
3333:
3329:
3324:
3321:
3314:
3313:
3312:
3310:
3306:
3302:
3298:
3291:
3285:
3272:
3266:
3259:
3253:
3244:
3240:
3233:
3227:
3225:
3216:
3207:
3201:
3190:
3158:
3152:
3149:
3146:
3140:
3137:
3132:
3128:
3124:
3114:
3113:
3112:
3110:
3106:
3103:For smaller,
3093:
3089:
3084:
3080:
3074:
3061:
3056:
3052:
3047:
3038:
3034:
3032:
3024:
3020:
3015:
3007:
3003:
2998:
2966:
2954:
2947:
2936:
2932:
2929:
2919:
2916:
2913:
2899:
2894:
2881:
2877:
2872:
2867:
2864:
2858:
2853:
2848:
2843:
2839:
2829:
2828:
2827:
2820:
2811:
2807:
2799:
2795:
2787:
2783:
2777:
2772:
2748:
2732:
2728:
2717:
2709:
2694:
2689:
2685:
2681:
2670:
2666:
2661:
2657:
2653:
2650:
2643:
2642:
2641:
2637:
2626:
2622:
2615:
2610:
2603:
2597:
2566:
2562:
2559:
2556:
2550:
2546:
2538:
2534:
2530:
2525:
2516:
2512:
2508:
2502:
2499:
2496:
2492:
2487:
2482:
2471:
2463:
2462:
2461:
2456:
2440:
2436:
2432:
2429:
2426:
2393:
2381:
2373:
2369:
2365:
2360:
2351:
2345:
2341:
2336:
2331:
2320:
2312:
2311:
2310:
2304:
2298:
2291:
2283:
2279:
2272:
2266:
2262:
2258:
2253:
2246:
2242:
2238:
2219:
2211:
2195:
2191:
2180:
2174:
2168:
2153:
2145:
2137:
2136:
2135:
2131:
2127:is finite at
2123:
2117:
2111:
2109:
2105:
2101:
2097:
2078:
2070:
2060:
2056:
2052:
2041:
2033:
2018:
2012:
2002:
1988:
1985:
1982:
1971:
1963:
1962:
1961:
1959:
1949:
1945:
1943:
1942:polylogarithm
1939:
1934:
1914:
1907:
1902:
1892:
1888:
1884:
1875:
1867:
1864:
1861:
1846:
1843:
1824:
1823:
1822:
1819:
1814:
1810:
1807:
1803:
1799:
1775:
1767:
1764:
1754:
1749:
1741:
1731:
1727:
1723:
1716:
1707:
1706:
1705:
1701:
1697:we will have
1696:
1690:
1685:
1681:
1658:
1649:
1646:
1640:
1632:
1616:
1612:
1605:
1596:
1595:
1594:
1590:
1585:
1581:
1574:
1570:
1551:
1548:
1538:
1527:
1521:
1518:
1515:
1510:
1497:
1487:
1482:
1479:
1476:
1469:
1468:
1467:
1465:
1446:
1440:
1437:
1434:
1429:
1423:
1420:
1417:
1413:
1409:
1406:
1403:
1399:
1395:
1392:
1382:
1378:
1374:
1355:
1354:
1353:
1317:
1304:
1300:
1292:
1287:
1278:
1275:
1270:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1237:
1215:
1204:
1199:
1194:
1191:
1184:
1183:
1182:
1180:
1159:
1156:
1152:
1148:
1142:
1139:
1136:
1130:
1123:
1122:
1121:
1120:by defining:
1119:
1116:
1112:
1107:
1103:
1098:
1094:
1089:
1085:
1066:
1062:
1054:
1050:
1046:
1043:
1039:
1035:
1032:
1029:
1025:
1021:
1018:
1013:
1009:
1003:
999:
995:
979:
976:
973:
970:
960:
959:
958:
956:
952:
942:
940:
936:
932:
928:
924:
920:
916:
911:
909:
905:
901:
897:
892:
890:
886:
882:
878:
874:
870:
866:
862:
861:absolute zero
858:
853:
851:
850:heat capacity
847:
843:
839:
835:
833:
828:
824:
820:
816:
811:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
749:
740:
738:
734:
730:
726:
722:
718:
714:
710:
706:
694:
689:
687:
682:
680:
675:
674:
672:
671:
665:
655:
652:
647:
641:
640:
639:
638:
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:
5254:
5251:Eur. J. Phys
5250:
5234:
5225:
5216:
5207:
5176:
5169:
5118:
5114:
5108:
5057:
5053:
5047:
5006:
5002:
4995:
4975:
4968:
4934:Bethe ansatz
4921:
4919:
4827:
4776:
4683:
4621:
3679:
3549:
3540:
3535:
3531:
3525:
3520:
3516:
3508:
3504:
3500:
3496:
3495:For a given
3494:
3369:
3364:
3360:
3358:
3308:
3304:
3300:
3296:
3289:
3283:
3270:
3264:
3257:
3242:
3238:
3231:
3228:
3223:
3214:
3205:
3199:
3188:
3185:
3108:
3102:
3091:
3087:
3082:
3078:
3072:
3059:
3054:
3050:
3035:
3022:
3018:
3005:
3001:
2993:
2818:
2809:
2805:
2797:
2793:
2785:
2781:
2778:
2770:
2768:
2635:
2632:
2620:
2613:
2608:
2607:diverges as
2601:
2595:
2592:
2454:
2418:
2302:
2296:
2289:
2281:
2277:
2270:
2264:
2260:
2256:
2251:
2240:
2236:
2234:
2129:
2121:
2115:
2112:
2107:
2103:
2099:
2095:
2093:
1955:
1946:
1937:
1932:
1929:
1820:
1812:
1808:
1805:
1801:
1797:
1795:
1699:
1688:
1683:
1677:
1588:
1579:
1572:
1568:
1566:
1463:
1461:
1299:gas in a box
1296:
1273:
1271:
1266:
1262:
1258:
1254:
1246:
1235:
1233:
1181:defined as:
1178:
1176:
1117:
1110:
1105:
1101:
1096:
1092:
1087:
1083:
1081:
948:
912:
904:Cooper pairs
893:
885:ground state
854:
842:normal modes
830:
823:Planck's law
812:
746:
704:
702:
569:von Klitzing
274:Kondo effect
134:Time crystal
114:Fermi liquid
98:
18:
4920:which, for
3311:defined by
1251:temperature
846:Peter Debye
838:Debye model
806:(quanta of
764:Higgs boson
391:Soft matter
312:Ferromagnet
5291:Categories
4960:References
3708:is shown.
3105:mesoscopic
2460:) we get:
1956:The total
1944:function.
869:superfluid
815:photon gas
729:photon gas
534:Louis NĆ©el
524:Schrieffer
432:Scientists
326:Spin glass
321:Metamagnet
303:Paramagnet
119:Supersolid
5302:Ideal gas
5279:250870934
5153:0002-9505
5100:119073938
5092:1612-2011
5039:118614846
5031:0022-4715
4895:α
4889:ζ
4883:α
4879:τ
4869:
4851:α
4809:−
4762:β
4746:α
4719:∂
4711:∂
4663:α
4654:β
4651:∂
4646:Ω
4643:∂
4600:α
4596:τ
4590:α
4577:α
4571:ζ
4565:−
4554:α
4550:τ
4542:α
4536:ζ
4526:
4520:≈
4464:
4427:
4388:α
4384:τ
4370:α
4357:α
4351:ζ
4345:−
4339:≈
4316:α
4312:τ
4301:α
4295:ζ
4279:α
4273:ζ
4244:α
4240:τ
4229:α
4223:ζ
4199:α
4157:Ω
4152:−
4140:β
4078:α
4074:τ
4047:α
4043:τ
4032:α
4026:ζ
4010:α
3956:−
3898:α
3891:τ
3885:α
3872:α
3860:ζ
3853:−
3846:α
3842:τ
3834:α
3828:ζ
3822:≈
3765:≫
3730:≪
3714:Quantity
3694:α
3690:τ
3661:α
3640:β
3613:α
3598:−
3589:−
3580:
3561:Ω
3478:α
3474:τ
3461:α
3455:ζ
3439:α
3409:−
3322:τ
3163:Ω
3150:−
3141:
3122:Ω
2962:otherwise
2914:α
2900:α
2868:−
2749:α
2729:β
2710:α
2560:π
2503:ζ
2427:α
2374:α
2352:α
2346:ζ
2243:) is the
2212:α
2192:β
2181:α
2175:ζ
2071:α
2053:β
2034:α
2010:∂
1999:Ω
1995:∂
1986:−
1940:) is the
1908:α
1889:β
1862:α
1847:−
1833:Ω
1768:β
1765:ω
1762:ℏ
1742:α
1724:β
1655:Λ
1633:α
1613:β
1539:α
1519:−
1516:α
1498:α
1492:Γ
1444:Ω
1441:≈
1421:β
1418:−
1407:−
1396:
1388:∞
1379:∫
1364:Ω
1340:Ω
1314:Ω
1291:Fermi gas
1192:β
1160:μ
1157:β
1143:μ
1137:β
1051:ϵ
1047:β
1044:−
1033:−
1022:
1000:∑
980:
974:−
968:Ω
939:ideal gas
927:Fermi gas
915:electrons
873:viscosity
792:deuterium
713:ideal gas
703:An ideal
614:Abrikosov
529:Josephson
499:Van Vleck
489:Luttinger
362:Polariton
294:Diamagnet
214:Conductor
209:Semimetal
194:Insulator
109:Fermi gas
4948:See also
3717:General
2909:if
1930:where Li
1580:critical
931:fermions
919:helium-3
857:helium-4
825:and the
804:plasmons
784:hydrogen
780:graviton
705:Bose gas
664:Category
619:Ginzburg
594:Laughlin
554:Kadanoff
509:Shockley
494:Anderson
449:von Laue
99:Bose gas
5259:Bibcode
5133:Bibcode
5072:Bibcode
5011:Bibcode
3236:, with
3219:
3196:
2247:(using
1265:), and
1249:is the
1241: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
5277:
5184:
5161:949741
5159:
5151:
5098:
5090:
5037:
5029:
4983:
3470:
3424:
3286:= 1000
3092:ε
3079:μ
3075:= 1000
3055:α
3051:τ
2769:where
2638:> 1
2235:where
2118:> 1
1796:where
1682:, and
1567:where
1546:
1234:where
953:. The
898:where
832:phonon
796:mesons
774:, the
770:, the
768:photon
766:, the
748:Bosons
727:for a
717:bosons
662:
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
5275:S2CID
5247:(PDF)
5157:S2CID
5123:arXiv
5096:S2CID
5062:arXiv
5035:S2CID
3234:= 3/2
3021:<
3004:<
2808:>
2784:<
2616:= 1/2
1591:= 3/2
1578:is a
772:gluon
539:Esaki
464:Bloch
459:Debye
454:Bragg
444:Onnes
377:Roton
74:Solid
5182:ISBN
5149:ISSN
5088:ISSN
5027:ISSN
4981:ISBN
3499:and
3268:for
3256:1 ā
3213:1 ā
2933:<
2917:>
2618:and
2593:For
2295:= 1/
2259:) =
2113:For
1804:) =
1261:(or
1245:and
1177:and
819:Bose
760:spin
750:are
609:Tsui
604:Yang
549:Kohn
474:Mott
5267:doi
5141:doi
5080:doi
5019:doi
5007:161
3528:= 0
3277:000
3248:= 1
3097:= 1
3066:000
2824:= 0
2623:= 1
2598:ā¤ 1
2132:= 1
1702:= 3
1691:= 1
917:or
834:gas
810:).
776:W/Z
164:QCP
84:Gas
54:QCP
5293::
5273:.
5265:.
5255:21
5253:.
5249:.
5155:.
5147:.
5139:.
5131:.
5119:71
5117:.
5094:.
5086:.
5078:.
5070:.
5056:.
5033:.
5025:.
5017:.
5005:.
4944:.
4866:ln
4523:ln
4461:ln
4424:ln
4193:Li
4004:Li
3607:Li
3577:ln
3433:Li
3275:10
3273:=
3241:=
3194:=
3138:ln
3090:=
3064:10
3062:=
2704:Li
2249:Li
2134::
2028:Li
1856:Li
1806:mĻ
1464:dg
1393:ln
1352::
1257:,
1109:;
1091:;
1019:ln
977:ln
941:.
891:.
794:,
739:.
5281:.
5269::
5261::
5190:.
5163:.
5143::
5135::
5125::
5102:.
5082::
5074::
5064::
5058:2
5041:.
5021::
5013::
4989:.
4922:Ī±
4904:)
4898:)
4892:(
4873:(
4863:+
4860:)
4857:1
4854:+
4848:(
4845:=
4842:S
4839:T
4812:G
4806:V
4803:P
4800:+
4797:U
4794:=
4791:S
4788:T
4759:U
4755:)
4752:1
4749:+
4743:(
4737:B
4732:k
4728:=
4722:T
4714:U
4705:=
4700:V
4696:C
4669:V
4666:P
4660:=
4637:=
4634:U
4586:2
4580:)
4574:(
4561:)
4545:)
4539:(
4530:(
4498:0
4495:=
4473:)
4470:z
4467:(
4458:=
4436:)
4433:z
4430:(
4421:=
4418:G
4378:1
4374:+
4366:2
4360:)
4354:(
4342:1
4304:)
4298:(
4290:)
4287:1
4283:+
4276:(
4267:=
4232:)
4226:(
4218:)
4215:z
4212:(
4207:1
4203:+
4184:=
4160:N
4149:=
4144:N
4137:V
4134:P
4105:1
4102:=
4070:=
4035:)
4029:(
4021:)
4018:z
4015:(
3995:=
3971:N
3966:0
3962:N
3953:1
3927:1
3924:=
3895:2
3881:2
3875:)
3869:(
3864:2
3837:)
3831:(
3799:z
3773:c
3769:T
3762:T
3738:c
3734:T
3727:T
3656:)
3649:c
3644:E
3636:(
3630:)
3627:z
3624:(
3619:1
3616:+
3595:)
3592:z
3586:1
3583:(
3572:0
3568:g
3564:=
3536:N
3532:T
3526:T
3521:Ļ
3517:Ļ
3509:z
3505:Ļ
3501:Ļ
3497:N
3464:)
3458:(
3450:)
3447:z
3444:(
3421:N
3418:+
3412:z
3406:1
3401:z
3395:0
3391:g
3384:=
3381:N
3365:Ļ
3361:Ļ
3339:c
3334:T
3330:T
3325:=
3309:Ļ
3305:z
3301:Ī¼
3297:N
3295:/
3293:0
3290:N
3284:N
3271:N
3265:N
3263:/
3261:0
3258:N
3246:c
3243:Īµ
3239:k
3232:Ī±
3224:z
3215:z
3210:/
3206:z
3203:0
3200:g
3192:0
3189:N
3168:m
3159:+
3156:)
3153:z
3147:1
3144:(
3133:0
3129:g
3125:=
3109:Īµ
3099:.
3095:c
3088:k
3083:z
3073:N
3060:N
3026:c
3023:T
3019:T
3009:c
3006:T
3002:T
2967:.
2955:0
2948:,
2942:c
2937:T
2930:T
2920:1
2895:)
2887:c
2882:T
2878:T
2873:(
2865:1
2859:{
2854:=
2849:N
2844:0
2840:N
2822:0
2819:N
2813:c
2810:T
2806:T
2801:0
2798:N
2794:z
2789:c
2786:T
2782:T
2774:0
2771:N
2745:)
2738:c
2733:E
2726:(
2721:)
2718:z
2715:(
2695:+
2690:0
2686:N
2682:=
2676:m
2671:N
2667:+
2662:0
2658:N
2654:=
2651:N
2636:Ī±
2621:Ī±
2614:Ī±
2609:z
2605:m
2602:N
2596:Ī±
2572:B
2567:k
2563:m
2557:2
2551:2
2547:h
2539:3
2535:/
2531:2
2526:)
2520:)
2517:2
2513:/
2509:3
2506:(
2500:f
2497:V
2493:N
2488:(
2483:=
2477:c
2472:T
2458:c
2455:E
2441:2
2437:/
2433:3
2430:=
2399:B
2394:k
2387:c
2382:E
2370:/
2366:1
2361:)
2355:)
2349:(
2342:N
2337:(
2332:=
2326:c
2321:T
2306:c
2303:Ī²
2300:B
2297:k
2293:c
2290:T
2285:c
2282:Ī²
2278:Ī²
2274:m
2271:N
2267:)
2265:Ī±
2263:(
2261:Ī¶
2257:1
2255:(
2252:Ī±
2241:Ī±
2239:(
2237:Ī¶
2220:,
2208:)
2201:c
2196:E
2189:(
2184:)
2178:(
2169:=
2163:x
2160:a
2157:m
2154:,
2151:m
2146:N
2130:z
2125:m
2122:N
2116:Ī±
2108:Ī±
2104:z
2100:z
2096:z
2079:.
2067:)
2061:c
2057:E
2050:(
2045:)
2042:z
2039:(
2019:=
2013:z
2003:m
1989:z
1983:=
1977:m
1972:N
1938:x
1936:(
1933:s
1915:,
1903:)
1897:c
1893:E
1885:(
1879:)
1876:z
1873:(
1868:1
1865:+
1844:=
1838:m
1816:c
1813:E
1809:r
1802:r
1800:(
1798:V
1776:3
1772:)
1759:(
1755:f
1750:=
1738:)
1732:c
1728:E
1721:(
1717:1
1700:Ī±
1689:f
1684:f
1659:3
1650:f
1647:V
1641:=
1629:)
1622:c
1617:E
1610:(
1606:1
1589:Ī±
1576:c
1573:E
1569:Ī±
1552:E
1549:d
1533:c
1528:E
1522:1
1511:E
1501:)
1495:(
1488:1
1483:=
1480:g
1477:d
1447:.
1438:g
1435:d
1430:)
1424:E
1414:e
1410:z
1404:1
1400:(
1383:0
1375:=
1369:m
1318:m
1274:z
1267:V
1263:T
1259:Ī²
1255:z
1247:T
1239:B
1236:k
1216:T
1210:B
1205:k
1200:1
1195:=
1179:Ī²
1153:e
1149:=
1146:)
1140:,
1134:(
1131:z
1118:Ī¼
1111:z
1106:i
1102:Īµ
1097:i
1093:g
1088:i
1084:Īµ
1067:.
1063:)
1055:i
1040:e
1036:z
1030:1
1026:(
1014:i
1010:g
1004:i
996:=
993:)
988:Z
983:(
971:=
865:K
788:O
692:e
685:t
678:v
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