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Bose-Einstein Condensation and Superfluidity Gordon Baym University of Illinois, Urbana January 2004. 東京大学. Fermions ( Fermi-Dirac, 1926 ): Particles that obey the exclusion principle ( Pauli, 1925 ). Can’t put two in same state at the same time.

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## Bose-Einstein Condensation and Superfluidity Gordon Baym University of Illinois, Urbana January 2004

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**Bose-Einstein Condensation and Superfluidity**Gordon Baym University of Illinois, Urbana January 2004 東京大学**Fermions (Fermi-Dirac, 1926):**Particles that obey the exclusion principle (Pauli, 1925). Can’t put two in same state at the same time. Bosons (Bose-Einstein, 1924-5): Particles that don’t obey the exclusion principle. Can put many in the same state at the same time**S. Bose**A. Einstein**S.N. Bose 1924: concept of light quanta as particles with**2 polarization states. New statistics => Planck distribution: A. Einstein 1924: Extension to monoatomic ideal gases: Condensation: Condensate**I maintain that in this case a number of molecules steadily**growing with increasing density goes over in the first quantum state ... a separation is affected; one part ‘condenses,’ the rest remains a saturated ideal gas. A. Einstein, 1925**Bose-Einstein Condensation**Hot atoms (bosons) in a box Cool below Bose-Einstein transition temperature At absolute zero temperature motion “ceases” Bose-Einstein condensate Gravity**Free Bose gas**Potential well (trap) Box In condensed system have macroscopic occupation of single (generally lowest) mode : ground state : flow state (vortex)**MANY-PARTICLE WAVE FUNCTION**= condensate wave function FINITE TEMPERATURE Thermal wavelength No. condensed particles**Which “statistics” apply to nature?**i.e., is ordinary matter made of fermions or bosons?**The [Fermi-Dirac] solution ... is probably the correct one**for gas molecules, since it is known to be the correct one for electrons in an atom, and one would expect molecules to resemble electrons more closely than light quanta. P.A.M. Dirac, 1926**With a heavy heart I have become converted to the idea that**Fermi … , not Einstein-Bose, is the correct statistics [for electrons]. W. Pauli to E. Schrödinger, Nov. 1926**Superfluid 4He:**The first Bose-Einstein condensate W.H. Keesom and Miss A.P. Keesom (1935): specific heat of liquid helium F.London (1938): Spectroscopic data => 4He obeys Bose-Einstein statistics: “The strange change of state of liquid helium at 2.19 o abs., even though it occurs in the liquid and not in the gaseous state, is due to the condensation mechanism of the Bose-Einstein gas.”**“It seems difficult not to imagine a connexion with**the condensation phenomenon of the Bose- Einstein statistics.” (London, 1938)**Superfluid Liquid Helium**1938 (Tony Leggett) Temperatures below “Lambda point” 2.17o above absolute zero Flows through tiny capillaries without friction Flows around a closed pipe forever**Spin bucket of superfluid**helium slowly. Helium liquid remains at rest! Spin fast enough. Form vortex in center of liquid!**L. Landau (1941): rejects suggestion “that**helium-II should be considered as a degenerate ideal Bose gas.” Importance of interactions! ROLE OF STATISTICS: Sydoriak, Grilley, and Hammel (1948) liquified 3He. Osborne, Winstock, and Abraham (1948): no superflow down to 1.05 K. Bose character critical to superfluidity**Order parameter of Bose-condensed system**-- 0 in normal system -- constant in BEC = complex order parameter Free particle state, |N> If |N> and |N-1> differ only by number of particles in condensate then In weakly interacting Bose gas:**Time dependent order parameter**condensate wave function condensate density superfluid velocity chemical potential superfluid acceleration eqn. Equilibrium:**Flow and superfluidity**Complex order parameter: => flow Superfluid velocity Superfluid mass density = Normal mass density = Momentum density of superfluid flow = rs vs Condensate density differs from superfluid mass density: At T=0 in 4He, rs = r, n0 = 0.09 n**BOSE CONDENSED SYSTEMS**Low temperature systems of bosons: liquid 4He trapped bosonic atoms excitons in semiconductors (?) Nuclear matter pion condensation kaon condensation Vacuum as Bose condensed state Chiral symmetry breaking Gluon condensation Higgs condensation Graviton condensation, gmn**PION CONDENSED MATTER**Softening of collective spin-isospin oscillation of nuclear matter Above critical density have transition to new state with nucleons rotated in isospin space: with formation of macroscopic pion field**Important, if it exists, for enhanced cooling of neutron**stars by neutrino emission Transition density very sensitive to effective particle-hole interactions (Landau g’) and D-hole interactions Analogous neutral pion condensate can coexist with**STRANGENESS (KAON) CONDENSATES**Analogous to p condensate Chiral SU(3) X SU(3) symmetry of strong interactions => effective low energy interaction Kaplan and Nelson (1986), Brown et al. (1994) “Effective mass” term lowers K energies in matter => condensation**Rotate u and s quark states:**Form condensate Admix in n; in p**Results very sensitive to K-**interactions in matter (Pandharipande, Pethick and Vesteinn, 1995) - * Would soften equation of state and lower maximum neutron star mass to ~ 1.5 solar masses * Would enhance neutrino luminosity and cooling of neutron stars Can also form condensate => macroscopic η field**EXPERIMENTAL BOSE-EINSTEIN**DECONDENSATION Ultrarelativistic heavy ion collisions: 2000: RHIC 100 GeV/A + 100 GeV/A colliding beams 2007?: LHC 2600 GeV/A + 2600 GeV/A**Break chiral symmetry in**different state? (Disordered chiral condensate?) Np ~104, V ~ 103 fm3 : p- BEC unlikely; entropy too high**Applications in Biology**A strong proponent of the idea that Bose-Einstein condensation may provide the “unitary sense of self” that seems to be characteristic of consciousness, in relation to Fröhlich’s ideas is Ian Marshall (1989) … R. Penrose, Shadows of the Mind (1994)**Application to the Movies**Information, Adaptive Contracting, Distributional Dynamics, Bayesian Choice, Bose-Einstein Statistics and the Movies

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