1 / 229

Phase transition in finite systems

Phase transition in finite systems. Philippe CHOMAZ, GANIL. Finite systems Zeroes of the partition sum Bimodal event distributions Negative heat capacity Abnormal fluctuations Experimental results Melting of Na clusters Superfluidity in atomic nuclei Fragmentation of nuclei

amara
Download Presentation

Phase transition in finite systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Phase transition in finite systems Philippe CHOMAZ, GANIL Finite systems Zeroes of the partition sum Bimodal event distributions Negative heat capacity Abnormal fluctuations Experimental results Melting of Na clusters Superfluidity in atomic nuclei Fragmentation of nuclei Fragmentation of H-clusters 2

  2. *****Chapitre 1

  3. IntroductionPhase transitionAnomaly in thermo - I -

  4. Order n: discontinuity in Ehrenfest’s definition EOS: Ex: first order: discontinuous Phase transition in infinite systems Thermodynamical potentials: Thermodynamical potentials: non analytical at L.E. Reichl, Texas Press (1980) Caloric curve Energy <E> EOS Temperature-1 R. Balian, Springer (1982) 4

  5. EOS: Ex: Continuous Phase transition in infinite systems Phase transition in infinite systems Thermodynamical potentials: non analytical at Thermodynamical potentials: Z >0, Caloric curve No phase transition continuous Energy <E> EOS Ehrenfest’s definition Temperature-1 6

  6. Caloric curve Energy <E> EOS Temperature-1 Caloric curve b Temperature E1 E2 Energy Phase transition in infinite systems Phase transition in infinite systems Thermodynamical potentials: Z >0, Thermodynamical potentials: non analytical at No phase transition continuous Ehrenfest’s definition But anomaly in the entropy Ex: negative heat capacity K. Binder, D.P. Landau Phys Rev B30 (1984) 1477; Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495.; W. Thirring, Z. Phys. 235, 339 (1970), D.H.E. Gross, Rep. Prog. Phys. 53, 605 (1990), 8

  7. Complex b Im(b) b PC & Gulminelli Phys A (2003) Re(b) • Free order param. (canonical) b E distribution at • Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404 • Bimodal distribution(P(E)) Energy K.C. Lee Phys Rev E 53 (1996) 6558 Caloric curve b Temperature E1 E2 • Fixed order para. (microcanonical) E1 E2 Energy • Back Bending in EOS(T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477 Lynden-Bell, D. & Wood, R. 1968, Mon. Not. R. Astr. Soc. 138, 495. sk /T2 Ck(3/2) • Abnormal fluctuation(sk(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153 E1 E2 Energy 1st order in finite systems 10

  8. *****Chapitre 2

  9. Origin of singularitiesZeroes of Z (b) Bimodal P(E) - II -

  10. , then logZ remains analytical If, when , no zeroes of Z converge on theReal b axis Imb => No phase transition Reb zeroes of Z Discontinuities from zeroes of Z Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z Complex bplane (C.N.Yang T.D.Lee 1952)

  11. , then logZ remains analytical If, when , no zeroes of Z converge on theReal b axis Imb => No phase transition Reb zeroes of Z Discontinuities from zeroes of Z Phase transitions are defined by the asymptotic distribution of zeroes of the partition sum Z Complex bplane (C.N.Yang T.D.Lee 1952) First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 Im(b) b0 Re(b) Yang - Lee theorem 12

  12. First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 Im(b) b0 Re(b) Yang - Lee theorem

  13. ∞ ∞ • A theoretical definition • Only valid asymptotically: V = • How to extend it, V ≠ ? • How to use it experimentally ? First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 Im(b) b0 Re(b) Yang - Lee theorem

  14. First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 Im(b) b0 Re(b) Yang - Lee theorem

  15. Z Laplace transform of P: Im(b) Re(b) Link with energy distribution P(E) Ph. Ch., F. Gulminelli, Physica A 2003 First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 b0

  16. Bimodal P with DEµN Im(b) Re(b) Link with energy distribution P(E) P1(E) P2(E) Z Laplace transform of P: <=> Ph. Ch., F. Gulminelli, Physica A 2003 First order phase transitions A uniform density of zero's on a line crossing the real baxis perpendicularly at b0 b0 14

  17. Fin Zero=>bimodalite

  18. Lattice gas model • Closest neighbors interaction • Metropolis Boltzmann weight e-E • Average volume <V> (pressure)

  19. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice gas model • <E> smooth fonction of (canonical caloric curve) • Closest neighbors interaction • Metropolis Boltzmann weight e-E • Average volume <V> (pressure) Lattice-gas Model 16

  20. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice gas model • <E> smooth function of (canonical caloric curve) • Closest neighbors interaction • Metropolis Boltzmann weight e-E • Average volume <V> (pressure) Lattice-gas Model

  21. Gas 100 Liquid 10 Energy Distribution 1 0.1 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 Canonical (Most Probable) 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice gas model • Energy distribution at a fix temperature Lattice-gas Model 18

  22. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 Canonical (Most Probable) 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice-gas model Gas 100 Liquid 10 Energy Distribution d 1 0.1 Lattice-gas Model • Energy distribution at a fixed temperature • Discontinuity of the most probable 18

  23. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 Canonical (Most Probable) 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice-gas model Gas Liquid 100 10 Distribution d ’énergie 1 0.1 Lattice-gas Model • Energy distribution at a fixed temperature • Bimodal distribution • Discontinuity of the most probable

  24. A first order phase transition in finite systems <=> A bimodal (event) distribution of energy If the energy is free to fluctuate (canonical - energy reservoire) 20

  25. A first order phase transition in finite systems <=> A bimodal (event) distribution of an observable (order parameter) If this observable is free to fluctuate (intensive ensemble)

  26. A3 A2 A1 • Access to order parameters • Nature of order parameter: • Is it collective? • Scaling of the jump with N: • What happen at the thermo limit (N = ∞) ? • Is it a macro. phase transition? A bimodal (event) distribution of an observable (order parameter) If this observable is free to fluctuate (intensive ensemble)

  27. Ising model Bimodal distributions Melting of Na Cluster Energy distribution M Liquid-gas in lattice-gas Multifragmentation of nuclei Temperature, Probability P (E) Fragment asymmetry (Zbig-Zsmall) Photo-dissociation 3hn 4hn 5hn INDRA Number of evaporated atoms Energy INDRA-2001 Haberland-PRL 2001

  28. ***** bébut Na

  29. E*(T) Cluster beam Variable T Helium Heat Bath + Melting of Na147. P(E) energy M.Schmidt et al, Nature 1999, PRL 2001

  30. E*(T)+ nhn E*(T) Cluster beam Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. P(E) energy +4hn M.Schmidt et al, Nature 1999, PRL 2001

  31. 3hn 5hn 6hn Photo-disintegration E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. 3hn +4hn 6hn P(E) energy counts +4hn Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  32. 3hn 5hn 6hn Photo-disintegration E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. 3hn 4hn 5hn 6hn P(E) energy counts 4hn Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  33. Melting 3hn 4hn 5hn 6hn Photo-disintegration Number of evaporated atoms E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. 3hn 4hn 5hn 6hn P(E) energy counts Temperature 3hn 4hn 5hn 6hn Number of evaporated atoms Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  34. Photo-disintegration E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. 3hn 4hn 5hn 6hn P(E) energy counts Temperature 3hn 4hn 5hn 6hn Number of evaporated atoms Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  35. Photo-disintegration E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. Energy distribution 3hn 4hn 5hn 6hn Expected Photo-dissociation P(E) energy counts Temperature 3hn 4hn 5hn 6hn Number of evaporated atoms Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  36. Photo-disintegration E*(T)+ nhn E*(T) Cluster beam detector Variable T Helium Heat Bath Laser E=nhn + Melting of Na147. Energy distribution 3hn 4hn 5hn 6hn Expected Photo-dissociation P(E) energy counts Temperature 3hn 4hn 5hn 6hn Number of evaporated atoms Observed Number of evaporated atoms M.Schmidt et al, Nature 1999, PRL 2001

  37. Ising model Bimodal distributions Melting of Na Cluster Energy distribution M Liquid-gas in lattice-gas Multifragmentation of nuclei Temperature, Probability P (E) Fragment asymmetry (Zbig-Zsmall) Photo-dissociation 3hn 4hn 5hn INDRA Number of evaporated atoms Energy INDRA-2001 Haberland-PRL 2001 24

  38. *****Fin bimodalité

  39. *****Chapitre 3

  40. Convex SBack bending TNegative heat capacity - III -

  41. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 Canonical (Most Probable) 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice-gas model Gas 100 Liquid 10 Energy Distribution 1 0.1 Lattice-gas Model • Energy distribution at a fixed temperature • Discontinuity of the most probable

  42. 3.6 3.5 Temperature (MeV) 3.4 Canonical (Average) 3.3 Canonical (Most Probable) 3.2 -2 -1 0 1 2 3 Energy (A MeV) Lattice-gas model Gas 100 Liquid 10 • E distribution • Pb(E) =W(E)e- bE/ Zb Energy Distribution 1 0.1 Lattice-gas Model

More Related