Random Integer Partitions and the Bose Gas Mathias Rafler 1 - PowerPoint PPT Presentation

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Random Integer Partitions and the Bose Gas Mathias Rafler 1 supervised by Sylvie Rœlly 1 and Hans Zessin 2 1 Universit¨ at Potsdam 2 Universit¨ at Bielefeld Disentis Summer School 2008

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Outline Classical Mechanics Free Particles in R d Quantum Mechanics Differences and the Bose Gas Partition Problem Extract from Bose Gas Quantum Mechanics Interpretation

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Classical Mechanics Free Particles in R d • given region G , place N point in G without interaction • given bigger region G ′ , repeat with N ′ points • is there a limit for G → R d , N | G | → u > 0?

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Classical Mechanics Free Particles in R d Answer: Yes, Poisson process on R d ! (Nguyen, Zessin, 1976) • #particles in G ∼ P u | G | • #particles in disjoint regions is independent

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Differences and the Bose Gas Basic Objects: Loops • x : [0 , β ] → R d , x (0) = x ( β ) • Brownian bridge of length β

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Differences and the Bose Gas Indistinguishable particles • loops may concatenate Composite loops • x : [0 , j β ] → R d , x (0) = x ( j β ) • Brownian bridge of length j β (Ginibre)

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Differences and the Bose Gas Indistinguishable particles • loops may concatenate Composite loops • x : [0 , j β ] → R d , x (0) = x ( j β ) • Brownian bridge of length j β (Ginibre)

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Differences and the Bose Gas Ginibre Gas • # j -loops in G ∼ P cj − (1+ d / 2) | G | • #loops of different lengths and in disjoint regions is independent Consequence • expected #loops j ≥ 1 j − (1+ d / 2) c | G | � • expected #particles j ≥ 1 j − d / 2 c | G | � N Limit as | G | → u > 0?

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Differences and the Bose Gas Theorem (Ginibre) Partition function of a system of free particles obeying Bose Statistics at inverse temperature β and fugacity z �� � z j � α G ( ω ) P j β Z G = exp xx ( d ω ) d x j j ≥ 1 [Ginibre J: Some Applications of functional Integration in Statistical Mechanics. Statist. Mech. and Quantum Field Theory, Les Houches Summer School Theoret. Phys, (Gordon and Breach), 1971, 327-427]

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas Suppose for α > 1 fixed, j = 1 , 2 , . . . , X j ∼ P tj − (1+ α ) independent. Put X = ( X 1 , X 2 , . . . ) � N ( X ) = jX j . j ≥ 1 Consider P t , N t = Law ( X | N ( X ) = N t ) . Partition problem: How does X N t behave as t → ∞ , N t t → u > 0?

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas By construction, 1 N ( X ) = 1 � jX j = 1 P t , N t -a.s. N t N t j ≥ 1 Let X Y := lim , N t t →∞ by Fatou’s lemma � N ( Y ) = jY j ≤ 1 . j ≥ 1 Is equality preserved in the Limit?

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas By construction, 1 N ( X ) = 1 � jX j = 1 P t , N t -a.s. N t N t j ≥ 1 Let X Y := lim , N t t →∞ by Fatou’s lemma � N ( Y ) = jY j ≤ 1 . j ≥ 1 Sometimes.

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas By construction, 1 N ( X ) = 1 � jX j = 1 P t , N t -a.s. N t N t j ≥ 1 Let X Y := lim , N t t →∞ by Fatou’s lemma � N ( Y ) = jY j ≤ 1 . j ≥ 1 But not always!

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas Theorem Subject to given conditions � z j if u ≤ u ∗ j α ujY j = 1 if u > u ∗ j α where z = z ( u ) is the solution of z j u ∧ u ∗ = � j α . j ≥ 1

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas Theorem Subject to given conditions � if u ≤ u ∗ u uN ( Y ) = u ∗ if u > u ∗ where z = z ( u ) is the solution of z j u ∧ u ∗ = � j α . j ≥ 1

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas Theorem Subject to given conditions � if u ≤ u ∗ 1 N ( Y ) = u ∗ if u > u ∗ u < 1 where z = z ( u ) is the solution of z j u ∧ u ∗ = � j α . j ≥ 1

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Partition Problem Extract from Bose Gas Theorem Subject to given conditions � if u ≤ u ∗ 1 N ( Y ) = u ∗ if u > u ∗ u < 1 Elements of proof • show LDP for X N t to get convergence and to obtain variational problem; • problems: poor continuity properties, small sets, minimisation problem with constraints

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Quantum Mechanics Interpretation Competition: density vs. Brownian bridges • low density. interparticle distance too large to build large composite loops • high density. interparticle distance small; even possibility to build infinitely long loops (Bose-Einstein Condensation)

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Ginibre J: Some Applications of functional Integration in Statistical Mechanics. Statist. Mech. and Quantum Field Theory, Les Houches Summer School Theoret. Phys, (Gordon and Breach), 1971, 327-427 Nguyen X X, Zessin H: Martin-Dynkin boundary of mixed Poisson processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 37 (1976/77), no. 3, 191–200 Kingman, J F C: The representation of partition structures, J. London Math. Soc. (2) 18 (1978), no. 2, 374–380 F¨ ollmer H: Phase transition and Martin Boundary, Seminaire de probabilites (Strasbourg) 9 (1975), 305–17 R: Martin-Dynkin boundaries of the Bose Gas, Preprint