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

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

Dec 16, 2022 483 likes •692 views

Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics Random Integer Partitions and the Bose Gas Mathias Rafler 1 supervised by Sylvie Rlly 1 and Hans Zessin 2 1 Universit at Potsdam 2 Universit at Bielefeld

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Random Integer Partitions and the Bose Gas

Mathias Rafler1 supervised by Sylvie Rœlly1 and Hans Zessin2

1Universit¨

at Potsdam

2Universit¨

at Bielefeld

Disentis Summer School 2008

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Outline

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

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Classical Mechanics

Free Particles in Rd

  • given region G, place N point

in G without interaction

  • given bigger region G ′, repeat

with N′ points

  • is there a limit for G → Rd,

N |G| → u > 0?

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Classical Mechanics

Free Particles in Rd

Answer: Yes, Poisson process on Rd! (Nguyen, Zessin, 1976)

  • #particles in G ∼ P

u|G|

  • #particles in disjoint regions is

independent

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Quantum Mechanics

Differences and the Bose Gas

Basic Objects: Loops

  • x : [0, β] → Rd, x(0) = x(β)
  • Brownian bridge of length β
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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, jβ] → Rd, x(0) = x(jβ)
  • Brownian bridge of length jβ

(Ginibre)

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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, jβ] → Rd, x(0) = x(jβ)
  • Brownian bridge of length jβ

(Ginibre)

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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

c|G|

j≥1 j−(1+d/2)

  • expected #particles

c|G|

j≥1 j−d/2

Limit as

N |G| → u > 0?

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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 ZG = exp

  • j≥1

zj j

  • αG(ω)Pjβ

xx(dω)dx

  • [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]

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

Suppose for α > 1 fixed, j = 1, 2, . . ., Xj ∼ P

tj−(1+α)

  • independent. Put

X = (X1, X2, . . .) N(X) =

  • j≥1

jXj. Consider Pt,Nt = Law(X|N(X) = Nt). Partition problem: How does X

Nt behave as t → ∞, Nt t → u > 0?

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

By construction, 1 Nt N(X) = 1 Nt

  • j≥1

jXj = 1 Pt,Nt-a.s. Let Y := lim

t→∞

X Nt , by Fatou’s lemma N(Y ) =

  • j≥1

jYj ≤ 1. Is equality preserved in the Limit?

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

By construction, 1 Nt N(X) = 1 Nt

  • j≥1

jXj = 1 Pt,Nt-a.s. Let Y := lim

t→∞

X Nt , by Fatou’s lemma N(Y ) =

  • j≥1

jYj ≤ 1. Sometimes.

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

By construction, 1 Nt N(X) = 1 Nt

  • j≥1

jXj = 1 Pt,Nt-a.s. Let Y := lim

t→∞

X Nt , by Fatou’s lemma N(Y ) =

  • j≥1

jYj ≤ 1. But not always!

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

Theorem

Subject to given conditions ujYj = zj

jα

if u ≤ u∗

1 jα

if u > u∗ where z = z(u) is the solution of u ∧ u∗ =

  • j≥1

zj jα .

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

Theorem

Subject to given conditions uN(Y ) =

  • u

if u ≤ u∗ u∗ if u > u∗ where z = z(u) is the solution of u ∧ u∗ =

  • j≥1

zj jα .

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

Theorem

Subject to given conditions N(Y ) =

  • 1

if u ≤ u∗

u∗ u < 1

if u > u∗ where z = z(u) is the solution of u ∧ u∗ =

  • j≥1

zj jα .

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Outline Classical Mechanics Quantum Mechanics Partition Problem Quantum Mechanics

Partition Problem

Extract from Bose Gas

Theorem

Subject to given conditions N(Y ) =

  • 1

if u ≤ u∗

u∗ u < 1

if u > u∗

Elements of proof

  • show LDP for X

Nt to get convergence and to obtain variational

problem;

  • problems: poor continuity properties, small sets, minimisation

problem with constraints

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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)

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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¨

  • llmer H: Phase transition and Martin Boundary, Seminaire

de probabilites (Strasbourg) 9 (1975), 305–17 R: Martin-Dynkin boundaries of the Bose Gas, Preprint