Boson random point processes and condensation
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LENARY SESSION
Boson Random Point Processes and Condensation1, 2 V. A. Zagrebnov Université de la Méditerranée(AixMarseille II) and Centre de Physique Théorique – UMR 6207 LuminyCase 907, 13288 Marseille Cedex 9, France Abstract—This is a short survey of the Boson Random Point Processes method and its application to the meanfield interacting boson gas trapped by a weak harmonic potential. DOI: 10.1134/S1063779610060122
(1) For any set D ⊂ E with finite Lebesgue measure ν(D) one puts:
1.2 1INTRODUCTION: RANDOM POINT PROCESSES We start by recall of some notations and definitions that we need to formulate our results. For details the reader may consult, for example, the book [1].
∫
⺠ { ND = n } =
n
Q(E)
( ην ( D ) ) –ην ( D ) π η ( dξ )δ n, ND ( ξ ) = e . n!
(a) Let E be a locally compact metric space serving as a state space of points, ᑜ the Borel σalgebra, ᑜ0 ⊆ ᑜ (relatively) compact Borel sets. Let ν be a (diffusive) locally finite reference measure on (E, ᑜ). The stan dard example: ν is the Lebesgue measure and E = ⺢d.
(2) For mutually disjoint subsets {Dn ⊂ Λ}n ≥ 1 the Poisson RPP πη is supposed to be uncorrelated:
(b) The space of the locally finite configurations of points in E is:
= ⺕ πη ( δ n1, ND ( ξ ) )… ⺕ πη ( δ nk, ND ( ξ ) )
⺕ πη ( δ n1, ND ( ξ ) …δ nk, ND ( ξ ) ) 1
1
Then Q(Λ) := {ξ ∈ Q : ξ ⊂ Λ} and the function: NΛ : ξ card(ξ ∩ Λ). (c) Each ξ ∈ Q can be identified with integervalued nonnegative Radon measure: λξ := δ on ᑜ, x∈ξ x i.e. λξ(D) := ND is the number of points that fall into the set D for the locallyfinite point configuration ξ ∈ Q(D).
∑
μ ( dξ )F ( ξ ).
Q(E)
• For a simple random point process the measure μ assigns a.s.: μ(x) ≤ 1, for any single point x ∈ Q(E).
(f) Definition: For any family of mutually disjoint subsets {Dn ⊂ Λ}n ≥ 1 the correlation functions (joint intensities) of the RPP μ are defined by the densities
1 Dedicated to the 100th anniversary of N.N. Bogolyubov’s birth. 2 The article is published in the original.
⺢ + } n ≥ 1 with respect to the measure ν: 1
{ρn : Λn
⺕ μ ⎛⎝
∫
=
∏⺙
1≤j≤n
⎞ ⎠
ξ ∩ Dj = 1
ν ( dx 1 )…ν ( dx n )ρ n ( x 1, …, x n ),
D1 × … × Dn
(g) Definition: A RPP is called determinantal/per manental with (a locally Trclass) kernel K, if it is sim ple and its correlation functions are:
• By K(x, y) we denote a kernel of nonnegative, selfadjoint, locally Trclass operator K ≥ 0 on L2(Λ). (e) Example: (The Poisson RPP πη with intensity η ≥ 0)
n
( ην ( D 1 ) ) –ην ( D1 ) ( ην ( k ) ) k –ην ( Dk ) = … e . e n 1! n k!
(d) Definition: A random point processes in a locally compact space E is a random probability Radon mea sure μ on the configuration space Q(E), with expecta tion that for any measurable function is defined by:
∫
k
n1
Q(E) := {ξ ⊂ E : card(ξ ∩ Λ) < ∞ for all Λ ∈ ᑜ0}.
⺕μ(F) :=
k
ρ n ( x 1, …, x n ) = det K ( x i, x j )
1 ≤ i, j ≤ n ,
ρ n ( x 1, …, x n ) = per K ( x i, x j )
1 ≤ i, j ≤ n .
For any n ≥ 1 and x1, …, xn ∈ Λ. detαA :=
∑
σ ∈
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