Phase-space approach to bosonic mean field dynamics

Transcription

1 field dynamics sept. 5th, 2013

2 Outline

3 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic N-particle space is L 2 s (X N, m N ; C) = N Z. (A, D(A)) is a s.a. operator on Z V (x) = V ( x) is a real function on X.

4 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic N-particle space is L 2 s (X N, m N ; C) = N Z. (A, D(A)) is a s.a. operator on Z V (x) = V ( x) is a real function on X.

5 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic N-particle space is L 2 s (X N, m N ; C) = N Z. (A, D(A)) is a s.a. operator on Z V (x) = V ( x) is a real function on X. N-body quantum dynamics { i t Ψ = N i=1 A iψ + 1 N i<j V (x i x j )Ψ Ψ(t = 0) = Ψ 0 N Z with A i = Id Z }{{} A Id Z. i

6 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic N-particle space is L 2 s (X N, m N ; C) = N Z. (A, D(A)) is a s.a. operator on Z V (x) = V ( x) is a real function on X. N-body quantum dynamics { i t Ψ = N i=1 A iψ + 1 N i<j V (x i x j )Ψ Ψ(t = 0) = Ψ 0 N Z with A i = Id Z }{{} A Id Z. Mean field limit N i

7 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic N-particle space is L 2 s (X N, m N ; C) = N Z. (A, D(A)) is a s.a. operator on Z V (x) = V ( x) is a real function on X. N-body quantum dynamics { iε t Ψ = ε N i=1 A iψ + ε 2 i<j V (x i x j )Ψ Ψ(t = 0) = Ψ 0 1 ε Z with A i = Id Z }{{} A Id Z. Mean field limit N ε 0. Divide by N and set ε = 1 N. i

8 Fock space formulation Rewriting in the bosonic Fock space Γ(Z) = n=0 n Z. { iε t Ψ ε = H ε Ψ ε Ψ ε (t = 0) = Ψ ε 0 Γ(Z), where H ε = A(x, y)a (x)a(y)dm(x)dm(y) X V (x y)a (x)a (y)a(x)a(y) dm(x)dm(y) 2 X 2 is the Wick quantization of the energy E(z, z) = z(x)a(x, y)z(y)dm(x)dm(y) X V (x y) z(x) 2 z(y) 2 dm(x)dm(y), 2 X 2 with the ε-dependent CCR s: [a(x), a(y)] = [a (x), a (y)] = 0, [a(x), a (y)] = εδ(x y).

9 Mean field=semiclassical The relation between iε t Ψ ε = H ε Ψ ε and the hamiltonian dynamics in the phase-space (Z, Im, Z ) i t z = z E in the limit is actually a semiclassical problem. ε 0, [a(x), a (y)] = εδ(x y), H ε = E Wick. Perfectly known when X = {1,..., K} = Z/(KZ), Z = C K R 2K, Γ(Z) L 2 (R K, dx; C), (semiclassical pseudo-differential calculus, Fourier integral operators, Egorov s theorem, higher order expansions are possible). Take a (k) = h xk + x k, a(k) = h xk + x k (k X, x k R) with ( 1 N ) ε = 2h.

10 Mean field=semiclassical The relation between iε t Ψ ε = H ε Ψ ε and the hamiltonian dynamics in the phase-space (Z, Im, Z ) i t z = z E in the limit is actually a semiclassical problem. ε 0, [a(x), a (y)] = εδ(x y), H ε = E Wick. Perfectly known when X = {1,..., K} = Z/(KZ), Z = C K R 2K, Γ(Z) L 2 (R K, dx; C), (semiclassical pseudo-differential calculus, Fourier integral operators, Egorov s theorem, higher order expansions are possible). Take a (k) = h xk + x k, a(k) = h xk + x k (k X, x k R) with ( 1 N ) ε = 2h.

11 Mean field=semiclassical The relation between iε t Ψ ε = H ε Ψ ε and the hamiltonian dynamics in the phase-space (Z, Im, Z ) i t z = z E in the limit is actually a semiclassical problem. ε 0, [a(x), a (y)] = εδ(x y), H ε = E Wick. Perfectly known when X = {1,..., K} = Z/(KZ), Z = C K R 2K, Γ(Z) L 2 (R K, dx; C), (semiclassical pseudo-differential calculus, Fourier integral operators, Egorov s theorem, higher order expansions are possible). Take a (k) = h xk + x k, a(k) = h xk + x k (k X, x k R) with ( 1 N ) ε = 2h.

12 Weyl observables For f Z, W (f ) = e iφ(f ). φ(f ) = 1 (a (f ) + a(f )) = X f 2 (x)a (x) dm(x)+ f (x)a(x) dm(x), X W (f 1 ) W (f 2 ) = e i ε 2 Im f1,f2 W (f 1 + f 2 ). Separation of variables: Γ(Z) = Γ(pZ) Γ((pZ) ) and b Weyl = b Weyl Id Γ((pZ) ). Can be extended to the Weyl-Hörmander symbol class p n S((1 + pz 2 ) n/2, d pz 2 ) which contains cylindrical polynomial functions and forms an algebra for the Moyal product with b Weyl 1 b Weyl 2 = (b 1 b 2 ) Weyl + εr(b 1, b 2, ε).

13 Weyl observables For f Z, W (f ) = e iφ(f ). Let p be a finite rank orthogonal projection. When b S cyl (Z) based on pz, i.e. b(z) = b(pz) with b S(pZ), b Weyl = Fb(ξ)W ( 2πξ) L pz (dξ), pz with Fb(ξ) = e 2iπ Re ξ,z b(z) L pz (dz). pz Separation of variables: Γ(Z) = Γ(pZ) Γ((pZ) ) and b Weyl = b Weyl Id Γ((pZ) ). Can be extended to the Weyl-Hörmander symbol class p n S((1 + pz 2 ) n/2, d pz 2 ) which contains cylindrical polynomial functions and forms an algebra for the Moyal product with b Weyl 1 b Weyl 2 = (b 1 b 2 ) Weyl + εr(b 1, b 2, ε).

14 Weyl observables For f Z, W (f ) = e iφ(f ). Let p be a finite rank orthogonal projection. When b S cyl (Z) based on pz, i.e. b(z) = b(pz) with b S(pZ), b Weyl = Fb(ξ)W ( 2πξ) L pz (dξ), pz with Fb(ξ) = e 2iπ Re ξ,z b(z) L pz (dz). pz Separation of variables: Γ(Z) = Γ(pZ) Γ((pZ) ) and b Weyl = b Weyl Id Γ((pZ) ). Can be extended to the Weyl-Hörmander symbol class p n S((1 + pz 2 ) n/2, d pz 2 ) which contains cylindrical polynomial functions and forms an algebra for the Moyal product with b Weyl 1 b Weyl 2 = (b 1 b 2 ) Weyl + εr(b 1, b 2, ε).

15 Weyl observables For f Z, W (f ) = e iφ(f ). Let p be a finite rank orthogonal projection. When b S cyl (Z) based on pz, i.e. b(z) = b(pz) with b S(pZ), b Weyl = Fb(ξ)W ( 2πξ) L pz (dξ), pz with Fb(ξ) = e 2iπ Re ξ,z b(z) L pz (dz). pz Separation of variables: Γ(Z) = Γ(pZ) Γ((pZ) ) and b Weyl = b Weyl Id Γ((pZ) ). Can be extended to the Weyl-Hörmander symbol class p n S((1 + pz 2 ) n/2, d pz 2 ) which contains cylindrical polynomial functions and forms an algebra for the Moyal product with b Weyl 1 b Weyl 2 = (b 1 b 2 ) Weyl + εr(b 1, b 2, ε).

16 Wick observables Monomials: b L( p Z; q Z), [ bu](x 1,..., x q ) = b(x, y)u(y 1,..., y p ) dm p (y), X p b(z) = z q, bz p, b Wick = b(x, y)a (x 1 )... a (x q )a(y 1 )... a(y p ) dm q (x)dm p (y). X p+q With the product Wick, b Wick 1 b Wick 2 = (b 1 Wick b 2 ) Wick, P(Z) is an algebra and b 1 Wick b 2 = min{p 1,q 2} k=0 ε k k! k z b 1. k z b 2. In particular ε 1 [ b1 Wick, b2 Wick ] = z b 1. z b 2 z b 2. z b 1 + εr(b 1, b 2, ε). When b has a finite rank there exists a cylindrical polynomial c such that

17 Wick observables Monomials: b L( p Z; q Z), Polynomials: b alg p,qp p,q (Z). b Pp,q(Z) r, 1 r, means b L r ( p Z; q Z). b(z) = z q, bz p for b P r p,q(z), P r (Z) = alg p,qp r p,q(z). With the product Wick, b Wick 1 b Wick 2 = (b 1 Wick b 2 ) Wick, P(Z) is an algebra and b 1 Wick b 2 = min{p 1,q 2} k=0 ε k k! k z b 1. k z b 2. In particular ε 1 [ b1 Wick, b2 Wick ] = z b 1. z b 2 z b 2. z b 1 + εr(b 1, b 2, ε). When b has a finite rank there exists a cylindrical polynomial c such that b Wick = c Weyl + εr(b, ε).

18 Wick observables Monomials: b L( p Z; q Z), Polynomials: b alg p,qp p,q (Z). b Pp,q(Z) r, 1 r, means b L r ( p Z; q Z). b(z) = z q, bz p for b P r p,q(z), P r (Z) = alg p,qp r p,q(z). With the product Wick, b Wick 1 b Wick 2 = (b 1 Wick b 2 ) Wick, P(Z) is an algebra and b 1 Wick b 2 = min{p 1,q 2} k=0 ε k k! k z b 1. k z b 2. In particular ε 1 [ b1 Wick, b2 Wick ] = z b 1. z b 2 z b 2. z b 1 + εr(b 1, b 2, ε). When b has a finite rank there exists a cylindrical polynomial c such that b Wick = c Weyl + εr(b, ε).

19 Wick observables Monomials: b L( p Z; q Z), Polynomials: b alg p,qp p,q (Z). b Pp,q(Z) r, 1 r, means b L r ( p Z; q Z). b(z) = z q, bz p for b P r p,q(z), P r (Z) = alg p,qp r p,q(z). With the product Wick, b Wick 1 b Wick 2 = (b 1 Wick b 2 ) Wick, P(Z) is an algebra and b 1 Wick b 2 = min{p 1,q 2} k=0 ε k k! k z b 1. k z b 2. In particular ε 1 [ b1 Wick, b2 Wick ] = z b 1. z b 2 z b 2. z b 1 + εr(b 1, b 2, ε). When b has a finite rank there exists a cylindrical polynomial c such that b Wick = c Weyl + εr(b, ε).

20 Wick observables Monomials: b L( p Z; q Z), Polynomials: b alg p,qp p,q (Z). b Pp,q(Z) r, 1 r, means b L r ( p Z; q Z). b(z) = z q, bz p for b P r p,q(z), P r (Z) = alg p,qp r p,q(z). With the product Wick, b Wick 1 b Wick 2 = (b 1 Wick b 2 ) Wick, P(Z) is an algebra and b 1 Wick b 2 = min{p 1,q 2} k=0 ε k k! k z b 1. k z b 2. In particular ε 1 [ b1 Wick, b2 Wick ] = z b 1. z b 2 z b 2. z b 1 + εr(b 1, b 2, ε). When b has a finite rank there exists a cylindrical polynomial c such that b Wick = c Weyl + εr(b, ε).

21 Main difficulty There is a natural quantization of cylindrical symbols (Weyl quantization) and of rather general polynomials (Wick quantization). Although these classes are algebra for the operator product (Moyal or Wick product), they are not preserved by the action of a nonlinear symplectic flow. Already at the symbolic level: No general Egorov theorem. The aim is thus to get a propagation result by the hamiltonian mean field flow with a flexible use of those Wick and Weyl observables. The small parameter helps.

22 Main difficulty There is a natural quantization of cylindrical symbols (Weyl quantization) and of rather general polynomials (Wick quantization). Although these classes are algebra for the operator product (Moyal or Wick product), they are not preserved by the action of a nonlinear symplectic flow. Already at the symbolic level: No general Egorov theorem. The aim is thus to get a propagation result by the hamiltonian mean field flow with a flexible use of those Wick and Weyl observables. The small parameter helps.

23 Main difficulty There is a natural quantization of cylindrical symbols (Weyl quantization) and of rather general polynomials (Wick quantization). Although these classes are algebra for the operator product (Moyal or Wick product), they are not preserved by the action of a nonlinear symplectic flow. Already at the symbolic level: No general Egorov theorem. The aim is thus to get a propagation result by the hamiltonian mean field flow with a flexible use of those Wick and Weyl observables. The small parameter helps.

24 Definition Consider the quantity Tr [Aϱ ε ] which extends the ψ ε, Aψ ε when ϱ ε is a general normal state (parametrized by ε). Admissible sets E for the parameter ε are bounded subsets of (0, + ) such that 0 E. Definition: Let (ϱ ε ) ε E be a family or normal states in Γ(Z), with Z separable and E admissible. The set of of (ϱ ε ) ε E M(ϱ ε, ε E) is the set of Borel probability measures µ on Z such that there exists an admissible subset E µ E for which ξ Z, lim ε 0 ε E µ Tr [ W ( 2πξ)ϱ ε] = e 2iπ Re ξ, z dµ(z). Z Theorem (Ammari N. Ann.IHP 08): With N = ( z 2 ) Wick = εn ε=1, the simple condition ensures M(ϱ ε, ε E). δ > 0, ε E, Tr [ N δ ϱ ε] C δ,

25 Definition Consider the quantity Tr [Aϱ ε ] which extends the ψ ε, Aψ ε when ϱ ε is a general normal state (parametrized by ε). Admissible sets E for the parameter ε are bounded subsets of (0, + ) such that 0 E. Definition: Let (ϱ ε ) ε E be a family or normal states in Γ(Z), with Z separable and E admissible. The set of of (ϱ ε ) ε E M(ϱ ε, ε E) is the set of Borel probability measures µ on Z such that there exists an admissible subset E µ E for which ξ Z, lim ε 0 ε E µ Tr [ W ( 2πξ)ϱ ε] = e 2iπ Re ξ, z dµ(z). Z Theorem (Ammari N. Ann.IHP 08): With N = ( z 2 ) Wick = εn ε=1, the simple condition ensures M(ϱ ε, ε E). δ > 0, ε E, Tr [ N δ ϱ ε] C δ,

26 Definition Consider the quantity Tr [Aϱ ε ] which extends the ψ ε, Aψ ε when ϱ ε is a general normal state (parametrized by ε). Admissible sets E for the parameter ε are bounded subsets of (0, + ) such that 0 E. Definition: Let (ϱ ε ) ε E be a family or normal states in Γ(Z), with Z separable and E admissible. The set of of (ϱ ε ) ε E M(ϱ ε, ε E) is the set of Borel probability measures µ on Z such that there exists an admissible subset E µ E for which ξ Z, lim ε 0 ε E µ Tr [ W ( 2πξ)ϱ ε] = e 2iπ Re ξ, z dµ(z). Z Theorem (Ammari N. Ann.IHP 08): With N = ( z 2 ) Wick = εn ε=1, the simple condition ensures M(ϱ ε, ε E). δ > 0, ε E, Tr [ N δ ϱ ε] C δ,

27 Definition Consider the quantity Tr [Aϱ ε ] which extends the ψ ε, Aψ ε when ϱ ε is a general normal state (parametrized by ε). Admissible sets E for the parameter ε are bounded subsets of (0, + ) such that 0 E. Definition: Let (ϱ ε ) ε E be a family or normal states in Γ(Z), with Z separable and E admissible. The set of of (ϱ ε ) ε E M(ϱ ε, ε E) is the set of Borel probability measures µ on Z such that there exists an admissible subset E µ E for which ξ Z, lim ε 0 ε E µ Tr [ W ( 2πξ)ϱ ε] = e 2iπ Re ξ, z dµ(z). Z Theorem (Ammari N. Ann.IHP 08): With N = ( z 2 ) Wick = εn ε=1, the simple condition ensures M(ϱ ε, ε E). δ > 0, ε E, Tr [ N δ ϱ ε] C δ,

28 Simple properties M(ϱ ε, ε E) = {µ} is characterized by b S cyl (Z), lim Tr [ b Weyl ϱ ε] = ε 0 ε E Moreover Z z 2δ dµ(z) C δ. Examples Coherent states: Ψ ε = E(z 0 ) = W ( 2 iε z 0) Ω, ϱ ε = E(z 0 ) E(z 0 ). M( E(z 0 ) E(z 0 ), ε E) = {δ z0 }. Z b(z) dµ(z). Hermite states: Ψ ε = z N 0 with z 0 = 1 and N = [ 1 ε]. } M( z0 N z N 0, ε E) = {δz S1 0 with δ S1 z 0 = 1 2π 2π 0 δ e iθ z 0 dθ.

29 Simple properties M(ϱ ε, ε E) = {µ} is characterized by b S cyl (Z), lim Tr [ b Weyl ϱ ε] = ε 0 ε E Moreover Z z 2δ dµ(z) C δ. Examples Coherent states: Ψ ε = E(z 0 ) = W ( 2 iε z 0) Ω, ϱ ε = E(z 0 ) E(z 0 ). M( E(z 0 ) E(z 0 ), ε E) = {δ z0 }. Z b(z) dµ(z). Hermite states: Ψ ε = z N 0 with z 0 = 1 and N = [ 1 ε]. } M( z0 N z N 0, ε E) = {δz S1 0 with δ S1 z 0 = 1 2π 2π 0 δ e iθ z 0 dθ.

30 Simple properties M(ϱ ε, ε E) = {µ} is characterized by b S cyl (Z), lim Tr [ b Weyl ϱ ε] = ε 0 ε E Moreover Z z 2δ dµ(z) C δ. Examples Coherent states: Ψ ε = E(z 0 ) = W ( 2 iε z 0) Ω, ϱ ε = E(z 0 ) E(z 0 ). M( E(z 0 ) E(z 0 ), ε E) = {δ z0 }. Z b(z) dµ(z). Hermite states: Ψ ε = z N 0 with z 0 = 1 and N = [ 1 ε]. } M( z0 N z N 0, ε E) = {δz S1 0 with δ S1 z 0 = 1 2π 2π 0 δ e iθ z 0 dθ.

31 An example with mean field correlations Take Ψ ε = 1 ε N 1+N 2N1!N 2! [a (φ 1 )] N1 [a (φ 2 )] N2 Ω, with lim εn 1 = lim εn 2 = 1 ε 0 ε 0 2, and φ 1 φ 2, φ 1 = φ 2 = 1. Γ(Z) = Γ(Cφ 1 ) Γ(Cφ 2 ) Γ({φ 1, φ 2 } ) Ψ ε = Ψ ε 1 Ψ 2 Ω, ϱ ε = ϱ ε 1 ϱ ε 2 Ω Ω, 1 2 µ = δ S1 2 δ S 2 φ1 2 φ2 δ 0. µ carried by a torus. Other writing: ψ 0 = 1 2 (φ 1 + φ 2 ), ψ π 2 = i 2 (φ 1 φ 2 ), ψ ϕ = cos(ϕ)ψ 0 + sin(ϕ)ψ π µ = 1 2π 2π 0 δ S1 ψ ϕ dϕ. 2,

32 An example with mean field correlations Take Ψ ε = 1 ε N 1+N 2N1!N 2! [a (φ 1 )] N1 [a (φ 2 )] N2 Ω, with lim εn 1 = lim εn 2 = 1 ε 0 ε 0 2, and φ 1 φ 2, φ 1 = φ 2 = 1. Γ(Z) = Γ(Cφ 1 ) Γ(Cφ 2 ) Γ({φ 1, φ 2 } ) Ψ ε = Ψ ε 1 Ψ 2 Ω, ϱ ε = ϱ ε 1 ϱ ε 2 Ω Ω, 1 2 µ = δ S1 2 δ S 2 φ1 2 φ2 δ 0. µ carried by a torus. Other writing: ψ 0 = 1 2 (φ 1 + φ 2 ), ψ π 2 = i 2 (φ 1 φ 2 ), ψ ϕ = cos(ϕ)ψ 0 + sin(ϕ)ψ π µ = 1 2π 2π 0 δ S1 ψ ϕ dϕ. 2,

33 An example with mean field correlations Take Ψ ε = 1 ε N 1+N 2N1!N 2! [a (φ 1 )] N1 [a (φ 2 )] N2 Ω, with lim εn 1 = lim εn 2 = 1 ε 0 ε 0 2, and φ 1 φ 2, φ 1 = φ 2 = 1. Γ(Z) = Γ(Cφ 1 ) Γ(Cφ 2 ) Γ({φ 1, φ 2 } ) Ψ ε = Ψ ε 1 Ψ 2 Ω, ϱ ε = ϱ ε 1 ϱ ε 2 Ω Ω, 1 2 µ = δ S1 2 δ S 2 φ1 2 φ2 δ 0. µ carried by a torus. Other writing: ψ 0 = 1 2 (φ 1 + φ 2 ), ψ π 2 = i 2 (φ 1 φ 2 ), ψ ϕ = cos(ϕ)ψ 0 + sin(ϕ)ψ π µ = 1 2π 2π 0 δ S1 ψ ϕ dϕ. 2,

34 and Wick observables Assume k N, C k > 0, ε E and M(ϱ ε, ε E) = {µ}. Then b P (Z), lim ε 0 ε E Tr [ ϱ ε N k] C k Tr [ b Wick ϱ ε] = b(z) dµ(z). Z

35 and Wick observables Assume k N, C k > 0, ε E and M(ϱ ε, ε E) = {µ}. Then b P (Z), lim ε 0 ε E Tr [ ϱ ε N k] C k Tr [ b Wick ϱ ε] = b(z) dµ(z). Z

36 and Wick observables Assume k N, C k > 0, ε E and M(ϱ ε, ε E) = {µ}. Then b P (Z), lim ε 0 ε E Tr [ ϱ ε N k] C k Tr [ b Wick ϱ ε] = Z b(z) dµ(z). But it is not true for general b P(Z) without additional assumptions.

37 and Wick observables Assume k N, C k > 0, ε E and M(ϱ ε, ε E) = {µ}. Then b P (Z), lim ε 0 ε E Tr [ ϱ ε N k] C k Tr [ b Wick ϱ ε] = Z b(z) dµ(z). But it is not true for general b P(Z) without additional assumptions. Example: (e n ) n N Hilbert basis of Z. Take ϱ ε = E(e [ε 1 ]) E(e [ε 1 ]). Then M(ϱ ε, ε E) = {δ 0 } and 0 = Z z 2 δ 0 (z) 1 = lim ε 0 ε E Tr [ϱ ε N] = lim ε 0 ε E Tr [ ϱ ε ( z 2 ) Wick].

38 Condition (PI) and consequences Assuming M(ϱ ε, ε E) = {µ}, the condition (PI ) says k N, lim Tr [ ϱ ε N k] = z 2k dµ(z). ε 0 Z ε E

39 Condition (PI) and consequences Assuming M(ϱ ε, ε E) = {µ}, the condition (PI ) says k N, lim Tr [ ϱ ε N k] = z 2k dµ(z). ε 0 Z ε E It is equivalent (Ammari-N. J.M.P.A 11) to b P(Z), lim Tr [ ϱ ε b Wick] = b(z) dµ(z). ε 0 Z ε E

40 Condition (PI) and consequences Assuming M(ϱ ε, ε E) = {µ}, the condition (PI ) says k N, lim Tr [ ϱ ε N k] = z 2k dµ(z). ε 0 Z ε E Define the reduced density matrices (see BBGKY hierarchy) by b L( p [ ] Z), Tr γp b ε = Tr [ ϱ ε b Wick] Tr [ϱ ε ( z 2p ) Wick ] The condition (PI) implies (Ammari-N. J.M.P.A 11) lim γε Z p z p z p dµ(z) ε 0 Z z 2p dµ(z) = 0. L 1 ( p Z) ε E

41 Condition (PI) and consequences Assuming M(ϱ ε, ε E) = {µ}, the condition (PI ) says k N, lim Tr [ ϱ ε N k] = z 2k dµ(z). ε 0 Z ε E Define the reduced density matrices (see BBGKY hierarchy) by b L( p [ ] Z), Tr γp b ε = Tr [ ϱ ε b Wick] Tr [ϱ ε ( z 2p ) Wick ] The condition (PI) implies (Ammari-N. J.M.P.A 11) lim γε Z p z p z p dµ(z) ε 0 Z z 2p dµ(z) = 0. L 1 ( p Z) ε E

42 Review of the framework Z = L 2 (X, dm; C) separable. (e.g. (X, +, m) meas. add. group) (A, D(A)) self-adjoint on Z, V = V (x) = V ( x). [a(x), a (y)] = εδ(x y). H ε = dγ(a) H ε = E(z) Wick E(z) = z, Az X X X X V (x y)a (x)a (y)a(x)a(y) dm(x)dm(y) V (x y) z(x) 2 z(y) 2 dm(x)dm(y). H ε, D(H ε ) s.a. in Γ(Z). ϱ ε (t) = e i t ε Hε ϱ ε e i t ε Hε. Φ(t) flow of i t z = z E(z).

43 Result for V L. See (Ammari-N. J.Math.Phys. 09, JMPA 11) Assume V L (X, dm), M(ϱ ε, ε (0, ε 0 )) = {µ 0 } and n N, lim Tr [ϱ ε N n ] = z 2n dµ 0 (z). ε 0 Then for all t R, the following result hold M(ϱ ε (t), ε (0, ε 0 )) = {µ(t) = Φ(t) µ 0 }, b P(Z), lim Tr [ ϱ ε (t)b Wick] = b(z) dµ t (z), ε 0 Z lim ε 0 γε Z p(t) = z p z p dµ t (z) Z = z p t zt p dµ 0 (z). Z z 2p dµ 0 (z) Z z 2p dµ 0 (z) Z

44 Singular case (± Coulombic case d = 3) See (Ammari-N. online Ann. Sc. Norm. di Pisa). Take X = R d and A = and assume V (1 ) 1 2 bounded and (1 ) 1 2 V (1 ) 1 2 compact. Under the sole condition δ > 0, ε (0, ε 0 ), Tr [ (dγ(1 )) δ ϱ ε] C δ and M(ϱ ε, ε (0, ε 0 )) = {µ 0 }, the following results hold for all t R: M(ϱ ε (t), ε (0, ε 0 )) = {µ(t)} where µ(t) is a Borel probability measure on Z 1 = H 1 (R d ; C) and µ(t) = Φ(t) µ 0 (Φ well-defined on Z 1 ). When (ϱ ε ) ε (0,ε0) satisfies the condition (PI ) then (ϱ ε (t)) ε (0,ε0) and b P(Z), lim Tr [ ϱ ε (t)b Wick] = b(z) dµ t (z), ε 0 lim ε γ ε p(t) = Z z p z p dµ t (z) Z z 2p dµ 0 (z) = Z Z z p t zt p dµ 0 (z) Z z 2p dµ 0 (z).

45 Singular case (± Coulombic case d = 3) See (Ammari-N. online Ann. Sc. Norm. di Pisa). Take X = R d and A = and assume V (1 ) 1 2 bounded and (1 ) 1 2 V (1 ) 1 2 compact. Under the sole condition δ > 0, ε (0, ε 0 ), Tr [ (dγ(1 )) δ ϱ ε] C δ and M(ϱ ε, ε (0, ε 0 )) = {µ 0 }, the following results hold for all t R: M(ϱ ε (t), ε (0, ε 0 )) = {µ(t)} where µ(t) is a Borel probability measure on Z 1 = H 1 (R d ; C) and µ(t) = Φ(t) µ 0 (Φ well-defined on Z 1 ). When (ϱ ε ) ε (0,ε0) satisfies the condition (PI ) then (ϱ ε (t)) ε (0,ε0) and b P(Z), lim Tr [ ϱ ε (t)b Wick] = b(z) dµ t (z), ε 0 lim ε γ ε p(t) = Z z p z p dµ t (z) Z z 2p dµ 0 (z) = Z Z z p t zt p dµ 0 (z) Z z 2p dµ 0 (z).

46 Evolution of mean field correlations.. Rψ π 2 ψ. ϕ 0.. Rψ ψ.rψ π 2. ψ ϕ(t).. Rψ 0 Fig.1: Evolution of the measure initially carried by a torus in Cψ 0 Cψ π. 2 The complex gauge parameter e iθ is represented by the small circle.