On The Weak-Coupling Limit for Bosons and Fermions

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Dario Benedetto, François Castella, Raffaele Esposito, Mario Pulvirenti

To cite this version:

Dario Benedetto, François Castella, Raffaele Esposito, Mario Pulvirenti. On The Weak-Coupling Limit

for Bosons and Fermions. Mathematical Models and Methods in Applied Sciences, World Scientific

Publishing, 2005, 15 (12), pp.1811-1843. �10.1142/S0218202505000984�. �hal-00004480�

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ccsd-00004480, version 1 - 16 Mar 2005

D. Benedetto

^∗

, F. Castella

⁺

, R. Esposito

^#

and M. Pulvirenti

^∗

3/10/2004

Abstract. In this paper we consider a large system of Bosons or Fermions. We start with an initial datum which is compatible with the Bose-Einstein, respectively Fermi-Dirac, statistics.

We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at timet, agrees with the analogous expansion for the solution to the Uehling-Uhlenbeck equation.

This paper follows in spirit the companion work [2], where the authors investigated the weak-coupling limit for particles obeying the Maxwell-Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

1. Introduction

In 1933 Uehling and Uhlenbeck in Ref. [17] proposed the following kinetic equation, called U-U in the sequel, for the time evolution of the one-particle Wigner function f (x, v; t) associated with a large system of weakly interacting Bosons or Fermions (see Ref. [18] for the definition of the Wigner function). The U-U equation is

∂

_t

f (x, v; t)+v · ∇

x

f(x, v; t) = Z

dv

_∗

Z

dv

_∗^′

Z

dv

^′

W (v, v

_∗

| v

^′

, v

_∗^′

) n

f

^′

f

_∗^′

(1 + 8π

³

θf )(1 + 8π

³

θf

∗

) − f f

∗

(1 + 8π

³

θf

^′

)(1 + 8π

³

θf

_∗^′

) o ,

(1.1)

where we use the standard short-hand notation

f = f(x, v; t), f

_∗

= f (x, v

_∗

; t), f

^′

= f (x, v

^′

; t), f

_∗^′

= f (x, v

_∗^′

; t).

Here, W denotes the transition kernel W (v, v

_∗

| v

^′

, v

_∗^′

) = 1

8π

²

b φ(v

^′

− v) + θ φ(v b

^′

− v

_∗

)

2

δ(v + v

_∗

− v

^′

− v

_∗^′

) δ 1

2 (v

²

+ v

²_∗

− v

^′2

− v

_∗^′²

) .

(1.2)

Key words and phrases. Uehling-Uhlenbeck equation, quantum systems, weak coupling.

∗ Dipartimento di Matematica Universit`a di Roma ‘La Sapienza’, P.le A. Moro, 5 - 00185 Roma - Italy

+ IRMAR, Universit´e de Rennes 1, Campus de Beaulieu - 35042 Rennes - Cedex - France

#Dipartimento di Matematica pura ed applicata, Universit`a di L’Aquila, Coppito - 67100 L’Aquila - Italy

1

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Finally,

φ(k) = b Z

dx e

^−ik·x

φ(x) (1.3)

is the Fourier transform of the two-body interaction potential φ, and θ = ± 1 for Bosons and Fermions respectively.

Note that the factors 8π

³

do not appear in the original U-U equation in Ref. [17], because there, the distribution function is normalized in such a way that its integral on the velocity variable equals the space density times 8π

³

. At variance, in (1.1), f is just the standard Wigner function, whose integral on the velocities equals the space density.

Let us mention that equation (1.1) actually is cubic (and not quartic) in the unknown f : apparent quartic terms have vanishing contribution, as shown by direct inspection.

Eq. (1.1) constitutes a natural modification of the usual quantum Boltzmann equation, in order to take into account statistics. In particular, there is a H -functional

H (f ) = Z

dx Z

dv

f log f − θ(1 + 8π

³

θf ) log(1 + 8π

³

θf ) (1.4) driving the system to the Bose-Einstein and Fermi-Dirac equilibrium distribution:

M (v) = 1

e

^βµ+βv²^/2

− 8π

³

θ (1.5)

outside the Bose condensation region. Here β and µ denote the inverse temperature and the chemical potential respectively.

Eq. (1.1) is largely studied (see for istance [1] and [12] for physical consideration, and [15], [7], [13], [14] ... for a more mathematically oriented analysis concerning the existence of solutions and asymptotic behavior), so that it is certainly of great relevance to derive this equation from the first principles, namely from the Schr¨ odinger equation.

As clearly explained by H. Spohn in [16], Eq (1.1) is indeed expected to hold in the so called weak-coupling limit, which consists in scaling space, time and the potential according to

x → εx, t → εt, φ → √

εφ, (1.6)

where ε is a small positive parameter.

A slightly different limit, usually called van Hove limit, scales t and φ as in (1.6) but leaves the microscopic space scale unchanged. Eq. (1.1) cannot be derived in the van Hove limit in general but, in case of translationally invariant states, we expect to achieve the homogeneous version of the U-U equation (for a large system). In fact Hugenholtz [11]

proved formally that this happens. Later on Ho and Landau [10] proved that the homogeneous U-U equation holds rigorously up to the second order expansion in the potential.

These approaches, as well as the recent contribution by Erd¨ os et al. [8] (where the quantum analog of the Boltzmann’s Stosszahlansatz is formulated), are based on the commutator expansion of the time evolution of the observables of the CCR and CAR algebras.

In the present paper we approach the problem from a different viewpoint. We start from the time evolution of a N particle quantum system in terms of the Wigner formalism.

Here the statistics enters only through the choice of the admissible states we take as initial

conditions. Such states, called quasi-free, must describe free Bosons and Fermions, so that

they cannot have any other correlations but those arising from the statistics. Therefore

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the first step is to characterize quasi-free states (see for example Ref. [4]) in terms of the Wigner functions. Then we apply the dynamics (in terms of the usual hierarchy) and represent the solution as a perturbative expansion. The truncation of this expansion up to the second order in the potential is shown to converge to the expansion associated to the U-U equation, up to the first order in the scattering cross section.

In other words we recover the result in Ref. [10] with the following main differences.

First we exploit the weak-coupling limit, so that we can deal with states which are not necessarily translationally invariant. Second, we work directly in terms of the Wigner formalism, in the same spirit of the Balescu book (see Ref. [1]). In doing so, we also follow a previous work [2] by the authors for the Maxwell-Boltzmann statistics. Hence the present work shows how the statistics can be handled in this formalism. Note in passing that the case of the Maxwell-Boltzmann statistics allows for a much stronger (but still partial) convergence result than the one presented here, see [2]. Note finally that the present formalism also allows to handle the low-density limit, see [3], see also the last section of this text.

It is also important to mention that a full rigorous derivation of the U-U equation (but also of the usual Boltzmann equation arising for the Maxwell-Boltzmann statistics) is still far beyond the present techniques and those of the previous references.

The plan of the paper is the following. In the next section we describe the particle system. In Section 3 we establish the result. The rest of the paper is devoted to the proofs.

2. The particle system.

We consider a Quantum particle system in R

³

. Let H = M

n≥0

L

2

( R

³

)

ⁿ

:= M

n≥0

H

_n

, (2.1)

be the Fock space. A state of the system is a self-adjoint, positive trace class operator acting on H :

σ = M

n≥0

σ

_n

. (2.2)

We assume

Tr σ = 1. (2.3)

The operator N , number of particles, is the multiplication by n on H

_n

and hence h N i = X

n≥0

n Tr σ

_n

, (2.4)

where the left hand side is the average number of particles in the state σ. If σ

_n

(X

_n

; Y

_n

) is the kernel of σ

n

, the Reduced Density Matrices (RDM) are defined by:

ρ

_n

(X

_n

; Y

_n

) = X

m≥0

(n + m)!

m!

Z

σ

_n+m

(X

_n

, Z

_m

; Y

_n

, Z

_m

) dZ

_m

. (2.5)

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Here X

n

= (x

1

, . . . , x

n

), x

i

∈ R

³

denotes the n-particle configuration. Note that Tr ρ

_n

=

Z

dZ

_n

ρ

_n

(Z

_n

; Z

_n

)

= X

m≥n

m(m − 1) . . . (m − n + 1)Tr σ

_m

= h N (N − 1) . . . (N − n + 1) i , (2.6) and hence the RDM are equivalent to the classical correlation functions.

The Hamiltonian of the system is the self-adjoint operator acting on H given by H =

M

∞ n=1

H

n

, (2.7)

where

H

_n

= − 1 2

X

n j=1

∆

_x_j

+ X

1≤i<j≤n

φ(x

_i

− x

_j

), (2.8)

and the potential φ is a smooth two-body interaction. Here, ~ as well as the mass of the particles are normalized to unity.

Under these circumstances, the time evolved state is given by the usual

σ(t) = e

^−iHt

σe

^iHt

. (2.9)

Now, quantum statistics is taken into account by suitable properties of the physically relevant states. Namely, for the Maxwell-Boltzmann (M-B) statistics we require symmetry of ρ

_n

(x

₁

, . . . , x

_n

; y

₁

. . . , y

_n

) in the exchange of particle names. For the Bose-Einstein (B-E) and Fermi-Dirac (F-D) statistics we require additionally

ρ

_n

(x

₁

, . . . , x

_n

; y

₁

. . . , y

_n

) = θ

^s(π)

ρ

_n

(x

₁

, . . . , x

_n

; y

_π(1)

. . . , y

_π(n)

), (2.10) where π ∈ P

n

is a permutation of n elements, and s(π) = 0 if the permutation is even, s(π) = 1 if it is odd.

Alternatively, the quantum statistics is automatically taken into account by considering states on the algebra generated by the annihilation and creation operators a(x) and a

^†

(x) (with the commutation and anti-commutation relations according to the B-E and F-D statistics respectively). Then the RDM are defined as

Tr

σa

^†

(x

n

) . . . a

^†

(x

1

)a(y

1

) . . . a(y

n

)

= ρ

n

(x

1

, . . . , x

n

; y

1

. . . , y

n

). (2.11) However we do not use here the second quantization formalism.

Given a state σ, we define the Wigner transform [18] by W

_n

(X

_n

; V

_n

) := 1

(2π)

³ⁿ

Z

dY

_n

e

^iYⁿ^·Vⁿ

σ

_n

X

_n

− 1

2 Y

_n

; X

_n

+ 1 2 Y

_n

. (2.12)

Therefore the analogous of the classical correlation functions are the j -particle Wigner functions defined through

F

_j

(X

_j

; V

_j

) = X

n≥0

(n + j)!

n!

Z dX

_n

Z

dV

_n

W

_j+n

(X

_j

, X

_n

; V

_j

, V

_n

). (2.13)

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Note that the F

j

’s are the Wigner transforms of the RDM ρ

j

, as one can easily check.

Due to the dynamics imposed by (2.9), it is a standard computation to check that the Wigner function W

_n

evolves according to the Wigner-Liouville equation

∂

_t

W

_n

+ X

n i=1

v

_i

· ∇

xi

W

_n

= T

_n

W

_n

. (2.14) As a consequence, the j-particle Wigner functions F

_j

’s satisfy the associated hierarchy

∂

_t

F

_j

+ X

j i=1

v

_i

· ∇

xi

F

_j

= T

_j

F

_j

+ C

_j+1

F

_j+1

, (2.15)

where T

_j

and C

_j+1

will be defined below after Eq.(2.22), and the index j takes any value between 1 and N . Equations (2.15) are analogous to the usual BBGKY hierarchy for the classical systems and are derived in a similar manner. Note that by the definition of the RDM the coefficient in front of C

_j+1

is one instead of N − j .

We now want to analyze (2.15) in the weak-coupling regime (1.6). Therefore, we set f

_j^ε

(X

_j

; V

_j

; t) := F

_j

(ε

⁻¹

X

_j

; V

_j

; ε

⁻¹

t), (2.16) where ε > 0 is a small parameter, and we scale the potential as well, by setting

φ → √

εφ. (2.17)

The resulting, scaled, equation is

∂

t

f

_j^ε

+ X

j i=1

v

i

· ∇

^xⁱ

f

_j^ε

= 1

√ ε T

_j^ε

f

_j^ε

+ 1

√ ε ε

⁻³

C

_j+1^ε

f

_j+1^ε

, (2.18) where

(T

_j^ε

f

_j^ε

)(X

_j

; V

_j

) = X

0<k<ℓ≤j

(T

_k,l^ε

f

_j^ε

)(X

_j

; V

_j

), (2.19) and the T

_k,l^ε

’s are defined as follows: if j = 1, we simply have T

₁^ε

= 0; otherwise,

(T

_k,l^ε

f

_j^ε

)(X

_j

; V

_j

) = − i X

σ=±1

σ

Z dh

(2π)

³

φ(h) e b

ⁱ^h^ε^·(x^k^−x^ℓ⁾

f

_j^ε

x

₁

, . . . , x

_j

; v

₁

, . . . , v

_k

− σ h

2 , . . . , v

_ℓ

+ σ h

2 , . . . , v

_j

.

(2.20)

On the other hand, the C

_j+1^ε

in (2.18) is computed as:

(C

_j+1^ε

f

_j+1^ε

)(X

_j

; V

_j

) = X

j k=1

(C

_k,j+1^ε

f

_j+1^ε

)(X

_j

; V

_j

), (2.21)

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where

(C

_k,j+1^ε

f

_j+1^ε

)(X

_j

; V

_j

) = − i X

σ=±1

σ

Z dh (2π)

³

Z

dx

_j+1

Z

dv

_j+1

φ(h) e b

ⁱ^h^ε^·(x^k^−x^j+1⁾

f

_j+1^ε

x

₁

, . . . , x

_j+1

; v

₁

, . . . , v

_k

− σ h

2 , . . . , v

_j+1

+ σ h 2

.

(2.22)

Note that T

_j

and C

_j+1

are T

_j^ε

and C

_j+1^ε

for ε = 1. Last, we fix an initial condition sequence

{ f

_0,j^ε

}

^∞j=1

(2.23)

according to the quantum statistics, and perform the limit ε → 0 in the resulting system.

Remark: Since Z

f

_0,1^ε

(x, v)dxdv = ε

³

h N i , (2.24) requiring k f

_0,1^ε

k

L1

= O(1), implies h N i = O(ε

⁻³

). In other words we are working in the Grand-canonical formalism and the density is automatically rescaled.

In the following we shall fix f

_0,1^ε

to be a given (independent of ε) probability density f

0

. This means that its inverse Wigner transform

ρ

^ε

(x, y) = Z

dv e

ⁱ^x−y^ε ^·v

f

₀

x + y 2 , v

, (2.25)

namely the one-particle rescaled RDM, is a superposition of WKB states.

We now make assumptions on the initial state to take into account the statistics. For the M-B statistics a suitable initial sequence can be chosen completely factorized, e.g.

f

_0,j^ε

= f

₀^⊗j

. (2.26)

Such a notion of statistical independence, which corresponds to a complete factorization of the RDM’s, is not compatible (but for the condensed Bose state) with the B-E and F- D statistics which exhibit intrinsic correlations even for non interacting particle systems.

States describing free Bosons or Fermions are usually called quasi-free and are defined in terms of the RDM’s by the following formula:

ρ

j

(x

1

, . . . , x

j

; y

1

, . . . , y

j

) = X

π∈Pj

θ

^s(π)

Y

j i=1

ρ(x

i

, y

_π(i)

), (2.27)

for some positive definite operator ρ on L

₂

( R

³

) with kernel ρ(x, y). We show in Appendix how to construct explicitly quasi-free states for Bosons.

From now on we assume that the initial sequence (2.23) for the rescaled problem (2.18) is given by the Wigner transform of a quasi-free state (2.27) generated by ρ(x, y) = ρ

^ε

(x, y) given by (2.25). As a consequence the initial sequence { f

_j⁰

}

^∞j=1

for the hierarchy (2.18) is of the form

f

_j⁰

(X

j

, V

j

) = X

π∈Pj

θ

^s(π)

f

_j^π

(X

j

, V

j

), (2.28)

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with

f

_j^π

(x

₁

, . . . , x

_j

, v

₁

, . . . , v

_j

) = 1 (2π)

^3j

Z

dy

₁

· · · Z

dy

_j

Z

dw

₁

· · · Z

dw

_j

Y

j

k=1

e

^iy^k^·v^k^+iw^k^·^xk

−xπ(k)

ε −iwk·^yk⁺^yπ(k)₂

f

₀

x

_k

+ x

_π(k)

2 − ε

4 (y

_k

− y

_π(k)

), w

_k

.

(2.29)

Eq. (2.29) follows from (2.27) and (2.25).

We underline once more that, in the present approach, the dynamics is given by the hierarchy of equations (2.18) which are completely equivalent to the Schr¨ odinger equation, while the statistics enters only in the structure of the initial state.

In the weak-coupling limit ε → 0, we expect that f

_j^ε

(t) converges to a factorized state (because the effects of statistics disappear in the macroscopic limit). On the more each factor should be solution to the U-U equation (the collisions being affected by the statistics because they involve microscopic scales).

3. The main result.

Let f = f (x, v, t) be a solution to the U-U equation and set f

_j

( · , · , t) = f

^⊗j

( · , · , t).

Then the sequence { f

j

}

^∞j=1

satisfies the following hierachy of equations:

(∂

_t

+ X

j

i=1

v

_i

· ∇

xi

)f

_j

= Q

_j,j+1

f

_j+1

+ Q

_j,j+2

f

_j+2

. (3.1) Here the Q

_j,j+1

contribution, a ”two particles term” in the terminology used below, is

(Q

j,j+1

f

j+1

)(X

j

, V

j

) = X

j k=1

Z dv

_k^′

Z

dv

j+1

Z

dv

_j+1^′

W (v

k

, v

j+1

| v

_k^′

, v

^′_j+1

) n

f

_j+1

(X

_j

, x

_k

; v

₁

, . . . , v

_k^′

, . . . v

_j+1^′

) − f

_j+1

(X

_j

, x

_k

; v

₁

, . . . , v

_j+1

) o ,

(3.2)

and the Q

_j,j+2

contribution, a ”three particles term”, is

(Q

_j,j+2

f

_j+2

)(X

_j

, V

_j

) = 8π

³

θ X

j k=1

Z dv

_k^′

Z

dv

_j+1

Z

dv

^′_j+1

W (v

_k

, v

_j+1

| v

^′_k

, v

_j+1^′

) (

f

_j+2

(X

_j

, x

_k

, x

_k

; v

₁

, . . . , v

_k^′

, . . . v

_j+1^′

, v

_k

) + f

_j+2

(X

_j

, x

_k

, x

_k

; v

₁

, . . . , v

_k^′

, . . . v

_j+1^′

, v

_j+1

)

− f

_j+2

(X

_j

, x

_k

x

_k

; v

₁

, . . . , v

_j+1

, v

_k^′

) − f

_j+2

(X

_j

x

_k

, x

_k

; v

₁

, . . . , v

_j+1

, v

^′_j+1

) )

.

(3.3)

Also, (X

_n

, y) denotes the (n + 1)-sequence (x

₁

, . . . , x

_n

, y).

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A formal solution to the hierarchy (3.1) is given by the following series expansion:

f

_j

(t) = X

n≥0

X

α1...αn αi=1,2

Z

t 0

dt

₁

Z

t1

0

dt

₂

· · · Z

tn−1

0

dt

_n

S(t − t

₁

)Q

_j,j+α₁

S (t

₁

− t

₂

)Q

_j+α₁_,j+α₁_+α₂

S(t

₂

− t

₃

) . . . Q

_j+α₁_+···+α_n−1,j+α1+···+αn

S(t

_n

)f

₀^⊗(j+α¹^+···+αⁿ⁾

,

(3.4)

where S(t) denotes the fream stream operator, namely,

(S(t)f

_j

)(X

_j

, V

_j

) = f

_j

(X

_j

− V

_j

t, V

_j

). (3.5) As for the solution to the ε-dependent hierarchy (2.18), we can also expand f

_ε^j

in the similar way, at least at the formal level. This gives

f

_j^ε

(t) = X

n≥0

X

γ1...γn γi=0,1

Z

t 0

dt

1

Z

t1

0

dt

2

· · · Z

tn−1

0

dt

n

S(t − t

₁

)P

_j,j+γ^ε ₁

S(t

₁

− t

₂

)P

_j+γ^ε ₁_,j+γ₁_+γ₂

S(t

₂

− t

₃

) . . . P

_j+γ^ε ₁_+···+γ_n−1,j+γ1+···+γn

S(t

n

)f

_j+γ⁰ ₁_+···+γ_n

,

(3.6)

where f

_j⁰

is an initial quasi-free state given by (2.29), and

P

_j,j+1^ε

= ε

⁻⁷²

C

_j+1^ε

, P

_j,j^ε

= ε

⁻¹²

T

_j^ε

. (3.7) We are not able to show the convergence of f

_j^ε

(t) to f

_j

(t) in the limit ε → 0 even for short times. However we are going to show that the two series agree up to the second order in the potential. Namely, defining the second order contributions

g(t) := S(t)f

₀

+ Z

t

0

dt

₁

S(t − t

₁

)Q

_1,2

S(t

₁

)f

₀^⊗2

+ Z

t

0

dt

₁

S(t − t

₁

)Q

_1,3

S(t

₁

)f

₀^⊗3

, (3.8) associated with f

_j

(t), and

g

^ε

(t) := S(t)f

₀

+ ε

⁻⁷²

Z

t

0

dt

₁

S(t − t

₁

)C

₂^ε

S(t

₁

)f

₂⁰

+ ε

⁻⁴

Z

t 0

dt

₁

Z

t1

0

dt

₂

S(t − t

₁

)C

₂^ε

S(t

₁

− t

₂

)T

₂^ε

S(t

₂

)f

₂⁰

+ ε

⁻⁷

Z

t 0

dt

₁

Z

t1

0

dt

₂

S(t − t

₁

)C

₂^ε

S(t

₁

− t

₂

)C

₃^ε

S(t

₂

)f

₃⁰

,

(3.9)

associated with f

_j^ε

(t), we rigorously prove below the convergence of g

^ε

(t) to g(t), under suitable assumptions on the data of the problem.

Assumptions: We require φ to be real and even, so that φ b is real. In particular

φ(k) = b φ( b − k) = φ( b − k).

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This the most important assumption we need in the analysis. Besides, we shall need to deal with ”smooth” data. Quantitatively, we assume the following regularity:

(1 + | ξ | )

^α

X

|β|≤2

| D

_ξ^β

φ(ξ) b | ∈ L

₁

, for a sufficiently large α, and

f

0

(x, v) ∈ L

1

,

(1 + | ξ | + | η | )

^α

X

0≤|β|≤2

X

0≤|γ|≤2

| (D

^β_ξ

+ D

_η^γ

) f b

₀

(ξ, η) | ∈ L

₁

, (3.10)

for a sufficiently large α as well. In (3.10), β and γ denote multi-indices, and D

^β_ξ

, D

^γ_η

denote derivatives with respect to the variables ξ and η. Note that throughout this paper we use the following normalization for the Fourier transform:

f b (h) = ( F

x

f )(h) = Z

Rⁿ

dx f (x) e

^−ih·x

, f (x) = ( F

_h⁻¹

f b )(h) = 1

(2π)

ⁿ

Z

Rⁿ

dh f b (h) e

^ih·x

.

(3.11)

Our main result is the

Theorem. Under the above assumptions, we have

ε→0

lim b g

^ε

(ξ, η, t) = b g(ξ, η, t), for any t > 0 and any (ξ, η) ∈ R

⁶

.

Remark: In the above statement (and the proofs given below), we found convenient to treat the terms in (3.8) and (3.9) in terms of their Fourier transforms, for which the convergence arises more naturally. However, we would like to stress that in the companion paper [2], a stronger, but analogous, result is formulated in terms of the pointwise convergence in the x − v space, hence without going to the Fourier space.

Before entering the details of the proof we first analyze all the contributions in the right hand side of (3.9).

The two-particle terms are (we skip the unessential operator R

t

0

dt

₁

S(t − t

₁

))

S

2,ε^π

:= ε

⁻⁷²

C

₂^ε

S (t

₁

)f

₂^π

, (3.12) where the permutation π may take the two values π = (1, 2) or π = (2, 1), together with

T

2,ε^π

:= ε

⁻⁴

Z

t1

0

dt

₂

C

₂^ε

S(t

₁

− t

₂

)T

₂^ε

S(t

₂

)f

₂^π

, (3.13)

with π taking the values π = (1, 2) or π = (2, 1). There are four such terms.

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The three-particle terms are twelve, namely:

W

3,ε^π

:= ε

⁻⁷

Z

t1

0

dt

₂

C

_1,2^ε

S(t

₁

− t

₂

)C

_1,3^ε

S(t

₂

)f

₃^π

, (3.14) and

V

3,ε^π

:= ε

⁻⁷

Z

t1

0

dt

2

C

_1,2^ε

S(t

1

− t

2

)C

_2,3^ε

S(t

2

)f

₃^π

, (3.15) with π ∈ P

3

, the set of the permutations of three objects, whose cardinality is six.

Note that all the above terms are funtions of (x, v) (and t

1

of course).

For further convenience, and in view of the proof of our main result, we readily express all these contributions in terms of their Fourier transforms.

We start with the following obvious three formulae for the basic operators S(t), T

₂

, and C

2

(see (3.5), (2.19)-(2.20), and (2.21)-(2.22), respectively):

T b

₂^ε

f b (ξ

₁

, ξ

₂

; η

₁

, η

₂

) = − i X

σ=±1

σ Z

dh φ(h) b

(2π)

³

e

^iσ^h²^·(η²^−η¹⁾

f b

ξ

₁

− h

ε , ξ

₂

+ h

ε ; η

₁

, η

₂

,

C b

₂^ε

f b (ξ; η) = T b

₂^ε

f b (ξ, 0; η, 0) = − i X

σ=±1

σ Z

dh φ(h) b

(2π)

³

e

^−iσ^h²^·η

f b

ξ − h ε , h

ε ; η, 0

, S(t) b f b (ξ; η) = f b (ξ, η + ξt).

These relations give in (3.12) through (3.15):

S b

2,ε^π

(ξ, η) = − i ε

⁻⁷²

(2π)

³

X

σ=±1

σ Z

dh φ(h) e b

⁻ⁱ^σ²^h·η

f b

₂^π

ξ − h ε , h

ε ; η + t

1

(ξ − h ε ), t

1

h ε

. (3.16) T b

2,ε^π

(ξ, η) = − ε

⁻⁴

(2π)

⁶

X

σ1,σ2=±1

σ

₁

σ

₂

Z

t1

0

dt

₂

Z

dh

₁

Z

dh

₂

φ(h b

₁

) φ(h b

₂

) e

⁻ⁱ^σ²¹^h¹^·η

e

⁻ⁱ^σ²²^h²^·(η+ξ(t¹^−t²⁾⁻²^ε^(t¹^−t²^)h¹⁾

f b

₂^π

ξ − 1

ε (h

₁

+ h

₂

), 1

ε (h

₁

+ h

₂

) ; η + t

₁

ξ − t

₁

h

₁

+ t

₂

h

₂

ε , t

₁

h

₁

+ t

₂

h

₂

ε

, (3.17) W c

3,ε^π

(ξ, η) = − ε

⁻⁷

(2π)

⁶

X

σ1,σ2=±1

σ

₁

σ

₂

Z

t1

0

dt

₂

Z

dh

₁

Z

dh

₂

φ(h b

₁

) φ(h b

₂

) e

⁻ⁱ^σ²¹^h¹^·η

e

⁻ⁱ^σ²²^h²^·(η+(ξ−^h^ε¹^)(t¹^−t²⁾⁾

f b

₃^π

ξ − 1

ε (h

1

+ h

2

), h

1

ε , h

2

ε ; η + t

1

ξ − t

1

h

1

+ t

2

h

2

ε , t

1

h

1

ε , t

2

h

2

ε ,

,

(3.18)

V b

3,ε^π

(ξ, η) = − ε

⁻⁷

(2π)

⁶

X

σ1,σ2=±1

σ

₁

σ

₂

Z

t1

0

dt

₂

Z

dh

₁

Z

dh

₂

φ(h b

₁

) φ(h b

₂

) e

⁻ⁱ^σ²¹^h¹^·η

e

⁻ⁱ^σ²²^h²^·^h^ε¹^(t¹^−t²⁾

f b

₃^π

ξ − h

₁

ε , h

₁

− h

₂

ε , h

₂

ε ; η + t

₁

ξ − t

₁

h

₁

ε , t

₁

h

₁

− t

₂

h

₂

ε , t

₂

h

₂

ε

.

(3.19)

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Starting form those expressions, the plan of the proof is the following. In Section 4 we evaluate the two particle terms S

2,ε^π

and T

2,ε^π

. We prove that they converge towards the associated two particles terms in the U-U equation. In Section 5 we deal with the three-particle terms associated to the permutations π with a fixed element. Those are shown to converge towards the associated three particles terms in the U-U equation, while contributing by the quantity φ(v b

^′

− v)

²

+ φ(v b

^′

− v

_∗

)

²

to the transition kernel W (see (1.2)).

Finally in Section 6 we treat the three particle terms relative to cyclic permutations. We recover in this way the missing contribution to the transition kernel, namely the cross term θ φ(v b

^′

− v) φ(v b

^′

− v

∗

).

For sake of simplicity we shall carry out the computations for Bosons (θ = 1), being clear that the Fermionic case is just the same with suitable changes of sign.

4. Two-particle terms.

We introduce the partial Fourier transform f e

_j^π

(x

₁

, . . . , x

_j

; η

₁

, . . . , η

_j

) :=

Z

dv

₁

· · · Z

dv

_j

e

⁻ⁱ

P

j

k=1vk·ηk

f

_j^π

(x

₁

, . . . , x

_j

; v

₁

, . . . , v

_j

).

(4.1) As a consequence of (2.29) we have

f e

_j^π

(x

₁

, . . . , x

_j

; η

₁

, . . . , η

_j

) = Y

j

k=1

f e

₀

x

_k

+ x

_π(k)

2 − ε η

_k

− η

_π(k)

4 , − x

_k

− x

_π(k)

ε + η

_k

+ η

_π(k)

2 . (4.2) In particular, we have the obvious

f e

₂^(1,2)

(x

₁

, x

₂

; η

₁

, η

₂

) = f e

₀

(x

₁

, η

₁

) f e

₀

(x

₂

, η

₂

), (4.3) together with

f e

₂^(2,1)

(x

1

, x

2

; η

1

, η

2

) = f e

0

x

1

+ x

2

2 − ε

4 (η

1

− η

2

); − x

1

− x

2

ε + η

1

+ η

2

2 f e

₀

x

₁

+ x

₂

2 + ε

4 (η

₁

− η

₂

); + x

₁

− x

₂

ε + η

₁

+ η

₂

2 .

(4.4)

Hence, upon now performing the complete Fourier transform, we obtain,

f b

₂^(1,2)

(ξ

₁

, ξ

₂

; η

₁

, η

₂

) = f b

₀

(ξ

₁

, η

₁

) f b

₀

(ξ

₂

, η

₂

), (4.5) together with

f b

₂^(2,1)

(ξ

1

, ξ

2

; η

1

, η

2

) =ε

³

Z

dy

1

Z

dy

2

e

^−iξ¹^·(y¹⁺^ε²^y²⁾

e

^−iξ²^·(y¹⁻^ε²^y²⁾

f e

₀

y

₁

− ε

4 (η

₁

− η

₂

); − y

₂

+ η

₁

+ η

₂

2 f e

₀

y

₁

+ ε

4 (η

₁

− η

₂

); +y

₂

+ η

₁

+ η

₂

2 .

(4.6)

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We are now in position to analyse the term S

2,ε^π

for π = (1, 2) and π = (2, 1). First, using the identity

X

σ=±1

σe

^−iσ^h²^·η

= − 2i sin h · η 2 , we get the the explicit expression:

S b

2,ε^π

= − 2ε

⁻⁷²

(2π)

³

Z

dh φ(h) sin b

h · η 2

f b

₂^π

ξ − h

ε , h

ε ; η + (ξ − h ε )t

₁

, h

ε t

₁

. (4.7)

In the case of S b

2,ε^(1,2)

, a change of variable h → εh then gives, using (4.5), the value S b

2,ε^(1,2)

(ξ, η) = − 2ε

⁻¹²

(2π)

³

Z

dh φ(εh) sin b

ε h · η 2

f b

₀

(ξ − h; η + (ξ − h)t

₁

) f b

₀

(h, ht

₁

). (4.8) Therefore, we may estimate

| S b

2,ε^(1,2)

| ≤

≤ C √

ε k φ b k

L^∞

Z

dh | h | ( | η + (ξ − h)t

₁

| + | ξ − h | t

₁

) | f b

₀

(h; ht

₁

) | | f b

₀

| (ξ − h; η + (ξ − h)t

₁

)

≤ C √

ε sup

ξ,η

| η || f b

₀

| (ξ; η) Z

dξ sup

η

| ξ || f b

₀

| (ξ; η) + t

₁

sup

ξ,η

| ξ || f b

₀

| (ξ; η) Z

dξ | ξ | sup

η

| f b

₀

(ξ; η) |

!

≤ C √ ε sup

ξ,η

( | ξ | + | η | ) | f b

₀

| (ξ; η) Z

dξ sup

η

( | ξ | + | η | ) | f b

₀

| (ξ; η) ,

(4.9) and the corresponding contribution vanishes with ε.

In the case of S b

2,ε^(2,1)

on the other hand, equations (4.7) and (4.6) give S b

2,ε^(2,1)

(ξ, η) = − 2ε

⁻¹²

(2π)

³

Z

dh Z

dy

1

Z

dy

2

φ(h) sin b h · η

2 e

^−i(y¹⁺^ε²^y²^)·ξ

e

^ih·y²

f e

0

y

1

− ε

4 η + ξt

1

− 2h εt

₁

; − y

2

+ η + ξt

1

2 f e

₀

y

₁

+ ε 4

η + ξt

₁

− 2h εt

₁

; y

₂

+ η + ξt

₁

2 = − 2ε

⁻¹²

(2π)

³

Z dh

Z dy

₁

Z

dy

₂

φ(h) sin b h · η

2 e

^−i(y¹^+ε^y²²^)·ξ

e

^ih·y²

f e

0

y

1

+ h

2 t

1

; − y

2

+ η + ξt

1

2 f e

0

y

1

− h

2 t

1

; y

2

+ η + ξt

1

2 + O( √ ε).

(4.10)

By the parity of φ, the first term in the right hand side is vanishing: Indeed, it is anti-

symmetric in the exchange h → − h and y

₂

→ − y

₂

. Note that the mechanism that makes

the dominant, O(ε

^−1/2

), contribution of S b

2,ε^(2,1)

, vanish in the limit, is very different from

the one involved in the vanishing of S b

2,ε^(1,2)

: here, antisymmetry plays a crucial role. This

aspect will play an even more important, and more intricate, role in the next two sections.

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There remains to prove that the O( √

ε) term in (4.10) indeed has the claimed size. It can be written as

− 2ε

¹²

(2π)

⁶

Z

dh dy

2

dξ

1

φ(h) sin b

h · η 2

f b

0

ξ

1

; η + ξt

₁

2 − y

2

f b

0

ξ − ξ

1

; η + ξt

₁

2 + y

2

e

^ih·y²⁺ⁱ^t²¹^h·(2ξ¹^−ξ)

1 − e

ⁱ^ε² ^−y²^{·ξ+(ξ−2ξ}¹^)·

η+ξt1 2

. It may be estimated by

Cε

¹²

Z

dh | φ b | (h) Z

dy

₂

dξ

₁

| f b

₀

|

ξ

₁

; η + ξt

₁

2 + y

₂

| f b

₀

|

ξ − ξ

₁

; η + ξt

₁

2 − y

₂

| ξ

₁

|

η + ξt

₁

2 + y

₂

+ | ξ − ξ

₁

|

η + ξt

₁

2 − y

₂

.

Therefore the term S

2,ε^(2,1)

vanishes as well.

As a conclusion, all terms S

2,ε^π

, which are the ones that are linear in φ, vanish in the limit ε → 0.

We now pass to the evaluation of the terms T

2,ε^π

.

The contribution T

2,ε^(1,2)

has been already considered in Ref. [2]. However, for sake of completeness, we analyze this term in the present context as well. Using (4.5) in (3.17), and performing the change of variables:

h

₁

+ h

₂

ε = k, h

₂

= h, t

₁

− t

₂

ε = s, (4.11)

we arrive at

T b

2,ε^(1,2)

(ξ, η) = − 1 (2π)

⁶

X

σ1,σ2=±1

σ

₁

σ

₂

Z

^t_ε¹

0

ds Z

dh Z

dk φ(h) b φ( b − h + εk) e

^−i[η·h

(

^σ²⁻2^σ¹

)

^+σ²^h²^s]

e

^−iε^σ²¹^k·η

e

^−iεσ²^h²^{·[ξs−2sk]}

f b

₀

(ξ − k; η + t

₁

ξ + hs − kt

₁

) f b

₀

(k; kt

₁

− hs).

(4.12)

This term converges formally to T b

2^(1,2)

(ξ, η) = − 1

(2π)

⁶

X

σ1,σ2=±1

σ

₁

σ

₂

Z

+∞

0

ds Z

dh Z

dk | φ(h) b |

²

e

^−i[η·h

(

^σ²⁻2^σ¹

)

^+σ²^h²^s]

f b

₀

(ξ − k; η + t

₁

ξ + hs − kt

₁

) f b

₀

(k; kt

₁

− hs).

(4.13)

To justify the limit we split the integration in ds over the two intervals [0, 1], [1, + ∞ ]. In the first interval we bound the integrand by

k φ b k

L^∞

k f b

₀

k

L^∞

| φ(h) b | sup

η

| f b

₀

(k, η) | , (4.14)