mean field asymptotics of generalized quantum kinetic equation

Vol. 70 (2012) REPORTS ON MATHEMATICAL PHYSICS No. 2

MEAN FIELD ASYMPTOTICSOF GENERALIZED QUANTUM KINETIC EQUATION

V. I. GERASIMENKO

Institute of Mathematics of NAS of Ukraine,

3, Tereshchenkivs’ka Str., 01601 Kyiv-4, Ukraine

(e-mail: [email protected])

and

ZH. A. TSVIR

Taras Shevchenko National University of Kyiv,

Department of Mechanics and Mathematics,

2, Academician Glushkov Av., 03187 Kyiv, Ukraine

(e-mail: [email protected])

(Received November 8, 2011 – Revised April 16, 2012)

We construct the mean field asymptotics of a solution of an initial value problem of the

generalized quantum kinetic equation. As a result we rigorously derive the Vlasov quantum

kinetic equation and the nonlinear Schrodinger equation. Moreover, the mean field asymptotics

of a sequence of explicitly defined functionals of a solution of the generalized quantum kinetic

equation, by which the average values of the nonadditive-type marginal observables are determined,

is established. The obtained results are extended to particles interacting via many-body potentials.

Keywords: quantum kinetic equation; nonlinear Schrodinger equation; scaling limit; cumulant

of scattering operators; quantum correlation.

1. Introduction

During the last decade a considerable progress in the rigorous derivation ofquantum kinetic equations, in particular the nonlinear Schrodinger equation andthe Gross–Pitaevskii equation, is observed [1–10]. The developed approach to thisproblem is based on the construction of the scaling limit (mean field) [11] ofa perturbation series of the initial value problem of the quantum BBGKY hierarchy.

In [12] we have established that, if the initial data is completely defined bya one-particle marginal density operator, then all possible states of infinite-particlesystems at an arbitrary moment of time can be described by the generalized quantumkinetic equation within the framework of a one-particle density operator withoutany approximations. The objective of this paper is to construct the mean field(self-consistent field) asymptotics of a solution of the initial value problem of thegeneralized quantum kinetic equation.

[135]

136 V. I. GERASIMENKO and ZH. A. TSVIR

We outline the structure of the paper. In Section 2 we recall some definitionsand preliminary facts on the rigorous description of quantum kinetic evolution. Thenin Section 3 the main results on the mean field scaling limit of a solution ofan initial value problem of the generalized quantum kinetic equation and marginalfunctionals of the state are stated. In Section 4 we prove the main results. Weestablish that the constructed asymptotics of a solution of the generalized quantumkinetic equation are governed by the quantum Vlasov kinetic equation and that thelimit marginal functionals of the state are the products of a solution of the derivedVlasov quantum kinetic equation which means the propagation of a chaos propertyin time. Finally, in Section 5, we conclude with some observations and perspectivesfor future research. We also consider the generalization of the obtained results tosystems of quantum particles interacting via many-body potentials.

2. The generalized quantum kinetic equation

We adduce some definitions and preliminary facts about the description ofquantum dynamics within the framework of a one-particle density operator governedby the generalized quantum kinetic equation.

We consider a quantum system of a nonfixed (i.e. arbitrary but finite) number ofidentical (spinless) particles obeying the Maxwell–Boltzmann statistics in the spaceRν, ν ≥ 1. We will use the units where h = 2πh = 1 is a Planck constant, and

m = 1 is the mass of particles. Let H be a one-particle Hilbert space, then then-particle space Hn is a tensor product of n Hilbert spaces H, and we adopt theusual convention that H0 = C. We denote by FH =

⊕∞n=0Hn the Fock space over

the Hilbert space H. The Hamiltonian Hn of an n-particle system is a self-adjointoperator with the domain D(Hn) ⊂ Hn by setting H0 = 0 and

Hn =

n∑i=1

K(i)+ ε

n∑i1<i2=1

�(i1, i2), (1)

where K(i) is the kinetic energy operator of the particle i, �(i1, i2) is the operatorof a two-body interaction potential and ε > 0 is a scaling parameter. The operatorK(i) acts on functions ψn that belong to the subspace L2

0(Rνn) ⊂ D(Hn) ⊂ L

2(Rνn)

of infinitely differentiable functions with compact supports according to the formula:K(i)ψn = −

12�qi

ψn. Correspondingly, we have �(i1, i2)ψn = �(qi1, qi2)ψn, and weassume that the function �(qi1, qi2) is symmetric with respect to permutations ofits arguments, and is translation-invariant and bounded.

Let L1(FH) =⊕∞

n=0L1(Hn) be the space of sequences f = (f0, f1, . . . , fn, . . .)

of trace class operators fn ≡ fn(1, . . . , n) ∈ L1(Hn) and f0 ∈ C, that satisfythe symmetry condition fn(1, . . . , n) = fn(i1, . . . , in) for arbitrary (i1, . . . , in) ∈

(1, . . . , n), equipped with the norm

‖f ‖L1(FH)

=

∞∑n=0

‖fn‖L1(Hn)=

∞∑n=0

Tr1,...,n|fn(1, . . . , n)|,

MEAN FIELD ASYMPTOTICS OF GENERALIZED QUANTUM KINETIC EQUATION 137

where Tr1,...,n are partial traces. We denote by L10(FH) =

⊕∞n=0L

10(Hn) the everywhere

dense set of finite sequences of degenerate operators with infinitely differentiablekernels with compact supports [13, 14].

On L1(FH) we define the group G(−t) = ⊕∞n=0Gn(−t) of operators of the von

Neumann equations [14]

Gn(−t)fn.= e−itHnfn e

itHn. (2)

The mapping (2) t → G(−t)f on L1(FH) is an isometric strongly continuous group

which preserves positivity and self-adjointness of operators. For f ∈ L10(FH) there

exists a limit in which the infinitesimal generator (−N ) = ⊕∞n=0(−Nn) of the groupof evolution operators (2) is determined in the sense of the norm convergence asfollows

limt→0

1

t

(Gn(−t)fn − fn

)= −i(Hnfn − fnHn)

.= −Nnfn, (3)

where Hn is the Hamiltonian (1) and the operator −i(Hnfn − fnHn) is defined onthe domain D(Hn) ⊂ Hn.

We also introduce the group of scattering operators

Gn(t).= Gn(−t, 1, . . . , n)

n∏i=1

G1(t, i), (4)

where G1(t, i) is adjoint to the operator G1(−t, i) defined by formula (2). Infinitesimalgenerator of the group (4) is the operator (−Nint(i, j)) defined on L

10(Hn) as follows,

(−Nint(i, j))fn.= −i

(�(i, j) fn − fn �(i, j)

).

Let us denote Y ≡ (1, . . . , s), X \ Y ≡ (s + 1, . . . , s + n), let {Y } be the setconsisting of one element Y = (1, . . . , s), and let θ be the declasterization mappingdefined by the formula θ({Y }, X \ Y) = X. We define the (1 + n)th-order (n ≥ 0)cumulant of the groups of operators (2) as follows,

A1+n(t, {Y }, X \ Y) =∑

P:({Y },X\Y)=⋃i Xi

(−1)|P|−1(|P| − 1)!∏Xi⊂P

G|θ(Xi)|(−t, θ(Xi)), (5)

where∑

P is the sum over all possible partitions P of the set ({Y }, X \ Y) into |P|nonempty mutually disjoint subsets Xi ⊂ ({Y }, X \ Y), for example,

A1(t, {Y }) = Gs(−t),

A2(t, {Y }, s + 1) = Gs+1(−t, Y, s + 1)− Gs(−t, Y )G1(−t, s + 1).

We indicate some properties of the operators (5). If n = 0, then the generatorof the first-order cumulant A1(t, {Y }) = Gs(−t) for fs ∈ L

10(Hs) ⊂ L1(Hs) is given

by the operator

limt→0

1

t

(A1(t, {Y })− I

)fs = −Nsfs,


in the sense of the norm convergence on L1(Hs). In the case of n = 1 we have,

in the sense of the norm convergence on L1(Hs+1),

limt→0

1

tA2(t, {Y }, s + 1)fs+1 = −ε

s∑i=1

Nint(i, s + 1)fs+1,

and for n ≥ 2, in the consequence of the fact that we consider a system of particlesinteracting by a two-body potential, it holds

limt→0

1

tA1+n(t)fs+n = 0.

We examine the mean field (self-consistent field) asymptotics of a solution of theCauchy problem of the generalized quantum kinetic equation [12] (see also [15])

d

dtF1(t, 1) = −N1(1)F1(t, 1) (6)

+ ε Tr2

(−Nint(1, 2)

) ∞∑n=0

1

n!Tr3,...,n+2V1+n

(t, {1, 2}, 3, . . . , n+ 2

) n+2∏i=1

F1(t, i),

F1(t, 1)|t=0 = F01 (1). (7)

In Eq. (6) the (n + 1)th-order evolution operator V1+n(t), n ≥ 0, is defined asfollows (in the case of {Y } = {1, 2}),

V1+n(t, {Y }, X \ Y).=

n∑k=0

(−1)kn∑

n1=1

. . .

n−n1−...−nk−1∑nk=1

n!

(n− n1 − . . .− nk)!

× A1+n−n1−...−nk(t, {Y }, s + 1, . . . , s + n− n1 − . . .− nk)

×

k∏j=1

∑Dj :Zj=

⋃ljXlj

,

|Dj |≤s+n−n1−···−nj

1

|Dj |!

s+n−n1−...−nj∑i1 �=... �=i|Dj |

=1

∏Xlj

⊂Dj

1

|Xlj |!A1+|Xlj

|(t, ilj , Xlj ), (8)

where∑

Dj :Zj=⋃ljXlj

is the sum over all possible dissections of the linearly ordered

set Zj ≡ (s + n − n1 − . . . − nj + 1, . . . , s + n − n1 − . . . − nj−1) on no more

than s + n− n1 − . . .− nj linearly ordered subsets, and we denote by A1+n(t) the(1+n)-th-order cumulant (5) of the groups of scattering operators (4), for example,

V1(t, {Y }) = A1(t, {Y }),

V2(t, {Y }, s + 1) = A2(t, {Y }, s + 1)− A1(t, {Y })

s∑i=1

A2(t, i, s + 1).

The series of the collision integral in Eq. (6) converges under the condition that‖F1(t)‖L1(H) < e−8, [12].


The global in time solution of the initial value problem (6)–(7) is determinedby the following expansion [12]

F1(t, 1) =

∞∑n=0

1

n!Tr2,...,1+n A1+n(t, 1, . . . , n+ 1)

n+1∏i=1

F 01 (i), (9)

where A1+n(t) is the (1+ n)-th-order cumulant (5) of the groups of operators (2).The series (9) converges under the condition that ‖F 0

1 ‖L1(H) < e−1.In the case of initial data given by means of a one-particle density operator,

the evolution of all possible states of quantum many-particle systems is describedby a solution of the initial value problem of the generalized quantum kineticequation (6)–(7) and by a sequence of explicitly defined functionals of a solutionof this quantum kinetic equation [12]

Fs(t, Y | F1(t)

) .=

∞∑n=0

1

n!Trs+1,...,s+nV1+n

(t, {Y }, X \ Y

) s+n∏i=1

F1(t, i), s ≥ 2, (10)

where the (n + 1)-th-order (n ≥ 0) evolution operator V1+n(t) is defined byformula (8). The series (10) of the marginal functionals of the state converges underthe condition that ‖F1(t)‖L1(H) < e−(3s+2).

3. The mean field limit theorems

The mean field scaling asymptotics of solution (9) of the initial value problemof the generalized kinetic equation (6) is described by the following limit theorem.

THEOREM 1. Let there exists the limit f 01 ∈ L1(H) of the initial data (7)

limε→0

∥∥ε F 01 − f

01

∥∥L1(H)

= 0,

then for the finite time interval t ∈ (−t0, t0), where t0 ≡(2 ‖�‖L(H2)

‖f 01 ‖L1(H)

)−1,

there exists the following limit of the solution (9) of the generalized quantum kineticequation (6),

limε→0

∥∥ε F1(t)− f1(t)∥∥L1(H)

= 0, (11)

where the limit one-particle marginal operator f1(t) is represented in the form

f1(t, 1)=

∞∑n=0

t∫0

dt1 . . .

tn−1∫0

dtnTr2,...,1+nG1(−t + t1, 1)(−Nint(1, 2)

)

×

2∏j1=1

G1(−t1 + t2, j1) . . .

n∏jn−1=1

G1(−tn−1 + tn, jn−1)

×

n∑in=1

(−Nint(in, 1+ n)

) 1+n∏jn=1

G1(−tn, jn)

1+n∏i=1

f 01 (i). (12)


For the bounded interaction potentials (1) the series (12) is norm convergent onthe space L1(H) under the condition that

t < t0 ≡(2 ‖�‖L(H2)

‖f 01 ‖L1(H)

)−1,

and for an initial data f 01 ∈ L

10(H) the mean field limit operator f1(t) is a strong

solution of the Cauchy problem of the Vlasov quantum kinetic equation

∂

∂tf1(t, 1)=−N1(1)f1(t, 1)+ Tr2

(−Nint(1, 2)

)f1(t, 1)f1(t, 2), (13)

f1(t)|t=0 = f01 . (14)

Since any solution of the initial value problem (6)–(7) of the generalized quantumkinetic equation converges to a solution of the initial value problem (13)–(14) of theVlasov quantum kinetic equation as (11), for marginal functionals (10) we obtainthe following result.

THEOREM 2. Under the conditions of Theorem 1 for functionals (10) it holds

limε→0

∥∥εsFs(t, 1, . . . , s | F1(t))−

s∏j=1

f1(t, j)∥∥L1(Hs )

= 0,

where f1(t) is the operator defined by (12).

This statement means that a chaos property propagates in time in the mean fieldscaling limit.

4. The proof of the limit theorems

For an asymptotic perturbation of first-order cumulant (5) the following statementis valid [13].

LEMMA 1. Let fs ∈ L1(Hs), then for an arbitrary finite time interval for thestrongly continuous group (2) it holds

limε→0

∥∥Gs(−t)fs − s∏j=1

G1(−t, j )fs∥∥L1(Hs )

= 0.

If an interaction potential is a bounded operator and fs ∈ L1(Hs), then forgroups (2) an analog of the Duhamel equation holds

(Gs(−t, 1, . . . , s)−

s∏l=1

G1(−t, l))fs

= ε

t∫0

dτ

s∏l=1

G1(−t + τ, l)(−

s∑i<j=1

Nint(i, j))Gs(−τ)fs. (15)


Indeed, the Duhamel equation (15) is valid for fs ∈ L10(Hs) ⊂ L

1(Hs). Since

the operators on both sides of equality (15) are bounded and the set L10(Hs) is

everywhere dense in L1(Hs), then equality (15) is valid for arbitrary fs ∈ L1(Hs) [13].We note that the integral in (15) exists in the strong sense and the statementfunction

s∏l=1

G1(−t + τ, l)(−

s∑i<j=1

Nint(i, j))Gs(−τ)

is strongly continuous over τ for every fs ∈ L1(Hs), and hence is integrable.

Thus, the validity of Lemma 1 follows from the estimate

∥∥∥(Gs(−t, 1, . . . , s)−

s∏l=1

G1(−t, l))fs

∥∥∥L1(Hs )

≤ ε

t∫0

dτ

∥∥∥ s∏l=1

G1(−t + τ, l)(−

s∑i<j=1

Nint(i, j))Gs(−τ)fs

∥∥∥L1(Hs )

≤ ε ts(s − 1)‖�‖L(H2)‖fs‖L1(Hs )

.

In the consequence of Lemma 1 for the scattering operators (4) or for thefirst-order cumulant (5) of operators (4) the equality holds

limε→0

∥∥Gs(t)fs − fs∥∥L1(Hs )= 0.

Analogously, for an asymptotic perturbation of cumulants (5) the followingstatement is true.

LEMMA 2. Let fs+n ∈ L1(Hs+n), then for an arbitrary finite time interval forthe (1+ n)-th-order cumulant of strongly continuous groups (2) it holds

limε→0

∥∥∥∥ 1

εn

1

n!A1+n(t, {Y }, X \ Y)fs+n −

t∫0

dt1 . . .

tn−1∫0

dtn

s∏j=1

G1(−t + t1, j)

×

s∑i1=1

(−Nint(i1, s + 1)

) s+1∏j1=1

G1(−t1 + t2, j1) . . .

s+n−1∏jn−1=1

G1(−tn−1 + tn, jn−1)

×

s+n−1∑in=1

(−Nint(in, s + n)

) s+n∏jn=1

G1(−tn, jn)fs+n

∥∥∥∥L1(Hs+n)

= 0.

The validity of Lemma 2 follows from the Duhamel formula for the cumulants (5)of strongly continuous groups (2)


1

n!Trs+1,...,s+nA1+n(t, {Y }, X \ Y)fs+n

= εn

t∫0

dt1 . . .

tn−1∫0

dtnTrs+1,...,s+nGs(−t + t1)

s∑i1=1

(−Nint(i1, s + 1)

)

× Gs+1(−t1 + t2) . . .Gs+n−1(−tn−1 + tn)

s+n−1∑in=1

(−Nint(in, s + n)

)Gs+n(−tn)fs+n,

and it is proved similar to the previous Lemma 1.

Taking into account Lemma 1 and Lemma 2, now we give a sketch of the proofof Theorem 1.

Because of the interaction potential is a bounded operator and fs+1 ∈ L1(Hs+1),

then for the second-order cumulant A2(t, {Y }, s + 1) of scattering operators (4) ananalog of the Duhamel equation holds

A2(t, {Y }, s + 1)fs+1 =

∫ t

0

dτ Gs(−τ, Y )G1(−τ, s + 1)

× ε

s∑i1=1

(−Nint(i1, s + 1)

)Gs+1(τ − t, Y, s + 1)

s+1∏i2=1

G1(τ, i2)fs+1,

and, consequently, for the second-order evolution operator V2(t, {Y }, s+ 1) we have

V2(t, {Y }, s + 1)fs+1

.=

(A2(t, {Y }, s + 1)− A1(t, {Y })

s∑i1=1

A2(t, i1, s + 1))fs+1

=

∫ t

0

dτ Gs(−τ, Y )G1(−τ, s + 1)(ε

s∑i1=1

(−Nint(i1, s + 1))Gs+1(τ − t, Y, s + 1)

− Gs(τ − t, Y )ε

s∑i1=1

(−Nint(i1, s + 1))G2(τ − t, i1, s + 1)) s+1∏i2=1

G1(τ, i2)fs+1.

Then according to Lemma 2, i.e. due to the formulae of an asymptotic perturbationof cumulants of groups of operators, and the definition (8) of the evolution operatorsV1+n(t, {Y }, X \ Y), n ≥ 0, we establish

limε→0

∥∥(V1(t, {Y })− I)fs

∥∥L1(Hs )

= 0,

and for n ≥ 1 it holds, respectively

limε→0

∥∥∥∥ 1

εnV1+n(t, {Y }, X \ Y)fs+n

∥∥∥∥L1(Hs+n)

= 0.


In view that the series for ε F1(t) converges under the condition that t < t0 ≡(2 ‖�‖L(H2)

‖ε F 01 ‖L1(H)

)−1, then for t < t0 the remainders of the solution series (9)

and (12) can be made arbitrary small for sufficiently large n = n0 independentlyof ε. Then for each integer n every term of these series converges term by term,according to Lemma 1, Lemma 2 and definition (8).

Now we construct an evolution equation, which satisfies the expression (12). Weprove that a solution of the initial value problem (13)–(14) of the Vlasov quantumkinetic equation is represented by the series (12).

Indeed, taking into account the validity of equality (3) for fn ∈ L10(Hn), we

differentiate (12) w.r.t. the time variable in the sense of pointwise convergence ofthe space L1(H),

d

dtf1(t, 1)= −N (1)f1(t, 1)

+ Tr2

(−Nint(1, 2)

) ∞∑n=0

t∫0

dt1 . . .

tn−1∫0

dtn Tr3,...,n+2

2∏i1=1

G1(−t + t1, i1)

×

2∑k1=1

(−Nint(k1, 3)

) 3∏j1=1

G1(−t1 + t2, j1) . . .

n+1∏in=1

G1(−tn + tn, in)

×

n+1∑kn=1

(−Nint(kn, n+ 2)

) n+2∏jn=1

G1(−tn, jn)

n+2∏i=1

f 01 (i). (16)

Using the product formula for the one-particle marginal density operator f1(t, i)

defined by (12),

k∏i=1

f1(t, i)=

∞∑n=0

t∫0

dt1 . . .

tn−1∫0

dtnTrk+1,...,k+n

k∏i1=1

G1(−t + t1, i1)

×

k∑k1=1

(−Nint(k1, k + 1)

) k+1∏j1=1

G1(−t1 + t2, j1) . . .

k+n−1∏in=1

G1(−tn−1 + tn, in)

×

k+n−1∑kn=1

(−Nint(kn, k + n)

) k+n∏jn=1

G1(−tn, jn)

k+n∏i=1

f 01 (i),

where the group property of one-parameter mapping (2) is applied, we express the

second summand on the right-hand side of equality (16) in terms of∏2i=1f1(t, i)

and, consequently, we derive Eq. (13).

To give a sketch of the proof of Theorem 2 we represent marginal functionals ofthe state (10) in terms of the marginal correlation functionals Gs

(t, Y | F1(t)

), s ≥ 2,


namely

Fs(t, Y | F1(t)

)=

∑P:Y=

⋃i Xi

∏Xi⊂P

G|Xi |(t, Xi | F1(t)

), s ≥ 2,

where∑

P:Y=⋃i Xi

is the sum over all possible partitions P of the set Y ≡ (1, . . . , s)

into |P| nonempty mutually disjoint subsets Xi ⊂ Y . The marginal correlationfunctionals Gs

(t, Y | F1(t)

), s ≥ 2, are represented by the following expansions [16]

Gs

(t, Y | F1(t)

)=

∞∑n=0

1

n!Trs+1,...,s+nV1+n

(t, θ({Y }), X \ Y

) s+n∏i=1

F1(t, i), (17)

where the notion of the declasterization mapping θ : {Y } → Y is introduced. Thismapping is defined by the formula θ({Y }) = Y , it means that the declasterizationof particle clusters in cumulants of the scattering operators, i.e. in contrast toexpansion (10) the n-th term of expansions (17) of the marginal correlation functionalsGs

(t, Y | F1(t)

)is governed by the (1 + n)-th-order evolution operator (8) of the

(s + n)-th-order, n ≥ 0, cumulants of the scattering operators. For example, ascompared to the evolution operators (8), the lower orders evolution operatorsV1+n

(t, θ({Y }), X \ Y

), n ≥ 0, have the form

V1(t, θ({Y })) = As(t, θ({Y }),

V2(t, θ({Y }), s + 1) = As+1(t, θ({Y }), s + 1)− As(t, θ({Y }))

s∑i=1

A2(t, i, s + 1),

or in the case of s = 2, we have V1(t, θ({1, 2})) = G2(t, 1, 2)− I.The statement of Theorem 2 is true because of the validity of the equality

limε→0

∥∥εsGs

(t, Y | F1(t)

)∥∥L1(Hs )

= 0

for the functionals (17). In view of the structure of the evolution operatorsV1+n

(t, θ({Y }), X\Y

), n ≥ 0, the last equality follows from Lemma 1 and Lemma 2.

5. Conclusion

In the mean field scaling limit we have derived the Vlasov quantum kineticequation (13) in the case of bounded potentials of interaction on the basis of thegeneralized quantum kinetic equation (6). Correspondingly, the Hartree equation orthe nonlinear Schrodinger equation for pure states of a quantum systems of particlesobeying the Maxwell–Boltzmann statistics can be justified.

If we consider pure states, i.e. f1(t) = |ψt 〉〈ψt | is a one-dimensional projectoronto a unit vector ψt ∈ H or in terms of the kernel of the marginal operatorf1(t): f1(t, q, q

′) = ψ(t, q)ψ(t, q ′), then for such states the Vlasov quantum kineticequation reduces to the Hartree equation

i∂

∂tψ(t, q) = −

1

2�qψ(t, q)+

∫dq ′�(q − q ′)|ψ(t, q ′)|2ψ(t, q). (18)


Moreover, if for pure states it holds

limε→0

∥∥ε F 01 − |ψ0〉〈ψ0|

∥∥L1(H)

= 0,

then the statement of Theorem 2 reads

limε→0

∥∥ εsFs(t | F1(t))− |ψt 〉〈ψt |

⊗s∥∥L1(Hs )

= 0,

where |ψt 〉 is the solution of the nonlinear Hartree equation (18) for initial data|ψ0〉.

We remark that for a system of particles, interacting by the potential whosekernel is the Dirac measure �(q) = δ(q), the Hartree equation (18) is reduced tothe cubic nonlinear Schrodinger equation [4]

i∂

∂tψ(t, q) = −

1

2�qψ(t, q)+ |ψ(t, q)|

2ψ(t, q).

The obtained results can be generalized to systems of quantum particles interactingvia many-body potentials, i.e. systems with the Hamilton operators

Hn =

n∑i=1

K(i)+

n∑k=2

εk−1n∑

i1<...<ik=1

�(k)(i1, . . . , ik),

where �(n) is an operator of an n-body interaction potential.

In the case under consideration the generalized quantum kinetic equation (6) hasthe form [12]

d

dtF1(t, 1) = −N1(1)F1(t, 1)

+

∞∑n=1

n∑k=1

εk1

(n− k)!

1

k!Tr2,...,n+1(−N

(k+1)int )(1, . . . ,

V1+n−k(t, {1, . . . , k + 1}, k + 2, . . . , n+ 1)

n+1∏i=1

F1(t, i), (19)

where V1+n−k(t) is the (1+ n− k)-th-order evolution operator (8),

(−N(n)

int )fn.= −i

(�(n) fn − fn �

(n)),

and the notation of formulae (4), (5) are used. The collision integral of the generalizedquantum kinetic equation (19) is given by the convergent series under the conditionthat ‖F1(t)‖L1(H) < e−8.

The mean field scaling limit f1(t) of the solution (9) of initial value problemfor the generalized kinetic equation (19) is described by the limit theorem similarto Theorem 1, and for an initial data f 0

1 ∈ L10(H) it is a strong solution of the


Cauchy problem of the following Vlasov quantum kinetic equation,

d

dtf1(t, 1)=−N (1)f1(t, 1)+

∞∑n=1

1

n!Tr2,...,n+1(−N

(n+1)int )(1, . . . , n+ 1)

n+1∏i=1

f1(t, i),

f1(t)|t=0 = f01 .

Then for an n-body interaction potential the Hartree equation takes the form

i∂

∂tψ(t, q1)=−

1

2�q1

ψ(t, q1)

+

∞∑n=1

1

n!

∫dq2 . . . dqn+1�

(n+1)(q1, . . . , qn+1)

n+1∏i=2

|ψ(t, qi)|2ψ(t, q1),

and correspondingly, we obtain a nonlinear Schrodinger equation with the 2n − 1power nonlinear term.

These results can be extended to quantum systems of bosons and fermionssimilarly to [17].

We note that one more approach to derive the Vlasov quantum kinetic equationconsists in constructing the scaling asymptotics of a solution of the dual BBGKYhierarchy for marginal observables [18]. One of the advantages of such approachas well as developed approach in this paper is the possibility to construct thekinetic equations in scaling limits in the presence of correlations of particle statesat the initial time, for instance, correlations characterizing the condensate states ofinteracting particles.

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mean field asymptotics of generalized quantum kinetic equation

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