On non-perturbative solution of quantum BBGKY hierarchy
We consider some approaches to the construction of a non-perturbative solution of the Cauchy problem of the quantum BBGKY hierarchy for a sequence of marginal density operators and analyze its properties for initial data from the space of sequences of trace class operators. One is represented in the...
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Інститут математики НАН України
2016
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552747326242816 |
|---|---|
| author | Gerasimenko, V. I. Krechko, V. V. Gerasimenko, V. I. Krechko, V. V. |
| author_facet | Gerasimenko, V. I. Krechko, V. V. Gerasimenko, V. I. Krechko, V. V. |
| author_institution_txt_mv | [
{
"author": "V. I. Gerasimenko",
"institution": "Institute of Mathematics, NAS of Ukraine"
},
{
"author": "V. V. Krechko",
"institution": "Institute of Mathematics, NAS of Ukraine"
}
] |
| author_sort | Gerasimenko, V. I. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T09:13:23Z |
| description | We consider some approaches to the construction of a non-perturbative solution of the Cauchy problem of the quantum BBGKY hierarchy for a sequence of marginal density operators and analyze its properties for initial data from the space of sequences of trace class operators. One is represented in the form of a series expansion over particle subsystems which generating operators are the corresponding-order cumulants of the groups of operators of systems of finitely many quantum particles. |
| first_indexed | 2026-08-04T01:04:21Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2016, т. 13, № 2, 7–26
УДК 517.956.223
V. I. Gerasimenko, V. V. Krechko
(Institute of Mathematics of NAS of Ukraine, Kyiv)
On non-perturbative solution of
quantum BBGKY hierarchy
gerasym@imath.kiev.ua, vi.kre4ko@gmail.com
We consider some approaches to the construction of a non-
perturbative solution of the Cauchy problem of the quantum BBGKY
hierarchy for a sequence of marginal density operators and analyze
its properties for initial data from the space of sequences of trace
class operators. One is represented in the form of a series expansi-
on over particle subsystems which generating operators are the
corresponding-order cumulants of the groups of operators of systems
of finitely many quantum particles.
Розглянуто пiдходи до побудови непертурбативного розв’язку за-
дачi Кошi для iєрархiї квантових рiвнянь ББҐКI для послiдовно-
стi маргiнальних операторiв густини i дослiджено його властиво-
стi у випадку початкових даних iз простору послiдовностей ядер-
них операторiв. Такий розв’язок зображується розкладом у ряд
за пiдсистемами частинок, твiрнi оператори якого є вiдповiдного
порядку кумулянтами груп операторiв систем скiнченої кiлькостi
квантових частинок.
c© V. I. Gerasimenko, V. V. Krechko, 2016
8 V. I. Gerasimenko, V. V. Krechko
1. Introduction
Nowadays the considerable advance in the rigorous derivation of
quantum kinetic equations in scaling limits, in particular the nonli-
near Schrödinger equation and the Gross–Pitaevskii equation [1]- [5]
as well as the quantum Boltzmann equation [6], [7], is observed.
This problem is closely related to the theory of the Bose–Einstein
condensation in systems of interacting bosons [8].
The approach to the derivation of kinetic equations is based on
the analysis of an asymptotic behavior of a solution of the quantum
BBGKY hierarchy for marginal density operators constructed by
the perturbation methods. The results were originally obtained for
bounded interaction potentials [9], [10] and then they have been
generalized to include the Coulomb interaction [11].
The aim of the paper is to develop rigorous approaches to the
construction of a non-perturbative solution of the Cauchy problem
of the quantum BBGKY hierarchy represented as an expansion over
particle clusters governed by the corresponding-order cumulant of
the groups of operators of finitely many particles stated in [12].
We now outline the structure of the paper and the main results.
In section 2 we introduce some preliminary facts about the Cauchy
problem of the quantum BBGKY hierarchy for marginal density
operators in case of initial data from the space of sequences of trace
class operators. The links of a perturbative solution of the quantum
BBGKY hierarchy and a non-perturbative solution represented in
the form of series expansion over particle subsystems which generati-
ng operators are corresponding-order cumulants (semi-invariant) of
groups of operators of finitely many quantum particles is established.
In section 3 we develop a cluster expansion approach to the justi-
fication of the structure of series expansions of a non-perturbative
solution. Section 4 deals with one more approach based on the defini-
tion of marginal density operators within the framework of dynamics
of correlations. Finally, in section 5 we conclude with some perspecti-
ves for future research.
On non-perturbative solution of quantum BBGKY hierarchy 9
2. The Cauchy problem of the quantum
BBGKY hierarchy
2.1. A non-perturbative solution of the quantum
BBGKY hierarchy
We consider a quantum system of a non-fixed (i.e. arbitrary but
finite) number of identical (spinless) particles obeying the Maxwell–
Boltzmann statistics in the space Rν , ν ≥ 1.
We will use units where h = 2π~ = 1 is a Planck constant, and
m = 1 is the mass of particles. Let H be a one-particle Hilbert space,
then the n-particle space Hn is a tensor product of n Hilbert spaces
H, and we adopt the usual convention that H0 = C. We denote by
FH =
⊕∞
n=0Hn the Fock space over the Hilbert space H.
Let L1
α(FH) =
⊕∞
n=0α
nL1(Hn) be the space of sequences f =
(f0, f1, . . . , fn, . . .) of trace class operators fn ≡ fn(1, . . . , n) ∈
L1(Hn) and f0 ∈ C, that satisfy the symmetry condition:
fn(1, . . . , n) = fn(i1, . . . , in) for arbitrary (i1, . . . , in) ∈ (1, . . . , n),
equipped with the norm: ‖f‖L1
α(FH) =
∑∞
n=0 α
n‖fn‖L1(Hn) =∑∞
n=0 α
n Tr1,...,n|fn(1, . . . , n)|, where the symbol Tr1,...,n denotes
partial traces and α > 1 is a real number. The everywhere dense
set in L1
α(FH) of finite sequences of degenerate operators with infi-
nitely differentiable kernels with compact supports we denote by
L1
0 ⊂ L1
α(FH) [10], [13].
The Hamiltonian Hn of a n-particle system is a self-adjoint
operator with the domain D(Hn) ⊂ Hn and Hn =
∑n
j=1K(j) +∑n
j1<j2=1 Φ(j1, j2), where K(j) is the operator of a kinetic energy
of the j particle, Φ(j1, j2) is the operator of a two-body interacti-
on potential. The operator K(j) acts on functions ψn, that
belong to the subspace L2
0(Rνn) ⊂ D(Hn) ⊂ L2(Rνn) of infi-
nitely differentiable functions with compact supports according
to the formula: K(j)ψn = −1
2∆qjψn. Correspondingly, we have:
Φ(j1, j2)ψn = Φ(qj1 , qj2)ψn, and we assume that the functi-
on Φ(qj1 , qj2) is symmetric with respect to permutations of its
10 V. I. Gerasimenko, V. V. Krechko
arguments, translation-invariant and bounded function.
To determine a solution of the Cauchy problem of the quantum
BBGKY hierarchy for marginal density operators we introduce some
necessary facts. For f ∈ L1
α(FH) let G(−t) =
⊕∞
n=0Gn(−t), where
Gn(−t)fn
.
= e−itHnfn e
itHn . (1)
In the space L1
α(FH) this mapping: t → G(−t)f , is an isometric
strongly continuous group which preserves positivity and self-
adjointness of operators [10]. For f ∈ L1
0(FH) ⊂ D(−N ) in the
sense of the norm convergence in the space L1
α(FH) there exists the
following limit of the group of operators (1) which is determined its
infinitesimal generator: −N =
⊕∞
n=0(−Nn)
lim
t→0
1
t
(
Gn(−t)fn − fn
)
= −i(Hnfn − fnHn) =
n∑
j=1
(
−N (j)
)
fn +
n∑
j1<j2=1
(
−Nint(j1, j2)
)
fn,
where the operators (−N (j)) and (−Nint(j1, j2)) are defined on the
subspace L1
0(Hs) as follows:
(−N (j))fs
.
= −i
(
K(j) fs − fsK(j)
)
, (2)
(−Nint(j1, j2))fs
.
= −i
(
Φ(j1, j2) fs − fs Φ(j1, j2)
)
. (3)
Let us denote: Y ≡ (1, . . . , s),X\Y ≡ (s+1, . . . , s+n) and {Y } is
the set consisting of one element Y = (1, . . . , s), the mapping θ is the
declusterization mapping defined by the formula: θ({Y }, X\Y ) = X,
the symbol
∑
P is the sum over all possible partitions P of the set
({Y }, X \Y ) into |P| nonempty disjoint subsets Xi ⊂ ({Y }, X \Y ).
The evolution of all possible states of quantum many-particle
systems is described by the sequences F (t) = (I, F1(t, 1), . . . ,
Fs(t, 1, . . . , s), . . .) of marginal density operators that satisfy the
On non-perturbative solution of quantum BBGKY hierarchy 11
Cauchy problem of the quantum BBGKY hierarchy [8]:
d
dt
Fs(t) =
( s∑
j=1
(
−N (j)
)
+
s∑
j1<j2=1
(
−Nint(j1, j2)
))
Fs(t) + (4)
s∑
j=1
Trs+1
(
−Nint(j, s+ 1)
)
Fs+1(t),
Fs(t) |t=0= F 0
s , s ≥ 1. (5)
For the Cauchy problem (4),(5) the following statement holds [12].
Theorem 1. If F (0) ∈ L1
α(FH) and α > e, then for t ∈ R there
exists a unique solution of the Cauchy problem (4),(5) given by the
series expansions
Fs(t, Y ) =
∞∑
n=0
1
n!
Trs+1,...,s+nA1+n(−t, {Y }, X \ Y )F 0
s+n(X), (6)
s ≥ 1,
where the (1 + n)th-order cumulant (semi-invariant) A1+n(−t) of
groups of operators (1) is defined by the expansion
A1+n(−t, {Y }, X \ Y ) = (7)∑
P :({Y }, X\Y )=
⋃
iXi
(−1)|P|−1(|P| − 1)!
∏
Xi⊂P
G|θ(Xi)|(−t, θ(Xi)).
For initial data F (0) ∈ L1
α,0 ⊂ L1
α(FH) it is a strong solution and
for arbitrary initial data from the space L1
α(FH) it is a weak solution.
We remark that, according to the estimate [14]∥∥A1+n(−t)fs+n
∥∥
L1(Hs+n) ≤ n!en+2
∥∥fs+n∥∥L1(Hs+n),
for F 0 ∈ L1
α(FH) series (6) converges in the norm of the space
L1
α(FH) provided that α > e, and the inequality holds
‖F (t)‖L1
α(FH) ≤ cα‖F (0)‖L1
α(FH),
12 V. I. Gerasimenko, V. V. Krechko
where cα = e2(1− e
α)−1. The parameter α is interpreted as the value
inverse to the average number of particles.
2.2. A perturbative solution of the quantum BBGKY
hierarchy
In paper [9] (see also [6], [11] and references cited therein) a solution
of the quantum BBGKY hierarchy was represented in the form of
the perturbation (iteration) series
Fs(t, Y ) =
∞∑
n=0
t∫
0
dt1 . . .
tn−1∫
0
dtn Trs+1,...,s+n Gs(−t+ t1)× (8)
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)F 0
s+n(X).
Series (8) converges in the norm of the space L1(Hs) on finite time
interval t ∈ (−t0, t0), where t0 ≡ (4‖Φ‖L(H2))
−1.
We establish the links of series expansion (6) and iteration series
(8). With this aim we shall formulate a preliminary statement.
Proposition 1. In case of a bounded interaction potential for arbi-
trary fn ∈ L1(Hn), n ≥ 2, the recursion relations for cumulants (7)
of groups of operators (1) are true
An(−t, 1, . . . , n)fn =
t∫
0
dt1
∏
k∈(1,...,n)
A1(−t+ t1, k)× (9)
∑
i<j∈(1,...,n)
(
−Nint(i, j)
)
An−1(−t1, I)fn,
where I ≡ ({i, j}, 1, . . . , i− 1, i+ 1, . . . , j− 1, j+ 1, . . . , n), i.e. |I| =
n− 1.
On non-perturbative solution of quantum BBGKY hierarchy 13
Proof. We introduce the operator
Tn(−t1)
.
=
n∏
k=1
A1(−t+ t1, k)An(−t1, 1, . . . , n), (10)
such that for fn ∈ L1(Hn), n ≥ 2, it holds
t∫
0
d
dt1
Tn(−t1)fn = An(−t, 1, . . . , n)fn.
As a result of the integration of expression (10) the last equality is
true due to the validity for n ≥ 2, of the identity
∑
P: (1,...,n)=
⋃
lZl
(−1)|P|−1(|P| − 1)! =
n∑
k=1
(−1)k−1s(n, k)(k − 1)! = 0,
where s(n, k) are the Stirling numbers of the second kind.
For fn ∈ L1
0(Hn) the mapping t1 → Tn(−t1)fn is differentiable
over the time variable and, according to definition (7) and formula
(2), for operator (10) the following equality holds
d
dt1
Tn(−t1)fn =
∑
P: (1,...,n)=
⋃
lZl,
|P|6=n
(−1)|P|−1(|P| − 1)!×
n∏
k=1
G1(−t+ t1, k)
(
−
∑
i<j∈Zl
Nint(i, j)
) ∏
Zl⊂P
G|Zl|(−t1, Zl)fn. (11)
Finally, gathering terms with the operator (−Nint)(i, j) for every
meaning of indexes (i, j) from right-hand side of equality (11), this
14 V. I. Gerasimenko, V. V. Krechko
expression is transformed to the form
t∫
0
d
dt1
Tn(−t1)fn = (12)
t∫
0
dt1
∏
k∈(1,...,n)
G1(−t+ t1, k)
(
−
∑
i<j∈(1,...,n)
Nint(i, j)
)
×
∑
P: I=
⋃
lZl
(−1)|P|−1(|P| − 1)!
∏
Zl⊂P
G|θ(Zl)|(−t1, θ(Zl)) =
t∫
0
dt1
∏
k∈(1,...,n)
G1(−t+ t1, k)
(
−
∑
i<j∈(1,...,n)
Nint(i, j)
)
An−1(−t1, I)fn,
where we used notations introduced above.
Thus, in consequence of equalities (11) and (12) we derive equality
(9) on the set L1
0(Hn). Since the operators from both sides of this
equality are bounded and the set L1
0(Hn) is everywhere dense set in
the space L1(Hn), equality (12) holds for arbitrary fn ∈ L1(Hn).
2
We remark that in case of n = 2 relationship (9) is the Duhamel
equation for the group of operators (1) of a system of two quantum
particles
A2(−t, 1, 2)
.
= G2(−t, 1, 2)− G1(−t, 1)G1(−t, 2) =
t∫
0
dt1G1(−t+ t1, 1)G1(−t+ t1, 2)
(
−Nint(1, 2)
)
G2(−t1, 1, 2).
Using equality (9), we can transform series (6) to the form
of the iteration (perturbation) series of the quantum BBGKY hi-
erarchy. Indeed, solving recursion relations (9), for the (1+n)th-order
On non-perturbative solution of quantum BBGKY hierarchy 15
cumulant (7) we obtain the following expansion:
A1+n(−t, {Y }, X \ Y ) = (13)
t∫
0
dt1 . . .
tn−1∫
0
dtn
∏
K1⊂I1
G|θ(K1)|(−t+ t1, θ(K1))×
(
−
∑
i1<j1⊂I1
Nint(i1, j1)
) ∏
K2⊂I2
G|θ(K2)|(−t1 + t2, θ(K2))×
(
−
∑
i2<j2⊂I2
Nint(i2, j2)
)
. . .
∏
Kn⊂In
G|θ(Kn)|(−tn−1 + tn, θ(Kn))×
(
−
∑
in<jn⊂In
Nint(in, jn)
)
Gs+n(−tn, X),
where I1 ≡ ({Y }, X \ Y ), I2 ≡ {i1, j1} ∪ (I1 \ (i1, j1)), . . . , In ≡
{in−1, jn−1} ∪ (In−1 \ (in−1, jn−1)) and we used notations accepted
above.
Therefore series expansion (6) reduces to iteration series (8).
Indeed, taking into account the Duhamel equation (13) for (1+n)th-
order cumulant (7) and the fact that the groups of operators (1) are
isometric in the spaces L1(Hn), n ≥ 1, series expansion (6) reduces
to series (8) in case of a two-body interaction potential.
We remark that representations (6) and (8) for a solution of the
Cauchy problem of the quantum BBGKY hierarchy are equivalent
for arbitrary initial data from the space L1
α(FH) in case of bounded
operators of the interaction potential. However, they are not equi-
valent in other operator spaces. As well known, by differentiating the
iteration series (8) we can not avoid the problems of getting out of
extra terms only in the space of trace class operators. This problem
disappears, if we use representation (13).
3. A cluster expansion approach
As known [14], for quantum systems of finitely many particles there
is an equivalent approach to the description of the evolution of states
16 V. I. Gerasimenko, V. V. Krechko
within the framework of sequences of the density operators governed
by the the groups of operators (1). To construct series expansions (6)
for a non-perturbative solution of the quantum BBGKY hierarchy,
using such an approach, we introduce the generating functional of a
sequence of marginal density operators [15], [16]
(F (t), u)
.
=
∞∑
n=0
1
n!
Tr1,...,n Fn(t, 1, . . . , n)
n∏
i=1
u(i) = (14)
∞∑
n=0
1
n!
∫
Fn(t, ξ1, . . . , ξn; ξ′1, . . . , ξ
′
n)×
n∏
i=1
u(ξ′i)
n∏
j=1
u∗(ξj)dξ
′
1 . . . dξ
′
ndξ1 . . . dξn,
where u = (I, u(1), . . . ,
∏n
i=1 u(i), . . .) is a sequence of the products
of the degenerate operators {u(i)}i≥1 with infinitely differentiable
kernels with compact supports. We refer to functional (14) as the
generating functional of marginal density operators Fn(t, 1, . . . , n) ∈
L1(Hn), n ≥ 1, by reason of the validity for their kernels of the
equalities
Fn(t, ξ1, . . . , ξn; ξ′1, . . . , ξ
′
n) = (15)
δ2n
δu(ξ1) . . . δu(ξn)δu∗(ξ′1) . . . δu
∗(ξ′n)
(F (t), u) |u=u∗=0,
where δ2n/δu(ξ1) . . . δu(ξn)δu∗(ξ′1) . . . δu
∗(ξ′n) is the 2nth order
functional derivative (the Gâteaux derivative [16]).
The generating functional of marginal density operators is defined
within the framework of a sequence of the groups of operators (1) of
the von Neuman equations for density operators [14] by the equality
(F (t), u) = (G(−t)D(0), I)−1(G(−t)D(0), u+ 1), (16)
where G(−t)D(0) = (I,G1(−t)D0
1, . . . ,Gn(−t)D0
n, . . .) is a sequence
of the density operators D(0) = (I,D0
1, . . . , D
0
n, . . .) ∈ L1
α(FH) and
On non-perturbative solution of quantum BBGKY hierarchy 17
(G(−t)D(0), I)
.
=
∑∞
n=0
1
n!Tr1,...,n Gn(−t)D0
n is a normalizing factor
(grand canonical partition function).
To determine a sequence of operators generated by the functional
(G(−t)D(0), I)−1(G(−t)D(0), u+ 1) we transform this functional to
canonical form (14).
Proposition 2. The equality is true
(G(−t)D(0), u+ 1) = (eaG(−t)D(0), u), (17)
where the operator a (an analog of the annihilation operator) is defi-
ned by the formula
(aD(0))n(1, . . . , n)
.
= Trn+1D
0
n+1(1, . . . , n, n+ 1). (18)
Proof. Indeed, according to definition (18), the following equalities
take place
(G(−t)D(0), u+ 1) =
∞∑
n=0
1
n!
Tr1,...,n Gn(−t)D0
n
n∏
i=1
(u(i) + 1) =
∞∑
n=0
1
n!
Tr1,...,n Gn(−t)D0
n
n∑
k=0
n∑
i1<...<ik=1
u(i1) . . . u(ik) =
∞∑
s=0
1
s!
∞∑
n=0
1
n!
Tr1,...,s+n Gs+n(−t)D0
s+n
s∏
i=1
u(i) = (eaG(−t)D(0), u).
2
Hence, in view of definition (15), from equalities (16) and (24)
we derive the series expansion for marginal density operators within
the framework of nonequilibrium grand canonical ensemble [19]
Fs(t, 1, . . . , s) = (G(−t)D(0), I)−1
∞∑
n=0
1
n!
Tr1,...,s+n Gs+n(−t)D0
s+n.
On the basis of relationship (16) for generating functionals, we
construct a non-perturbative solution of the Cauchy problem of the
quantum BBGKY hierarchy (4),(5).
18 V. I. Gerasimenko, V. V. Krechko
In the functional (eaG(−t)D(0), u) we expand operators (1) over
their cumulants as the following cluster expansions
Gs+n(−t, Y, X \ Y ) =
∑
P: ({Y }, X\Y )=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi), (19)
where as above Y ≡ (1, . . . , s), ({Y }) is the set consisting of one
element Y = (1, . . . , s), X ≡ (1, . . . , s+ n), and
∑
P:({Y }, X\Y )=
⋃
iXi
is the sum over all possible partitions P of the set ({Y }, X \Y ) into
|P| nonempty mutually disjoint subsets Xi ⊂ ({Y }, X \ Y ).
Owing to the equality∑
P: ({Y }, X\Y )=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi) =
∑
Z⊂X\Y
A1+|Z|(−t, {Y }, Z)
∑
P:X\Y \Z=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi),
and, according to the symmetry property of the integrand, the vali-
dity of the following equality∑
Z⊂X\Y
A1+|Z|(−t, {Y }, Z)
∑
P:X\Y \Z=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi) =
n∑
k=0
n∑
i1<...<ik=1
A1+k(−t, {Y }, s+ 1, . . . , s+ k)×
∑
P: (s+k+1,...,s+n)=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi),
as a result we obtain
(eaG(−t)D(0), u) = (20)
∞∑
s=0
1
s!
∞∑
n=0
1
n!
Tr1,...,s+n+k A1+|X\Y |(−t, {Y }, X \ Y )×
∞∑
k=0
1
k!
∑
P: (s+n+1,...,s+n+k)=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi)D
0
s+n+k
s∏
i=1
u(i),
On non-perturbative solution of quantum BBGKY hierarchy 19
where the generating operators A1+n(−t), n ≥ 0, of series (20) are
solutions of cluster expansions (19).
According to the validity of the equality [19]
Trs+n+1,...,s+n+k
∑
P:Z=
⋃
iXi
∏
Xi⊂P
A|Xi|(−t,Xi)D
0
s+n+k =
Trs+n+1,...,s+n+kD
0
s+n+k,
where Z ≡ (s + n + 1, . . . , s + n + k), and a similar equality for
the normalizing factor: (G(−t)D(0), I) = (D(0), I), and, taking into
account the definition of initial marginal density operators, from
relation (16) and representation (20) we derive series expansion (6).
In fact, there is such a criterion. Series expansion (6) is a solution
of the Cauchy problem of the quantum BBGKY hierarchy (4),(5)
if and only if its generating operators A1+n(−t), n ≥ 0, satisfy
recurrence relations (19).
4. An approach based on dynamics of correla-
tions
In addition to an approach to the description of the evolution of
states of quantum many-particle systems within the framework of
a sequence of the groups of operators (1) one more an equivalent
approach is given by means of the groups of nonlinear operators of
the von Neuman hierarchy for correlation operators [17].
The generating functional (g(0), u) of a sequence of the correlati-
on operators g0s , s ≥ 1, is defined by means of the generating functi-
onal of the density operators as follows [17]
(D(0), u) = e(g(0),u),
i.e., the correlation operators are determined by cluster expansions
of the density operators.
20 V. I. Gerasimenko, V. V. Krechko
Then the following equality holds
(G(−t)D(0), u) = e(G(t|g(0)),u), (21)
where the sequence of correlation operators is determined by the
following expansions:
G(t;Y | g(0))
.
= (22)∑
P:Y=
⋃
j Xj
A|P|(t, {X1}, . . . , {X|P|})
∏
Xj⊂P
g0|Xj |(Xj), s ≥ 1,
and we used notations introduced above.
Thus, according to relationships (16) and (21), the generating
functional of marginal density operators is determined by means of
generating functional of correlation operators as follows
(F (t), u) = e(G(t|g(0)),u+1)−(G(t|g(0)),I). (23)
To determine a sequence of operators generated by the functional
e(G(t|g(0)),u+1)−(G(t|g(0)),I) we transform this functional to canonical
form (14).
Proposition 3. The equality is true
e(G(t|g(0)),u+1)−(G(t|g(0)),I) = (24)
(Exp∗G(t | g(0)), I)−1(eaExp∗G(t | g(0)), u),
where the mapping Exp∗ is defined by the formula
(Exp∗ f)|Y |(Y ) = 1δ|Y |,0 +
∑
P:Y=
⋃
iXi
∏
Xi⊂P
f|Xi|(Xi), (25)
and the notations accepted above are used, δ|Y |,0 is the Kronecker
symbol.
On non-perturbative solution of quantum BBGKY hierarchy 21
Proof. On sequences of operators f, f̃ ∈ L1
α(FH) we define the ∗-
product
(f ∗ f̃)|Y |(Y ) =
∑
Z⊂Y
f|Z|(Z) f̃|Y \Z|(Y \ Z),
where
∑
Z⊂Y is the sum over all subsets Z of the set Y ≡ (1, . . . , s).
By means of this definition on sequences f = (0, f1, . . . , fn, . . .) we
introduce mapping (25) by the expansions
(Exp∗ f)|Y |(Y ) =
(
I +
∞∑
n=1
1
n!
f∗n
)
|Y |(Y ) =
= 1δ|Y |,0 +
∑
P:Y=
⋃
iXi
∏
Xi⊂P
f|Xi|(Xi),
where we use the notations accepted above.
Then, observing the validity of the equality
(f ∗ f̃ , u) = (f, u)(f̃ , u), (26)
in view of definition (25) we justify equality (24), i.e.
e(G(t|g(0)),u) = (Exp∗G(t | g(0)), u),
and as a result the following equality holds
e(G(t|g(0)),u+1) = (eaExp∗G(t | g(0)), u).
2
To write down the sequence (Exp∗G(t | g(0)), I)−1(eaExp∗G(t |
g(0)), u) in the component-wise form we introduce the following
mappings:
(dY f)n
.
= f|Y |+n(Y, s+ 1, . . . , s+ n),
(d{Y }f)n
.
= f1+n({Y }, s+ 1, . . . , s+ n), n ≥ 0.
22 V. I. Gerasimenko, V. V. Krechko
Then we have
(eaExp∗G(t | g(0)))s(Y ) = (dY Exp∗G(t | g(0)), I).
Owing to the validity of the following equalities [14]:
dY Exp∗G(t | g(0)) = d{Y }Exp∗G(t | g(0)),
d{Y }Exp∗G(t | g(0)) = Exp∗g(t) ∗ d{Y }G(t | g(0)),
and according to equality (26), we finally derive
(Exp∗G(t | g(0)), I)−1(dY Exp∗G(t | g(0)), I) = (d{Y }G(t | g(0)), I).
>From this representation we obtain the series expansion for
marginal density operators by means of the correlation operators
Fs(t, Y ) = (27)
∞∑
n=0
1
n!
Tr1,...,s+n G(t, {Y }, s+ 1, . . . , s+ n | g(0)), s ≥ 1,
where the set, consisting from one element Y = (1, . . . , s), we
denoted by {Y } and the correlation operators G(t, {Y }, s+1, . . . , s+
n | g(0)), n ≥ 0, are defined by expansions (22).
We remark that, according to the estimate
Tr1,...,n
∣∣G(t, 1, . . . , n | g(0))
∣∣ ≤ n!e2ncn,
where c ≡ maxP:Y=
⋃
iXi
(TrXi |g0|Xi|(Xi)|), series (27) exists and the
following inequality holds:
Tr1,...,s
∣∣Fs(t, 1, . . . , s)∣∣ ≤ e3c ∞∑
n=0
e3ncn.
On the basis of relationship (23) for generating functionals,
i.e. representation (27), we derive the series expansion for a non-
perturbative solution of the Cauchy problem of the quantum
BBGKY hierarchy (4),(5).
On non-perturbative solution of quantum BBGKY hierarchy 23
The following equality holds:
Trs+1,...,s+n
∑
P : ({Y }, X \ Y ) =
⋃
iXi
A|P|(−t, {θ(X1)}, . . . ,
{θ(X|P|)})
∏
Xi ⊂ P
g0|Xi|(Xi) =
Trs+1,...,s+n
∑
Z ⊂ X \ Y
A1+|Z|(−t, {Y }, Z)g01+|X\Y \Z|({Y,Z},
X \ Y \ Z),
Taking into account this equality, for series expansion (27),(22)
we successively derive
Fs(t, Y ) =
∞∑
n=0
1
n!
Trs+1,...,s+n
∑
Z ⊆ X \ Y
A1+|Z|(−t, {Y }, Z)×
g01+|X\Y \Z|({Y,Z}, X \ Y \ Z) =
∞∑
n=0
1
n!
Trs+1,...,s+n
n∑
k=0
n!
k!(n− k)!
A1+n(−t, {Y }, s+ 1, . . . ,
s+ k)g01+n−k({Y, s+ 1, . . . , s+ k}, s+ k + 1, . . . , s+ n) =
∞∑
n=0
1
n!
∞∑
k=0
1
k!
Trs+1,...,s+n+k A1+n(−t, {Y }, X \ Y )×
g01+k({X}, s+ n+ 1, . . . , s+ n+ k).
According to definition (27) at initial instant, i.e.
F 0
s+n(X) =
∞∑
k=0
1
k!
Trs+n+1,...,s+n+k g
0
1+n+k({X}, s+ n+ 1, . . . , s+ n+ k),
we finally derive the series expansions (6) for marginal density
operators with generating operators which are corresponding-order
cumulant (7) of groups of operators (1) of a system of finitely many
quantum particles.
24 V. I. Gerasimenko, V. V. Krechko
5. Conclusion
In the paper we developed two approaches to the construction of
a non-perturbative solution of the Cauchy problem of the quantum
BBGKY hierarchy for the marginal density operators and for ini-
tial data from the space of sequences of trace class operators its
properties were analyzed.
It was established that a non-perturbative solution of the Cauchy
problem of the quantum BBGKY hierarchy (4),(5) for a sequence
of marginal density operators is represented in the form of series
expansion (6) over particle subsystems which generating operators
are corresponding-order cumulant (7) of the groups of operators (1)
of finitely many quantum particles. One of the advantages of such
a representation of the solution is an opportunity to construct the
quantum kinetic equations, in particular, kinetic equations for large
particle systems in condensed states [14].
We also emphasize that the natural Banach spaces for the descri-
ption of states of large particle quantum systems, for instance,
containing equilibrium states [10], are different from the used Banach
space of sequences of trace class operators [14].
This paper deals with a quantum system of a non-fixed, i.e.,
arbitrary but finite, number of identical (spinless) particles obeying
Maxwell–Boltzmann statistics. The obtained results can be extended
to large particle quantum systems of bosons and fermions [18]. In
these cases corresponding series expansions have the same structure,
as in case of the Maxwell–Boltzmann statistics [14] which caused by
the fact that symmetrization and anti-symmetrization operators are
integrals of motion.
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26 V. I. Gerasimenko, V. V. Krechko
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|
| id | oai:trim.imath.kiev.ua:article-250 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:04:21Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/46/14fd6df3abe2e95a1476d21aea9db946.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2502018-01-23T09:13:23Z On non-perturbative solution of quantum BBGKY hierarchy Gerasimenko, V. I. Krechko, V. V. Gerasimenko, V. I. Krechko, V. V. We consider some approaches to the construction of a non-perturbative solution of the Cauchy problem of the quantum BBGKY hierarchy for a sequence of marginal density operators and analyze its properties for initial data from the space of sequences of trace class operators. One is represented in the form of a series expansion over particle subsystems which generating operators are the corresponding-order cumulants of the groups of operators of systems of finitely many quantum particles. Розглянуто підходи до побудови непертурбативного розв'язку задачі Коші для ієрархії квантових рівнянь ББҐКІ для послідовності маргінальних операторів густини і досліджено його властивості у випадку початкових даних із простору послідовностей ядерних операторів. Такий розв'язок зображується розкладом у ряд за підсистемами частинок, твірні оператори якого є відповідного порядку кумулянтами груп операторів систем скінченої кількості квантових частинок. Інститут математики НАН України 2016-08-22 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/250 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 13 No. 2 (2016): Differential equations and related problems of analysis; 7-26 Сборник Трудов Института математики НАН Украины; Том 13 № 2 (2016): Диференціальні рівняння і суміжні питання аналізу; 7-26 Збірник Праць Інституту математики НАН України; Том 13 № 2 (2016): Диференціальні рівняння і суміжні питання аналізу; 7-26 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/250/225 Авторське право (c) 2016 V. I. Gerasimenko, V. V. Krechko |
| spellingShingle | Gerasimenko, V. I. Krechko, V. V. Gerasimenko, V. I. Krechko, V. V. On non-perturbative solution of quantum BBGKY hierarchy |
| title | On non-perturbative solution of quantum BBGKY hierarchy |
| title_full | On non-perturbative solution of quantum BBGKY hierarchy |
| title_fullStr | On non-perturbative solution of quantum BBGKY hierarchy |
| title_full_unstemmed | On non-perturbative solution of quantum BBGKY hierarchy |
| title_short | On non-perturbative solution of quantum BBGKY hierarchy |
| title_sort | on non-perturbative solution of quantum bbgky hierarchy |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/250 |
| work_keys_str_mv | AT gerasimenkovi onnonperturbativesolutionofquantumbbgkyhierarchy AT krechkovv onnonperturbativesolutionofquantumbbgkyhierarchy AT gerasimenkovi onnonperturbativesolutionofquantumbbgkyhierarchy AT krechkovv onnonperturbativesolutionofquantumbbgkyhierarchy |