On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials
We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2010
|
| Назва видання: | Український математичний журнал |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/166294 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials / W.I. Skrypnik // Український математичний журнал. — 2010. — Т. 62, № 12. — С. 1687–1704. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-166294 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1662942020-02-19T01:28:32Z On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials Skrypnik, W.I. Статті We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials. Розглядається ґраткове рівняння Кірквуда-Зальцбурга (КС) осциляторного типу з загальним фазовим одночастинковим вимірним простором та багаточастинковими потенціалами взаємодії. При певному виборі цього вимірного простору розв'язки КС рівняння описують кореляційні функції великого канонічного ансамблю ґраткових рівноважних класичних та квантових систем осциляторів. Доведено існування розв'язку симетризованого КС рівняння для багаточастинкових потенціалів взаємодії, які або притягувальні (недодатні) та мають скінченну дію, або відштовхувальні (додатні) та мають нескінченну дію. Розглядувана симетризація нова і ґрунтується на умові суперстійкості для багаточастинкових потенціалів. 2010 Article On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials / W.I. Skrypnik // Український математичний журнал. — 2010. — Т. 62, № 12. — С. 1687–1704. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166294 517.9 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Skrypnik, W.I. On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials Український математичний журнал |
| description |
We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials. |
| format |
Article |
| author |
Skrypnik, W.I. |
| author_facet |
Skrypnik, W.I. |
| author_sort |
Skrypnik, W.I. |
| title |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials |
| title_short |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials |
| title_full |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials |
| title_fullStr |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials |
| title_full_unstemmed |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials |
| title_sort |
on the lattice oscillator-type kirkwood–salsburg equation with attractive many-body potentials |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166294 |
| citation_txt |
On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials / W.I. Skrypnik // Український математичний журнал. — 2010. — Т. 62, № 12. — С. 1687–1704. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT skrypnikwi onthelatticeoscillatortypekirkwoodsalsburgequationwithattractivemanybodypotentials |
| first_indexed |
2025-07-14T21:07:30Z |
| last_indexed |
2025-07-14T21:07:30Z |
| _version_ |
1837658016017547264 |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG
EQUATION WITH ATTRACTIVE MANYBODY POTENTIALS
ПРО ҐРАТКОВЕ РIВНЯННЯ КIРКВУДА – ЗАЛЬЦБУРГА
ОСЦИЛЯТОРНОГО ТИПУ З ПРИТЯГУВАЛЬНИМИ
БАГАТОЧАСТИНКОВИМИ ПОТЕНЦIАЛАМИ
A lattice oscillator-type Kirkwood – Salsburg (KS) equation with a general one-body phase measurable space
and manybody interaction potentials is considered. For special choices of the measurable space, its soluti-
ons describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator
systems. We prove the existence of a solution of the symmetrized KS equation for manybody interaction
potentials which are either attractive (non-positive) and finite-range or infinite-range and repulsive (positive).
The considered symmetrization of the KS equation is new and is based on the superstability of manybody
potentials.
Розглядається ґраткове рiвняння Кiрквуда – Зальцбурга (КС) осциляторного типу з загальним фазовим
одночастинковим вимiрним простором та багаточастинковими потенцiалами взаємодiї. При певному
виборi цього вимiрного простору розв’язки КС рiвняння описують кореляцiйнi функцiї великого ка-
нонiчного ансамблю ґраткових рiвноважних класичних та квантових систем осциляторiв. Доведено
iснування розв’язку симетризованого КС рiвняння для багаточастинкових потенцiалiв взаємодiї, якi або
притягувальнi (недодатнi) та мають скiнченну дiю, або вiдштовхувальнi (додатнi) та мають нескiнчен-
ну дiю. Розглядувана симетризацiя нова i ґрунтується на умовi суперстiйкостi для багаточастинкових
потенцiалiв.
1. Introduction and main result. In this paper we consider the oscillator-type lattice
Kirkwood – Salsburg (KS) equation with the one-body phase measurable space (Ω, P 0),
an interaction potential energy U(ωX), X ⊂ Zd and an external potential u(ω), where
ωX = (ωx ∈ Ω, x ∈ X). It is an resolvent-type equation satisfied by a sequence
of correlation functions ρ = {ρ(ωX), X ∈ Zd} and may describe grand canonical
classical and quantum oscillator systems with a potential energy generated by a pair and
manybody potentials uY (ωY ), Y ⊂ Zd (|Y | = 2 corresponds to a pair potential, |Y | is
a number of sites in Y ), that is
U(ωX) =
∑
|Y |≥2,Y⊆X
uY (ωY ).
The potential energy is an unbounded function and P 0(Ω) = ∞ for oscillator-type or
abstract unbounded spin systems. The space Ω can be considered as a metric space (σ-
algebra is associated with Borel sets), which is a discrete union of finite balls, and the
measure P 0 is finite on them.
The KS equation is written as follows
ρ = zKρ+ zα,
where α(ωX) = δ|X|,1, δk,l is the Kroneker symbol. The KS operator K in its turn is
given by
(KF )(ωX) =
∑
Z⊆Xc
∫
K(ωx|X\x;ωZ)
[
F (ωX\x∪Z)−
∫
P (dωx)F (ωX∪Z)
]
P (dωZ),
c© W. I. SKRYPNIK, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12 1687
1688 W. I. SKRYPNIK
where the integrations are performed over the cartesian |Y |-fold product Ω|Y | of the
measurable space and Ω, for X = x the first term in the square bracket corresponding to
Z = ∅ is equal to zero and P (dωY ) =
∏
y∈Y
e−βu(ωx)P 0(dωy). The KS kernels are
connected with the potential energy U(ωX) in the following way (x ∈ X,X ∩ Y = ∅)
e−βW (ωx|ωX\x∪Y ) =
∑
S⊆Y
K(ωx|ωX\x;ωS), W (ωx|ωY ) = U(ωY ∪x)− U(ωY ).
(1.1)
The expression for the kernel K will be derived in the beginning of the next section and
is given by (2.2) or (2.3). It has the same structure as the KS kernel for particle systems
[1, 2].
A derivation of the KS equation with manybody potentials is very close to its deriva-
tion in the case of lattice gas proposed in [3]. We give it in the Appendix starting from
the expression of the grand canonical correlation functions in a compact set which is
enlarged to the whole Zd. That is, the KS equation is related to the correlation functions
in the thermodynamic limit.
Classical lattice oscillator systems are described by ω = q ∈ R = Ω, P 0(dq) = dq
and quantum lattice oscillator systems by ω = w ∈ Ω, P 0(dw) = dqP βq,q(dw) and
u(w) = β−1
β∫
0
u(w(τ))dτ, U(wX) = β−1
β∫
0
U(wX(τ))dτ, wX(0) = qX ,
where Ω is the space of all continuous paths, P βq,q(dw) is the conditional Wiener mea-
sure concentrated on continuous paths, starting from q and arriving into q at a “time”
β (see [4, 5]). This Wiener measure is generated by the probability transition den-
sity P t0(q − q′) = (4πt)−1/2 exp
{
− (q − q′)2
4t
}
which coincides with the kernel of
exp{t∂2}, where ∂ is the operator of differentiation in the oscillator variable q. Ω can
be represented as the Cartesian product of R and the space Ω0 of continuous paths start-
ing from the origin due to the translation invariance (with respect to the starting point)
of the conditional Wiener measure (see the remark in the end of the paper). We assume
that the mass of an oscillator is equal to
1
2
and the Plank constant is equal to the unity.
The lattice oscillator-type (or unbounded spin) KS equation is not well known even
for pair interaction contrary to the case of the particle KS equation considered by Ruelle
and Ginibre in [3, 5]. Classical and quantum systems of oscillators with pair interaction
were usually considered in the canonical ensemble (see [6 – 9]). A short-range ternary
interaction between quantum oscillators was considered in [10] also in the canonical
ensemble. The lattice KS equation for unbounded spins appeared earlier for the integer-
spin Ising systems with pair interaction in [11] and systems of classical and quantum
oscillators with finite-range positive (repulsive) manybody potentials in [12, 13].
It is known [1, 3] that to solve the particle KS equation at low activities one needs
to symmetrize it with respect to the stability condition if short-range pair potentials are
not positive. In this paper we show that in order to solve the lattice oscillator-type KS
equation one needs to symmetrize it with respect to a super-stability condition introduced
in [14] for classical lattice oscillator systems. In a general case it can be formulated as
follows: there exists a non-negative function v on Ω such that
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1689
|uY (ωY )| ≤ JY
∑
y∈Y
v(ωy), N0 =
∫
eβγv
1+ζ(ω)P (dω) <∞, (1.2)
where the constants β, ζ are non-negative, the constant γ is positive, ‖J‖1 =
= max
x
∑
Y,x∈Y
JY < ∞ and the summation is performed over subsets of Zd con-
taining a site x. In the case of positive many-body finite-range potentials we used the
symmetrization with respect to the superstability condition for a pair potential in [12].
The idea of the symmetrization was used in [11] in a special way. For finite-range
manybody potentials we will be able to put ζ = 0 if∣∣ux,y(ωx, ωy)
∣∣ ≤ Jx−y√v(ωx)v(ωx). (1.3)
Such the condition was postulated by Kunz for proving of a convergence of a polymer
cluster expansion for gibbsian canonical correlation functions of a lattice system of os-
cillators interacting via a pair potential ux,y. He employed this condition for an estimate
of the cluster functions, satisfying the KS recursion relation, in a way reminiscent of its
symmetrization.
We will consider the following four cases: (A) finite-range potentials; (B1) infinite-
range positive potentials; (B2) finite-range manybody potentials and infinite-range pair
potentials; (C) ζ = 0, finite-range manybody potentials and infinite-range pair potentials
satisfying (1.3). The range of the potentials will be denoted by R.
We will find solutions of the KS equation for positive finite-range potentials and
symmetrized KS equation in other cases in the Banach spaces Eξ, Eξ,f (Eξ = Eξ,0),
respectively. Eξ,f is the linear space of sequences of measurable functions with the norm
‖F‖ξ,f = max
X
ξ−|X| ess sup
wX
exp
{
−
∑
x∈X
f(ωx)
}
|FX(ωX)|, f(ω) = γβv1+ζ(ω).
We will use the following notations: P ′(dω) = ef(ω)P (dω), N ′0 = N−1
0 Ñ0, Ñ0 =
=
∫
v(ω)P ′(dω), N1 =
∫
eβc0vP ′(dω), N2 =
∫
v(ω)eβ‖J2‖1v(ω)P ′(dω); c0 = ‖J‖1
(c0 = ‖J2‖1) for general (positive finite-range manybody) potentials, the norm ‖J‖1
will be denoted by ‖J2‖1 if manybody potentials are zero and
|J |l = max
x
|J |l(x), |J |l(x) =
∑
Z⊆(x)c
Jx∪Zσ
|Z|(|Z|+ 1)l−1, l ≥ 1, σ ≥ 1,
where the summation is performed over Zd\x. For the symmetrization we employ the
function
W ′(x|ωX) =
∑
y∈X\x
v(ωy)
∑
Y⊆(x∪y)c
Jx∪y∪Y σ
|Y |, σ ≥ 1.
We have the following inequality:∑
x∈X
W ′(x|ωX) =
∑
x∈X
v(ωx)
∑
y∈X\x
∑
Y⊆(x∪y)c
Jy∪x∪Y σ
|Y | ≤
≤
∑
x∈X
v(ωx)
∑
y∈(x)c
∑
Y⊆(x∪y)c
Jy∪x∪Y σ
|Y | ≤ |J |1
∑
x∈X
v(ωx). (1.4)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1690 W. I. SKRYPNIK
It makes possible to symmetrize the KS equation with its respect with the help of the
related inequality
W ′(x|ωX) ≤ |J |1v(ωx) (1.5)
in the following way. Let χx(ωX) be the characteristic(indication) function of the set
Dx where this inequality holds. Then (1.4) implies that ∪x∈XDx = Ω|X| or∑
x∈X
χx(ωX) ≥ 1
since Dx may intersect for different x. It is more convenient to deal with χ∗x
χ∗x(ωX) =
∑
y∈X
χy(ωX)
−1
χx(ωX),
∑
x∈X
χ∗x(ωX) = 1. (1.6)
The symmetrized KS operator K̃ is given by
(K̃F )(ωX) =
∑
x∈X
χ∗x(ωX)
∑
Z⊆Xc
∫
K(ωx|X\x;ωZ)×
×
[
F (ωX\x∪Z)−
∫
P (dωx)F (ωX∪Z)
]
P (dωZ)
where for X = x the first term in the square bracket corresponding to Z = ∅ is equal
to zero. The symmetrized KS equation
ρ = zK̃ρ+ zα
is derived after multiplying both sides of the KS equation for fixed X,x by the charac-
teristic functions χ∗x(ωX) and applying (1.6). For all the cases except B1 one can put
σ = 1. Our main result is formulated in the following theorem.
Theorem 1.1. Let either ζ > 0 or ζ = 0 and γ − c0 − |J |1 ≥ 0, γ − ξ(3|J |1 +
+ ‖J‖1) ≥ 0, γ − c0 − |J |1 − ξN1‖J2‖1 ≥ 0, γ > |J |1 for the four cases A, B1, B2, C,
respectively. Moreover, let σ ≥ 1 + N0 in the case B1. Then there exists a continuous
positive function G(ξ) such that for the norm of the symmetrized KS operator in the
Banach space Eξ,f the following bound holds ‖K̃‖ξ,f ≤ (ξ−1 + N0)eG(ξ) and the
vector ρ from the space Eξ,f
ρ =
∑
n≥0
zn+1K̃nα (1.7)
determines the unique solution of the symmetrized KS equations in Ef,ξ, respectively,
if |z| < ‖K̃‖−1
ξ,f . If the potentials are positive and finite-range then these conclusions
hold for f = 0, the KS operator and the solution of the KS equation if K is substituted
instead of K̃ in the right-hand side of (1.7).
This theorem will be proven with the help of our basic bound for the function
K̄x(ωX) =
∑
Y⊆Xc
ξ|Y |
∫
|K(ωx|ωX\x;ωY )P ′(dωY ),
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1691
P ′(dωY ) = exp
{∑
x∈Y
f(ωx)
}
P (dωY ).
For the norm of the KS operator we have the following inequality:
‖K̃‖ξ,f ≤ (ξ−1 +N0) max
X
ess sup
ωX
K̄(ωX |f),
K̄(ωX |f) =
∑
x∈X
χ∗x(ωX)e−f(ωx)K̄x(ωX).
(1.8)
We will show that N0 is divergent at zero β for classical and quantum oscillator systems
with mild restrictions on the potential energy (see the proof of Proposition 4.2). This
implies that the convergence radius of series in (1.7) shrinks to zero at zero β. This is a
result of the facts that U, u are unbounded functions and P 0 is an unbounded measure.
The results of the proposed paper generalize the results of our papers [12, 13].
The case corresponding to positive infinite-range potentials (B1) will be treated by
us separately starting from the second representation of the KS kernels. Our result for
this case may be considered as a nontrivial generalization of the Ruelle’s result in [3]
concerning the existence of a solution of the KS equation for lattice gas with many-
body interaction potentials. Remark that The lattice gas is equivalent to the Ising model,
i.e., the simplest lattice system of finite-valued spins. Oscillator systems are systems
with unbounded spins which are more complicated than the bounded spin systems. All
our results presented in this paper and previous ones in [12, 13] lead to the conclu-
sion: to have a solution in Eξ,f the lattice oscillator-type KS equation needs a special
symmetrization if potentials are not positive and finite-range.
Our paper is organized as follows. In the next section we write down two expressions
for the KS kernels and the basic bound in Theorem 2.1, which prove Theorem 1.1, and
comment on an optimal choice of ξ which trivializes the expression for G. In the third
section we prove the basic bound. In the fourth section we adapt our result for quantum
lattice oscillator systems and establish the character of dependence of G at the zero β.
2. KS kernels and the basic bound. The first representation for the KS kernels cab
be derived with the help of the purely algebraic relation
F (ωX) =
∑
S⊆X
∑
S′⊆S
(−1)|S\S
′|F (ωS′). (2.1)
It is derived from the simple equality
n = |X|,
∑
S∈X
(−1)|S| =
n∑
l=0
(−1)lCln = 0. Cln =
n!
l!(n− l)!
.
Indeed, let’s consider the coefficient before F (ωX\x) in the right-hand side of the pre-
vious equality for arbitrary x. It corresponds either for the case S = X or S = X\x
and S′ = X\x . The signs are different before F for these options and this coefficient is
equal to zero. Further one has to take S = X, S = X\x1, S = X\x2, S = X\x1 ∪ x2,
S′ = X\x1 ∪ x2 and check that the coefficient before F (ωX\x1∪x2
), i.e., the last equal-
ity for n = 2, is equal to zero. In the same fashion one has to calculate the coefficients
before F (ωX\x1∪x2...∪xn), corresponding for the choice S′ = X\x1 ∪ x2 ∪ . . . ∪ xn,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1692 W. I. SKRYPNIK
and check that it coincides with the above sum with the binomial coefficients. As a
result (1.1) follows from (2.1) with
K(ωx|ωX\x;ωY ) =
∑
S⊆Y
(−1)|Y \S|e−βW (ωx|ωX\x∪S), (2.2)
The second representation for the KS kernels is found from the standard arguments
whose analog in the case of the lattice gas can be seen in [3]. Let
W (ωX ;ωY |x) =
∑
x∈Z⊆X
uZ∪Y (ωZ∪Y ),
W̃ (ωX ;ωY |x) =
∑
x∈Z⊆X
∑
∅ 6=S⊆Y
uZ∪S(ωZ∪S) =
∑
∅ 6=S⊆Y
W (ωX ;ωS |x).
Then
W (ωx|ωX\x, ωY ) = W (ωx|ωX\x) + W̃ (ωX ;ωY |x)
and
e−βW̃ (ωX ;ωY |x) =
∏
∅ 6=S⊆Y
(
1 + (e−βW (ωX ;ωS |x) − 1)
)
=
∑
S⊆Y
Kx(ωX ;ωS),
Kx(ωX ;ω∅) = 1,
where
Kx(ωX ;ωY ) =
|Y |∑
n=1
∑
∪Yj=Y,Yj 6=∅
n∏
j=1
(
e−βW (ωX ;ωYj |x) − 1
)
.
As a result we obtain the second representation for the KS kernels
K(ωx|ωX\x;ωY ) = e−βW (ωx|ωX\x)Kx(ωX ;ωY ) (2.3)
which will be used only for positive infinite-range potentials.
Proposition 2.1. Let all the potentials be finite-range except the pair one and
have the range R. Then the following equality holds for X ∩ Y = ∅, x ∈ X
K(ωx|ωX\x;ωY ) =
∑
S′⊆Y
K(ωx|ωX\x;ωS′)χBx(R)(S
′)G(ωx|ωY \S′)χBcx(R)(Y \S′),
(2.4)
where Bx(R) is the hyper-ball with the radius R centered at x, Bcx(R) = Zd\Bx(R),
G(ωx|ωS) =
∑
S′⊆S
(−1)|S\S
′|e−βW2(ωx|ωS′ ) =
∏
y∈S
(e−βu(x,y)(ωx,ωy) − 1)
and W2(ωx|ωS′) =
∑
y∈S′
ux,y(ωx, ωy).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1693
Proof. The manybody potentials have the finite range R, that is for an arbitrary
x ∈ X, |X| > 2 the following quality holds uX(ωX) = 0, |x− x′| ≥ R, x′ ∈ X\x and
|x− x′| is the Euclidean distance between two lattice sites. We demand also
W (ωx|X\x, ωS) = W (ωx|ωX\x, ωS\S2
) +W2(ωx|ωS2
), (2.5)
where y 6∈ Bx(R) if y ∈ S2. Here one has to take also into account the equality
W2(ωx|ωS) = W2(ωx|ωS2
) +W2(ωx|ωS\S2
).
Let’s substitute the equality
1 =
∏
y∈Y
(χBcx(R)(y) + χBx(R)(y)) =
∑
S′⊆Y
χBx(R)(S
′)χBcx(R)(Y \S′)
into the expression for the KS kernel and apply (3.8). This results in∑
S⊆Y
(−1)|Y \S|e−βW (ωx|ωX\x,ωS)
∑
S′⊆Y
χBx(R)(S
′)χBcx(R)(Y \S′) =
=
∑
S′⊆Y
∑
S⊆Y
(−1)|Y \S|e−βW (ωx|ωX\x,ωS)χBx(R)(S
′)χBcx(R)(Y \S′) =
=
∑
S′⊆Y
∑
S2⊆Y \S′
∑
S1⊆S′
(−1)(|Y |−|S1|−|S2|)×
×e−β[W (ωx|ωX\x,ωS1 )+W2(x|ωS2 )]χBx(R)(S
′)χBcx(R)(Y \S′) =
=
∑
S′⊆Y
χBx(R)(S
′)χBcx(R)(Y \S′)
∑
S1⊆S′
(−1)(|S′|−|S1|)e−βW (ωx|ωX\x,ωS1 )×
×
∑
S2⊆Y \S′
(−1)(Y−|S′|−|S2|)e−βW2(ωx|ωS2 ).
Proposition 2.1 is proved.
Theorem 1.1 will be proven with the help of (1.5), (1.6) and the following theorem.
Theorem 2.1. The following inequality holds:
K̄x(ωX) ≤ exp{ξc1 + βc2v(ωx) + ξc3
√
βv(ωx) + βc4W
′(x|ωX)}, cj ≥ 0, (2.6)
where c2 = c3 = c4 = 0, c1 = 2|B0(R)|N0 hold for positive finite-range potentials;
c3 6= 0 except for C; c4 = 2ξ, c2 = ξ(|J |1 + ‖J‖1), for B1 and c4 = 1 for the rest
three cases. The constant c1 takes the following values in the four remaining cases,
respectively,
2|B0(R)|N1, βN ′0|J |2, 2|B0(R)|N1 +N2β‖J2‖1, 2|B0(R)|N1.
For the constant c2 the following expression is true c2 = c0 + ξc′, where c0 = c0, for
A, B2; c′ = 0 for A, C; c′ = N1‖J2‖1 for B2 and γ − |J |1 > c0 if γ − |J |1 > 0 for C.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1694 W. I. SKRYPNIK
For positive finite-range potentials Theorem 2.1 and (1.8) yield immediately that
G(ξ) = ξc1 in Theorem 1.1. That is, the KS operator is bounded in Eξ. The inequali-
ties (2.6) and (1.5), (1.6) yield the bound
K̄(ωX |f) ≤ ec1ξ ess sup
ω
exp
{
−f(ω) + β(c2 + c4|J |1)v(ω) + ξc3
√
βv(ω)
}
. (2.7)
This bound and (1.8) imply that the symmetrized KS operator is bounded in the Banach
space Eξ,f for ζ, γ > 0 and G in Theorem 1.1 is given by
G(ξ) = ξc1 + ζβγ−1/ζ
(
c5
1 + ζ
)(1+ζ)/ζ
, c5 = c2 + c4|J |1.
Here we used the formula max
v≥0
e−v
1+ζ+av = exp
{
ζ
(
a
1 + ζ
)(1+ζ)/ζ
}
. If ζ = 0 then
the condition γ − c2 − c4|J |1 ≥ 0 corresponds to the conditions stated in Theorem 1.1
for the three first cases and then
G(ξ) = ξc1, ζ = 0. (2.8)
For the fourth case C we have γ − c2 − c4|J |1 = γ − c0 − |J |1 and derive, taking
into account the last statement in Theorem 2.1 (c0 depends on γ), for γ > |J |1 (this
condition is required in Theorem 1.1 for ζ = 0)
G(ξ) = ξc1 + ξ2c6. (2.9)
The optimal choice of ξ will be such that G(ξ) is less than a constant or a function with
a simple dependence on β. Such a choice is obvious if G(ξ) is given either by (2.8)
or (2.9). Then either ξ = c−1
1 and G(ξ) = 1 or ξ =
(
c1 +
√
c6
)−1
and G(ξ) ≤ 2.
The same choice is possible for ζ > 0 and A with G proportional to β (c5 does not
depend on ξ). For B1, ζ > 0 one can put ξ =
(
c1 + 3|J |1 +‖J‖1
)−1
and obtain G(ξ) ≤
≤ 1 + ζβγ−1/ζ
(
1
1 + ζ
)(1+ζ)/ζ
. For B2, ζ > 0 one can put ξ =
(
N1‖J2‖1 + c1
)−1
and this means that G(ξ) ≤ 1 + ζβγ−1/ζ
(
c0 + |J |1
1 + ζ
)(1+ζ)/ζ
.
3. Proof of Theorem 2.1. We have the following inequalities which are analogs of
the inequalities for the KS kernels for the lattice gas systems from [1]
∑
Y⊆Xc
ξ|Y |
∫
|Kx(ωX ;ωY )|P ′(dωY ) ≤
≤
∑
Y⊆Xc
ξ|Y |
|Y |∑
l=1
∑
∪Yj=Y
l∏
j=1
∫
|e−βW (ωX ;ωYj |x) − 1|P ′(dωYj ) =
=
∑
n≥0
ξn
∑
|Y |=n,Y⊆Xc
n∑
l=1
∑
∪Yj=Y
l∏
j=1
∫
|e−βW (ωX ;ωYj |x) − 1|P ′(dωYj ) ≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1695
≤
∑
n≥0
ξn
n!
∑
Y⊆Xc
∫
|e−βW
(
ωX ;ωY |x
)
− 1|P ′(dωY )
n =
= exp
ξ ∑
Y⊆Xc
∫
|e−βW
(
ωX ;ωY |x
)
− 1|P ′(dωY )
.
Hence for positive potentials uY ≥ 0 one derives from (2.3) the following estimate:∑
Y⊆Xc
ξ|Y |
∫
|K(ωx|ωX\x;ωY )|P ′(dωY ) ≤
≤ exp
ξβ ∑
Y⊆Xc
∫
|W (ωX ;ωY |x)|P ′(dωY )
. (3.1)
For positive short-range potentials (case B1) we have to make estimates of W (ωx|ωX\x)
and the integral in (3.1) using (1.2)∣∣W (ωX ;ωY |x)
∣∣ ≤ ∑
x∈Z⊆X
|uZ∪Y (ωZ∪Y )| ≤
∑
x∈Z⊆X
JZ∪Y
∑
y⊆Z∪Y
v(ωy) =
=
∑
x∈Z⊆X
JZ∪Y
∑
y∈Y
v(ωy) +
∑
y∈Z\x
v(ωy) + v(ωx)
. (3.2)
The last inequality yields ∫
|W (ωX ;ωY |x)|P ′(dωY ) ≤
≤ N |Y |0
∑
Z⊆X\x
Jx∪Z∪Y
N ′0|Y |+ ∑
y∈Z\x
v(ωy) + v(ωx)
=
= N
|Y |
0
∑
Z⊆X\x
Jx∪Z∪Y
[
N ′0|Y |+ v(ωx)
]
+
+N
|Y |
0
∑
y∈X\x
v(ωy)
∑
Z⊆X\(x∪y)
Jy∪x∪Z∪Y .
Here we utilized the equality∑
Y⊆Λ
∑
y∈Y
F (Y ; y) =
∑
y∈Λ\X
∑
Y⊆Λ\y
F (Y ∪ y; y). (3.2′)
As a result∑
Y⊆Xc
∫
|W (ωX ;ωY | x)|P ′(dωY ) ≤ N ′0|J |2+|J |1v(ωx)+W ′(x | ωX), σ ≥ 1+N0.
(3.3)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1696 W. I. SKRYPNIK
We also obtain from (1.2) and (3.2′)∣∣W (ωx | ωX\x)
∣∣ ≤ ∑
x∈Z⊆X
|uZ(ωZ)| ≤
≤
∑
Z⊆X\x
JZ∪x
∑
y∈Z∪x
v(ωy) ≤W ′(ωx | ωX\x) + ‖J‖1v(ωx). (3.4)
The previous two inequalities prove Theorem 2.1 for the case B1. The proof of the
remaining part of Theorem 1.1 will be based on the inequality
K̄x(ωX) ≤ eβW
′(ωx|ωX)+c0v(ωx)
∏
y,y 6=x
(1 + ξK̄x,y), (3.5)
where he product is taken over Zd\x. If all the potentials have range R then
K(ωx | ωX\x;ωY ) = K(ωx | ωX\x;ωY )χBx(R)(Y ). (3.6)
Here one has to apply (2.5) with y = S2, W2(ωx|ωy) = 0 and check that the left-hand
side of the last equality is zero since the term with S = S′ in (2.2) has the opposite sign
to the term with S = S′\y. For positive finite-range potentials (3.6) and (2.2) result in
K(ωx | ωX\x;ωY ) ≤ 2|Y |χBx(R)(Y ),
K̄x(ωX) ≤
∑
Y,x 6∈Y
(2ξN0)|Y |χBx(R)(Y ) =
=
∏
y,x6=y
(1 + 2ξN0χBx(R)(y)) ≤ e2|B0(R)|ξN0 .
Hence Theorem 2.1 holds for positive finite-range manybody potentials potentials (the
simplest subcase of A).
For positive (infinite-range) many-body potentials we have∣∣K(ωx | ωX\x;ωY )
∣∣ ≤ eβ|W2(ωx|ωX\x)|+‖J2‖1v(ωx)
∏
y∈Y
(1 + eβ‖J2‖1v(ωy)). (3.6′)
It is a consequence of the inequality∣∣K(ωx | ωX\x;ωY )
∣∣ ≤ ∑
S⊆Y
e−βW2(ωx|ωX\x∪S) ≤
≤ e−βW2(ωx|ωX\x∪S)
∏
y∈Y
(1 + e−βux,y(ωx,ωy))
derived from the first representation (2.2) for the KS kernels and (1.2) for the pair
potential.
Now let the potentials be non-positive. From (2.2) it follows that∣∣K(ωx | ωX\x;ωY )
∣∣ ≤ ∑
S⊆Y
eβ|W (ωx|ωX\x,ωS)|. (3.7)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1697
Let’s estimate the function under the sign of the exponent of the last inequality starting
from the equality
W (ωx | ωX\x∪Y ) =
∑
Z⊆X∪Y
uZ(ωZ)−
∑
x 6∈Z⊆X∪Y
uZ(ωZ) =
∑
x∈Z⊆X∪Y
uZ(ωZ).
An employment of (1.2) and changing orders of summations yield∣∣W (ωx | ωX\x∪Y )
∣∣ ≤ ∑
x∈Z⊆X∪Y
|uZ(ωZ)| ≤
≤
∑
x∈Z⊆X∪Y
JZ
∑
y⊆Z
v(ωy) =
∑
Z⊆X∪Y \x
Jx∪Z
∑
y⊆Z∪x
v(ωy) =
=
∑
y∈Y ∪X\x
v(ωy)
∑
Z⊆X∪Y \(x∪y)
Jx∪y∪Z + v(ωx)
∑
Z⊆X∪Y \x
Jx∪Z =
=
∑
y∈X\x
v(ωy)
∑
Z⊆X∪Y \(x∪y)
Jx∪y∪Z +
∑
y∈Y
v(ωy)
∑
Z⊆X∪Y \(x∪y)
Jx∪y∪Z+
+v(ωx)
∑
Z⊆X∪Y \x
Jx∪Z .
For the third summand in the right-hand side of the last inequality we have
v(ωx)
∑
Z⊆X∪Y \x
Jx∪Z ≤ v(ωx)
∑
Z⊆(x)c
Jx∪Z ≤ v(ωx)‖J‖1.
For the second and first summands we have, respectively∑
y∈Y
v(ωy)
∑
Z⊆X∪Y \(x∪y)
Jx∪y∪Z ≤
∑
y∈Y
v(ωy)
∑
Z⊆(y∪x)c
Jx∪y∪Z ≤ ‖J‖1
∑
y∈Y
v(ωy),
∑
y∈X\x
v(ωy)
∑
Z⊆X∪Y \(x∪y)
Jx∪y∪Z ≤
∑
y∈X\x
v(ωy)
∑
Z⊆(x∪y)c
Jx∪y∪Z .
As a result∣∣W (ωx | ωX\x∪Y )
∣∣ ≤ ‖J‖1v(ωx) +W ′(ωx|ωX) + ‖J‖1
∑
y∈Y
v(ωy). (3.8)
From (3.7), (3.8) one deduces the analog of (3.6′)∣∣K(ωx | ωX\x;ωY )
∣∣ ≤ eβW ′(ωx|ωX)+‖J‖1v(ωx)
∏
y∈Y
(1 + eβ‖J‖1v(ωy)). (3.9)
So, let the potentials have the range R and be non-positive (case A). From (3.6) and
(3.9) it follows that Theorem 2.1 is true (the series in the expression for K̄x into a
product as in the formula before (3.6′)) since (3.5) is true with
K̄x,y = 2χBx(R)(y)N1.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1698 W. I. SKRYPNIK
Let the pair potential be infinite-range. Then taking absolute values of the summands
and dropping χBcx(R)(Y \S′) in (2.4) one deduce from it and (3.9) that∣∣K(ωx | ωX\x, ωY )
∣∣ ≤
≤ eβW
′(ωx|ωX)+‖J‖1v(ωx)
∏
y∈Y
[G(ωx|ωy) + χBx(R)(y)(1 + eβ‖J‖1v(ωy))]. (3.10)
From (3.6′) it follows that for positive manybody potentials the analogous inequality
holds with J2 substituted instead of J. The last inequality gives rise to (3.5) with
K̄x,y = Kx,y + 2χBx(R)(y)N1, (3.11)
where Kx,y =
∫
|e−βux,y(ωx,ωy) − 1|P ′(dωy). But using (1.2) for the pair potential
with Jx,y = Jx−y one obtains
Kx,y ≤ β
∫
|ux,y(ωx, ωy)|eβ|ux,y(ωx,ωy)|P ′(dωy) ≤
≤ βJx−y(N1v(wx) +N2)eβJx−yv(ωx). (3.12)
The last inequality and (3.5) show that Theorem 2.1 is true for B2. Here one has to use
the inequality 1 + ξβJx−y(N1v(wx) +N2) ≤ exp
{
ξβJx−y(N1v(wx) +N2)
}
.
Let the pair potential satisfy (1.3) (case C). This inequality implies
Kx,y ≤
∫
P ′(dω)(eβJx−y
√
v(ωx)
√
v(ω) − 1)dω.
For arbitrary a > 0 we have
Kx,y ≤
∫ (
eβJx−y
√
v(ωx)
√
v(ω) − e−βJx−y
√
v(ωx)
√
v(ω)
)
P ′(dω) =
= eβ(2a)−2J2
x−yv(ωx)
∫
eβa
2v(ω)
(
e−β(a
√
v(ω)−(2a)−1Jx−y
√
v(ωx))2−
−e−β(a
√
v(ω)+(2a)−1Jx−y
√
v(ωx))2
)
P ′(dω).
For the function in the round brackets we have the bound
(
b = (2a)−1Jx−y
√
v(ωx)
)
∣∣∣e−β(a
√
v(ω)−b)2 − e−β(a
√
v(ω)+b))2
∣∣∣ =
=
∣∣∣∣π−1/2
∫
e−k
2
e2ika
√
βv(ω)
(
e2ib
√
β − e−2ib
√
β
)∣∣∣∣ dk ≤
≤ π−1/2
∫
e−k
2
(∣∣e2ibk
√
β − 1
∣∣+
∣∣e−2ibk
√
β − 1
∣∣) dk ≤ π−1/28b
√
βκ0,
κ0 =
∫
e−k
2
|k|dk.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1699
Here we used the inequality |eiz − 1| ≤ 2|z|, z ∈ R. Let B0(a) =
∫
eβa
2v(ω)P ′(dω).
Then the last and previous bounds yield the generalized IN bound [11]
Kx,y ≤ 8κ0B0(a)(2a)−1eβ(2a)−2J2
x−yv(ωx)Jx−y
√
βv(ωx),
1 + ξ(Kx,y + 2χBx(R)(y)N1) ≤
≤ eβ(2a)−2J2
x−yv(ωx)+8κ0(2a)−1Jx−y
√
βv(ωx)B0(a)ξ+2χBx(R)(y)N1ξ.
(3.5), (3.11) and the last bound prove the basic bound (2.6) with
c2 = c0 = (2a)−2‖J2
2‖1, c3 = 8κ0(2a)−1‖J2‖1B0(a). (3.13)
We determine a from the equality
γ − |J |1 − c0 = γ − |J |1 − (2a)−2‖J2
2‖1 = θ(2a)−2, θ > 0,
and put B0(a) = N ′ for this choice. This means that for γ − |J |1 > 0
(2a)2 = (γ − |J |1)−1(θ + ‖J2
2‖1), c6 = θ−1(4κ0‖J2‖1N ′)2. (3.14)
If in addition manybody finite-range potentials are positive then all the above equalities
for cj are true if one substitutes ‖J2‖1 instead of ‖J‖1. Note that for classical systems,
the choice θ = β−s,
1
n
≤ s ≤ 1− n0
n
, where 2n, 2n0 are the degrees of the polynomials
u, v, respectively, allows one to prove that c6 tends to a finite limit (zero) if β tends to
zero
(
s >
1
n
)
.
4. Quantum systems. In this section we show that all the integrals in our theorems
are well defined for quantum lattice oscillator systems.
Proposition 4.1. Let (1.2), (1.3) hold for classical systems. Then (1.2), (1.3) hold
for quantum systems and the norms Nj , N ′0, N
′ are finite.
Proof. Obviously, the first inequality in (1.2) for quantum systems is satisfied with
βv(w) =
∫ β
0
v(w(τ))dτ. From the Schwartz inequality it follows that (1.3) holds for
quantum systems. From the Helder inequality
βv1+ζ(w) = β−ζ
β∫
0
v(w(τ))dτ
1+ζ
≤
β∫
0
v1+ζ(w(τ))dτ
and the Golden – Thompson inequality Tr eA+B ≤ Tr eAeB one derives the bound
N0 =
∫
dq
∫
e−βu(w)−f(w)P βq,q(dw) =
= Tr eβ[∂2−u+γv1+ζ ] ≤ (4πβ)−1/2
∫
e−β[u(q)−γv1+ζ(q)]dq. (4.1)
Here we took into account that eβ∂
2
(q, q) = (4πβ)−1/2. By the same arguments
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1700 W. I. SKRYPNIK
N1 ≤ (4πβ)−1/2
∫
e−β[u(q)−γv1+ζ(q)−c0v(q)]dq,
N2 ≤ (4πβ)−1/2
∫
e−β[u(q)−γv1+ζ(q)−c0v(q)]+v(q)dq,
(4.2)
B0(a) ≤ (4πβ)−1/2
∫
e−β[u(q)−(γ+a2)v(q)]dq, (4.3)
where we used the elementary inequality v(w) ≤ ev(w). The bound N0 > 0 is derived
from the bounds v ≥ 0, u(w(τ)) ≤ ūR, |w(τ)| ≤ R, 0 ≤ τ ≤ β, where ūR is a
constant. These bounds show that N0, N1 <∞, Ñ0 ≤ N2 <∞ and N ′0, N
′ <∞.
Proposition 4.1 is proved.
The reduced density matrices ρ(qX |q′X) are given by
ρ(qX |q′X) =
∫
e−β
∑
x∈X u(wx)ρ(wX)P βqX ,q′X
(dwX),
where the sequence
{
ρ(wX), X ⊂ Zd
}
is the solution of the (symmetrized) KS equation
determined in Theorem 1.1. The following bound is valid for the them
ρ(qX |q′X) ≤ ξ|X|‖ρ‖ξ,f
∏
x∈X
∫
e−βu(wx)+f(wx)P βqx,q′x(dwx) =
= ξ|X|‖ρ‖ξ,f
∏
x∈X
P β(qx; q′x),
where P β(qx; q′x) is the kernel of the semigroup whose infinitesimal generator is ∂2 −
− u+ γv1+ζ .
The following proposition clarifies a dependence of the above norms on β in a
neighborhood of the origin.
Proposition 4.2. Let η−q2n−η̄ ≤ u(q) ≤ η+q
2n+η̄, v(q) = q2n0 +1, 1+ζ <
n
n0
.
Let also θ = β−s and s ≤ 1− n0
n
. Then
Nj ≤ β−(1+n)/2nN̄j , N2 ≤ β−1/2−n0/n−1/2nN̄2,
N ′0 ≤ β−n0/nN̄ ′0, N ′ ≤ β−(1+n)/2nN̄ ′,
where j = 0, 1 and all the norms in the right-hand sides of the inequalities are finite on
a finite interval in β.
Proof. The proof is based on an application of the Helder inequality
β∫
0
w2n0(τ)dτ ≤ β1−n0/n
β∫
0
w2n(τ)dτ
n0/n
.
It and the bound |x|r ≤ c0re|x| give
N2 −N1 ≤ β−n0/n(4−1η−)−n0/nc0n0/n
∫
e−βũ(w)dqP βq,q(dw),
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1701
where ũ(w) = u(w)− c0v(w))− f(w)− η−
4
∫ β
0
w2n(τ)dτ. By rescaling the variables
by β−1/2n in the integrals in the bounds (4.1), (4.2) for N0, N1 one sees the inequalities
for N0, N1 in the proposition are true. They imply also the similar inequality for N2
after an application for the Golden – Thompson inequality. The inequality for N ′ follows
from (4.3) and the inequality (4.1) for N1.
To estimate accurately N ′0 on has to obtain an accurate bound from above for N0
starting from the inequalities
u(w) ≤ η′β−1
β∫
0
w2n(τ)dτ + a′, 0 < η′ < η+,
N0 ≥ e−βa
′
∫
dq
∫
exp
−η′
β∫
0
w2n(τ)dτ
P βq,q(dw),
where a′ is a constant. The first inequality is a result of the inequality ql ≤ εqk + ε−l,
k > l, q ≥ 0, ε ≤ 1. From the Jensen inequality it follows that
N0 ≥ e−βa
′
(4πβ)−1/2
∫
dq exp
−η′√4πβ
∫
P βq.q(dw)
β∫
0
w2n(τ)dτ
.
Here we took into account
∫
P βq,q(dw) = (4πβ)−1/2. Further
∫
P βq.q(dw)
β∫
0
w2n(τ)dτ =
β∫
0
∫
P τ0 (q − q′)q′2nP β−τ0 (q′ − q)dq′dτ ≤
≤ 22n
β∫
0
∫
P τ0 (q′)(q′2n + q2n)P β−τ0 (q′)dq′dτ = (4π)−1/2
√
β(2q)2n + c′n.
Here we used the inequality (q′ − q + q)2n ≤ 22n(q2n + (q − q′)2n), the semigroup
property of P t0(q) and the equality P t0(0) = (4πβ)−1/2. c′n does not depend on q and
is finite for a finite β since
c′n = 22n
β∫
0
∫
P τ0 (q′)q′2nP β−τ0 (q′)dq′dτ ≤
≤ 22n
β∫
0
(∫
(P τ0 (q′))2q′2ndq′
)1/2(∫
(P β−τ0 (q′))2q′2ndq′
)1/2
dτ =
= (4π)−122n
(∫
eq
2/2q2ndq
) β∫
0
[
τ(β − τ)
]n/2
dτ.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1702 W. I. SKRYPNIK
Here we applied the Schwartz inequality. A a result
N−1
0 ≤
√
4πβe
√
4πβc′n+βa′
(∫
dqe−η
′β(2q)2n
)−1
=
=
√
4πβ1/2+1/2ne
√
4πβc′n+βa′
(∫
dqe−η
′(2q)2n
)−1
.
Taking into account that Ñ0 ≤ N2, the bound for N2 and the last bound one derives
N ′0 ≤ β−n0/nN̄ ′0. This concludes the proof of the proposition.
Remark. For the choice P 0(dw) = dqPq(dw), where Pq(dw) is the conditional
Wiener measure, concentrated on continuous paths starting from q, solutions of the
KS equation may correspond to correlation functions of a stochastic dynamics of lattice
oscillators. The result of the proposed paper may be applied without difficulty for a proof
of an existence of solutions of the BBGKY-type hierarchy for the stochastic dynamics
of oscillators interacting via manybody potentials (in [15] only pair interaction was
considered). A scheme for a proof of the local convergence of the finite-volume grand-
canonical correlation functions to the solution of the KS equation can be found in [15].
5. Appendix. To derive the KS equation one has to start from the expressions for
the grand canonical correlation functions in a compact set Λ
ρΛ(ωX) = χΛ(X)Ξ−1
Λ
∑
Y⊆Λ\X
z|Y ∪X|
∫
P (dωY )e−βU(ωX∪Y ), (5.1)
where the grand partition function ΞΛ coincides with the numerator of the right-hand
side of (5.1) for empty set X. Substituting U(ωX∪Y ) = U(ωX∪Y \x)+W (ωx | ωX\x∪Y )
and the first equality in (1.1) into the expression of the finite volume grand canonical
correlation functions one obtains
ρΛ(ωX) =
= Ξ−1
Λ χΛ(X)
∑
Y⊆Λ\X
z|Y ∪X|
∫
P (dωY )e−βU(ωX∪Y \x)
∑
S⊆Y
K(ωx | ωX\x;ωS) =
= Ξ−1
Λ χΛ(X)
∑
Y⊆Λ\X
z|Y ∪X|
∑
S⊆Y
∫
P (dωY )K(ωx | ωX\x;ωS)e−βU(ωX∪Y \x) =
= z
∑
Z⊆Λ\X
∫
P (dωZ)K(ωx | ωX\x;ωZ)Ξ−1
Λ χΛ(X ∪ Z)×
×
∑
Y⊆Λ\(Z∪X)
z|Y ∪X∪Z|−1
∫
P (dωY )e−βU(ωX\x,ωY ).
The equality
ρΛ(ωX\x) = Ξ−1
Λ χΛ(X\x)
∑
Y⊆(Λ\X)∪x
z|Y ∪X|−1
∫
P (dωY )e−βU(ωX\x,ωY )
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
ON LATTICE OSCILLATOR-TYPE KIRKWOOD – SALSBURG EQUATION WITH ATTRACTIVE . . . 1703
leads to
Ξ−1
Λ χΛ(X ∪ Z)
∑
Y⊆Λ\(Z∪X)
z|Y ∪X∪Z|−1
∫
P (dωY )e−βU(ωX\x,ωY ) =
= χΛ(x)(ρΛ(ωX\x∪Z)−
∫
P (dωx)ρΛ(ωX∪Z)).
It’s clear that the terms with x ∈ Y in the sum, representing the first term in the round
brackets, are canceled by the same terms in the sum representing the second term in the
brackets. This concludes the derivation of the KS equation if one takes also into account
that ρΛ(ω∅) = 1. That, is the KS equation is given for x ∈ X, |X| > 1 by
ρΛ(ωX) = zχΛ(x)
∑
Z⊆Λ\X
∫
K(ωx|ωX\x;ωZ)[ρΛ(ωX\x∪Z)−
−
∫
P (dωx)ρΛ(ωX∪Z)]P (dωZ) (5.2)
and for X = x by
ρΛ(ωx) = zχΛ(x)
{
1−
∫
ρΛ(ωx)P (dωx)+
+
∑
|Z|≥1,Z⊆Λ\x
∫
K(ωx|ωZ)
[
ρΛ(ωZ)−
∫
P (dωx)ρΛ(ωZ∪x)
]
P (dωZ)
}
.
Let α(ωX) = δ|X|,1. Let, also, the KS operator K be given for Λ = Zd by the right-
hand side of (5.2), if |X| > 1 and the right-hand side of without the unity if X = x. As
a result the finite volume and infinite volume KS equations in an abstract form look like
ρΛ = zKΛρΛ + zχΛα, ρ = zKρ+ zα,
where KΛ = χΛKχΛ, χΛ is the operator of multiplication by the characteristic function
of Λ: (χΛF )X(ωX) = χΛ(X)FX(ωX).
1. Skrypnik W. Solutions of Kirkwood – Salsburg equation for particles with non-pair finite-range repulsion
// Ukr. Math. J. – 2008. – 60, № 8. – P. 1138 – 1143.
2. Moraal H. The Kirkwood – Saltsburg equation and the virial expansion for many-body potentials // Phys.
Lett. A. – 1976. – 59, № 1. – P. 9 – 10.
3. Ruelle D. Statistical mechanics. Rigorous results. – New York; Amsterdam: W. A. Benjamin, Inc., 1969.
– 219 p.
4. Reed M., Simon B. Methods of modern mathematical physics. – Acad. Press, 1975. – Vol. 2.
5. Ginibre J. Reduced density matrices of quantum gases. I. Limit of infinite systems // J. Math. Phys. –
1965. – 6, № 2. – P. 238 – 251.
6. Kunz H. Analiticity and clustering properties of unbounded spin systems // Communs Math. Phys. –
1978. – 59. – P. 53 – 69.
7. Park Y. M., Yoo H. J. Uniqueness and clustering properties of Gibbs states for classical and quantum
unbounded spin systems // J. Statist. Phys. – 1995. – 80, № 1/2. – P. 223 – 271.
8. Minlos R. A., Verbeure A., Zagrebnov V. A. A quantum crystal model in the light-mass limit: Gibbs
states. – Preprint KUL-TP-97/16.
9. Albeverio S., Kondratiev Yu. G., Minlos R. A., Rebenko O. L. Small mass behaviour of quantum Gibbs
states for lattice models with unbounded spins. – Preprint UMa-CCM 22/97.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
1704 W. I. SKRYPNIK
10. Skrypnik W. On polymer expansions for equilibrium systems of oscillators with ternary interaction //
Ukr. Math. J. – 2001. – 53, № 11. – P. 1532 – 1544.
11. Israel R., Nappi C. Quark confinement in the two dimensional lattice Higgs – Villain model // Сommuns
Math. Phys. – 1978/1979. – 64, № 2. – P. 177 – 189.
12. Skrypnik W. Solutions of Kirkwood – Salsburg equation for lattice systems of one-dimensional oscillators
with positive finite-range manybody interaction potentials // Ukr. Math. J. – 2008. – 60, № 10. – P. 1427 –
1433.
13. Skrypnik W. Kirkwood – Salsburg equation for lattice quantum systems of oscillators with manybody
interaction potentials // Ibid. – 2009. – 61, № 5. – P. 689 – 700.
14. Ruelle D. Probability estimates for continuous spin systems // Communs Math. Phys. – 1976. – 50. –
P. 189 – 193.
15. Skrypnik W. On the evolution of Gibbs states of the lattice gradient stochastic dynamics of interacting
oscillators // Theory Stochast. Processes. – 2009. – 15(31), № 1. – P. 61 – 82.
Received 28.12.09,
after revision — 25.06.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 12
|