On Gibbs quantum and classical particle systems with three-body forces
For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.
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| author | Skrypnik, W. I. Скрипник, В. І. |
| author_facet | Skrypnik, W. I. Скрипник, В. І. |
| author_sort | Skrypnik, W. I. |
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| description | For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit. |
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UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS
WITH THREE-BODY FORCES
PRO HIBBSIVS\KI KVANTOVI TA KLASYçNI SYSTEMY
ÇASTYNOK Z TRYÇASTYNKOVYMY SYLAMY
For equilibrium quantum and classical systems of particles, interacting via ternary and pair (nonpositive) infinite-
range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices
and correlation functions are constructed in the thermodynamic limit.
Dlq rivnovaΩnyx kvantovyx ta klasyçnyx system çastynok, wo vza[modigt\ zavdqky ternarnomu i
parnomu (nepozytyvnym) dalekosqΩnym potencialam, pobudovano klasternyj rozklad dlq ]x reduko-
vanyx matryc\ wil\nosti ta korelqcijnyx funkcij velykoho kanoniçnoho ansamblg, zbiΩnyj pry
nyz\kyx aktyvnostqx u termodynamiçnij hranyci.
1. Introduction and main result. We consider classical and quantum systems of parti-
cles with the Maxwell – Boltzmann statistics that are characterized by the n-particle po-
tential energy (see Remark in the end of the paper)
U(x(n)) =
∑
1≤j1<j2≤n
φ0(xj2 − xj1) +
∑
1≤j1 �=j2 �=j3≤n
φ1(xj2 − xj1 , xj3 − xj1), (1.1)
where x(n) = (x1, . . . , xn) ∈ R
dn, xj = (x1
j , . . . , x
d
j ) he three-body translation invariant
potential φ1 is given by φ1(x, y) = 2
∑d′
l=1
φl(x)φl(y). The pair (two-body) and three-
body potentials are Euclidean invariant functions. U can be rewritten as
U(x(n)) =
∑
1≤j1<j2≤n
φ0(xj2 − xj1) + U ′(x(n)),
where φ0 = −2
∑d′
l=1
φ2
l + φ0, and
U ′(x(n)) =
d′∑
l=1
n∑
j1=1
n∑
j2 �=j1,j2=1
φl(xj2 − xj1)
2
.
The conditions of stability (see [1 – 7]) for it is reduced to the stability condition for the
pair potential φ0:
U0(x(n)) =
∑
1≤j1<j2≤n
φ0(xj2 − xj1) ≥ −Bn, (1.2)
where B is a constant. U is superstable [2 – 4] if φ0 is superstable.
The equilibrium systems of d-dimensional classical and quantum particles, enclosed
in a compact domain Λ ∈ R
d, are described by the sequence of the grand canonical
correlation functions ρΛ = {ρΛ(x(m)), m ∈ Z
+} and reduced density matrices (RDMs)
ρΛ = {ρΛ(x(m); y(m)), m ∈ Z
+}. The former are given by
ρΛ(x(m)) = Ξ−1
Λ
∑
n≥0
zn+m
n!
∫
e−βU(x(m),x
′
(n))χΛ(x(m), x
′
(n))dx
′
(n),
x(m) = (x1, . . . , xm),
where the integration is performed over R
nd, β is the inverse temperature, z is the particle
activity, the grand partition function ΞΛ is given by the denominator in which m = 0 and
c© W. I. SKRYPNIK, 2006
976 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 977
χΛ(x) is the characteristic(indicator) function of the compact domain Λ ∈ R
d. The
RDMs are expressed (see [1, 6, 7])
ρΛ(x(m)|y(m)) =
∫
P βx(m),y(m)
(dw(m))ρΛ(w(m)), (1.3)
where the path (quantum) correlation functions ρΛ(w(m)) for the Dirichlet boundary con-
ditions are determined by
ρΛ(w(m)) = Ξ−1
Λ
∑
n≥0
zn+m
n!
∫
exp{−βU(w(m), w
′
(n))}×
×χΛ(w(m), w
′
(n))dx
′
(n)P
β
x′(n),x
′
(n)
(dw′
(n)), (1.4)
U(w(m)) = β−1
β∫
0
U(w(m)(τ))dτ,
ΞΛ is the grand partition function, coinciding with the numerator in the r.h.s. of the ex-
pression when m = 0, P tx,y(dw) is the conditional Wiener measure, concentrated on
paths, starting from x and arriving into y at the time t, χΛ(w(m)) is the product of the
characteristic functions of paths wj localized in Λ on all the interval [0, β],
P βx(m),y(m)
(dw(m)) = Px(m)(dw(m)|w(m)(β) = y(m)) =
= Px(m)(dw(m))
m∏
j=1
δ(yj − wj(β)),
δ(x) is the point measure concentrated in x, Px(dw) is the Wiener measure concentrated
on paths starting from the the point x. These measures are defined on the probability
space Ωd0 which may be considered as the Banach space of continuous functions [8] with
the σ-algebra of the Borel sets (see also [9]). One easily checks with the help of the Feyn-
man – Kac formula (see [6, 7, 10]) that every term in the sum in (1.3) after substitution
of (1.4) in it is equal to the integral over R
dn of the kernel of the (n + m)-particle quan-
tum Hamiltonain with the Dirichlet boundary condition, the usual kinetic term and the
potential energy U.
There is a standard derivation of the KS equation for the sequence of the correla-
tion functions and path correlation functions corresponding to a pair interaction potential
(see [1 – 7])
ρΛ = zKΛρ
Λ + zαΛ, (1.5)
where KΛ = χ̂ΛKχ̂Λ, αΛ = χ̂Λα, α is the sequence whose first component is the unity
and other components are zero, χ̂Λ is the operator of multiplication by the characteristic
function of Λ, (χ̂ΛF )(ω(n)) = χΛ(ω(n))F (ω(n)) and ω is either a vector from R
d or
a Wiener path. The KS operator K will be introduced by us in the end of this section.
The denominator and numerator in the expression for correlation function diverge in the
thermodynamic limit Λ → R
d but the equation (1.5) is well defined in the limit
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
978 W. I. SKRYPNIK
ρ = zKρ+ zα. (1.6)
Its solution determines the equilibrium state of the infinite-particle system.
It is well known that the norm of the classical KS operator K [11, 12] is bounded in
the Banach space of sequences of bounded functions Eξ for a positive short-range inter-
action (many-body potentials have finite supports). This Banach space is determined by
the following norm ‖F‖ξ = maxn≥1 ξ
−n ess sup
ω(n)
∣∣F (ω(n))
∣∣, where ω = x ∈ R
d for
classical systems. In this case the solutions of (1.5), (1.6) are expressed in terms of the
perturbation expansion of the resolvent of K acting on zα. The norm of the classical
KS operator is also bounded for a potential energy generated by an infinite range positive
integrable pair potential [1 – 3]. If the interaction is mediated only by an integrable and
stable pair potential φ then Ruelle proposed to symmetrize the classical KS operator to
in order to make it bounded in Eξ. There were no examples of potential energies with
nonpositive many-body (nonpair) potentials yielding the bounded KS operator in the Ba-
nach space Eξ. For quantum systems there were no results at all concerning solutions of
the KS equation in Eξ for the case of many-body potentials (in the definition of its norm
one has to put ω = w, where w is a Wiener path). In this paper we consider such the
systems, i.e., classical and quantum systems for which the potential φ0, φ1 are not neces-
sarily positive and have finite supports, and construct their (path) correlation function in
the thermodynamic limit.
Let’s deal with quantum systems at first.If one drops the ternary term U ′ in the expres-
sion of U then by the standard technique of [1, 5 – 7] the series in powers of z for the cor-
relation functions is found converging if |z| < c−1(β), where c(β) = ess supw cβ(w) =
= |cβ |0, cβ(w) =
∫
dxP βx,x(dw
′)
∣∣e−βφ0(w−w′) − 1
∣∣. It’ll be shown that the presence of
U ′ leads to the necessity of dealing with the additional function
c∗β(w) =
d′∑
l=1
lθ
∫
dxP βx,x(dw
′)
β∫
0
φ2
l (w(τ) − w′(τ))dτ
1
2
, θ ≥ 0. (1.7)
The functions
Cβ(w) = cβ(w) + 16
√
2
d′∑
l=1
l−θ
c∗β(w), C(β) = |Cβ |0,
play a prominent role in the convergence of a cluster expansion for the correlation func-
tions when U ′ is non-zero. The function CΛ
β (w) = cΛβ (w) + 16
√
2
(∑d′
l=1
l−θ
)
cΛ∗β(w)
will determine the character of a convergence to the thermodynamic limit, where the ex-
pressions for the functions cΛβ , c
Λ
∗β are obtained from the expressions of the corresponding
functions without dependence on Λ by inserting (1−χΛ(w′)) under the sign of the Wiener
integral. The following theorem will enable us to prove our main result formulated in the
subsequent two theorems (expression for the constant c in Theorem 1.1 is given in Sec-
tion 5).
Theorem 1.1. Let φ0 be an integrable stable potential and the condition (A)
l2θ|φl(x)| ≤ h(|x|), l ≥ 1, hold, where h is a monotone integrable function and θ ≥ 0.
Then
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 979
c(β) ≤ βe2βB‖φ0‖1, c∗(β) ≤
√
β(4πβ)−
d
2
d′∑
l=1
l−θ
c,
where ‖ · ‖1 is the norm of the space of integrable functions L1(Rd) and c is a positive
constant. Moreover, if the condition (B) h(|x|) ≤ h̄|x|−d−2ε, |x| ≥ R, holds, where
R, h̄, ε are positive constants then there exists a positive function C ′(β) independent of w
such that
CΛ
β (w) ≤ δ−εC ′(β), w ∈ Λ(δ), δ ≥ 2R, (1.8)
where Λ(δ) = {x ∈ Λ: dist(x, ∂Λ) ≥ δ}, ∂Λ is a boundary of Λ.
The second theorem is the first step towards proving an existence of the thermody-
namic limit of the path correlation functions.
Theorem 1.2. Let the condition (A) of Theorem 1.1 be satisfied. Then for an arbi-
trary bounded or unbounded Λ there exist functions ρΛ
n(w(m)) such that
ρΛ(w(m)) =
∑
n≥0
zn+mρΛ
n(w(m)),
∣∣ρΛ
n(w(m))
∣∣ ≤ ξm−1(e0e2βBC(β))n+m−1
for ξ−1 = 4C(β) and the series converges in the norm of the space of bounded func-
tions if
|z| ≤ (e0e2βBC(β))−1, e0 = 4
∑
n≥1
(n!)−
1
2 .
It is natural to expect the correlation functions ρ(w(m)) = ρΛ(w(m)), Λ = R
d, de-
termined in Theorem 1.2, to be the thermodynamic limit of the finite volume correlation
functions. The third theorem confirms this establishing the character of a convergence of
the finite volume correlation functions to the functions ((1.8) plays a crucial role).
Theorem 1.3. Let the conditions (A), (B) of Theorem 1.1 be true then there exist
positive functions ε(λ), ε′(λ), decreasing at infinity, such that if ρn(w(m)) = ρΛ
n(w(m))
for Λ = R
d and
ρ(w(m)) =
∑
n≥0
zn+mρn(w(m)) (1.9)
then for ξ−1 = 4C(β) and λ = max
j
dist(xj ∈ A, ∂Λ), A ⊂ Λ, the following inequalities
are true: ∣∣∣ρ(w(m)) − ρΛ(w(m))
∣∣∣ ≤ ξmε(λ), w(m) ∈ Am, (1.10)
∣∣∣ρ(x(m) | y(m)) − ρΛ(x(m) | y(m))
∣∣∣ ≤ (4πβ)−
dm
2 mξmε′(λ), xj , yj ∈ A, (1.11)
where ρ(xm | ym) are given by (1.3) with Λ = R
d.
Corollary 1.1. Let ρn(x(m)|y(n)) =
∫
ρn(w(m))P βx(m),y(m)
(dw(m)). Then the ther-
modynamic limits of the RDMs of the quantum system with the potential energy (1.1),
satisfying conditions of Theorem 1.1, are given by
ρ(x(m)|y(m)) =
∑
n≥0
zn+mρn(x(m)|y(m)),
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
980 W. I. SKRYPNIK
∣∣ρn(x(m)|y(m))
∣∣ ≤ ξm−1ξ′
n+m−1
m∏
j=1
P β(xj − yj),
for |z| ≤ ξ′−1, ξ′ = e0e
2βBC(β), P β(x) = (4πβ)−
d
2 e−
|x|2
4β .
In our approach the Gibbs factor containing the quadratic term U ′ is transformed
into the Gibbs factor with an imaginary potential energy, generated by a pair potential,
depending on additional d′ Wiener paths, labeled by a star, with the help of the Fourier-
type transformation (see (2.1)). In this way we simplify our system and have to deal
with a new Gibbs path system with a potential energy expressed through a complex pair
potential. It easy to write down the KS equation and R-symmetrized KS equation for
the correlation functions but it is not obvious that the operator in its right-hand side (KS
operator)is bounded in the Banach space of sequences of bounded functions.
It turns out that the bounds of the norm of the Ruelle symmetrized (R-symmetrized)
KS operator in the Banach space of sequences of bounded functions is difficult to obtain
because of the presence of a stochastic integral in the expression of the imaginary pair
potential. We circumvent this difficulty by introducing a new Lp-type norm ‖Ψ‖ξ,q on
sequences of functions with an increasing number of variables (integrating out depen-
dence in the star paths) and establish the expected boundedness of the R-symmetrized
KS operator in the corresponding Banach space Eξ,q . This implies, as usual, existence of
the thermodynamic limit of the correlation functions. The corner stone for the results of
this paper is boundedness of the function
∫
dx
[ ∫ β
0
h2
(
|w(τ) − x|
)
dτ
] 1
2
, where w is a
Wiener path and h is a monotonic integrable function. Our main tool is the generalized
Holder inequality (3.7) and an inequality for the weighted convolution of two monotone
functions (5.3). Without difficulty our result can be generalized to the case of infinite d′
since for θ > 1 our bounds are uniform in d′. The proposed approach is inspired by the
results of the papers [13 – 16] in which diffusion Gibbs path particle systems with three-
body forces were introduced. The thermodynamic limit of their path correlation functions
allow to calculate the thermodynamic limit of correlation functions of the nonequilibrium
systems of interacting Brownian particles. In this papers only KS recursion relation was
used for this purpose. The results of this paper can be applied without difficulty for the
systems.
The similar simplified technique we develop for classical systems. The new variables
which are finite-dimensional vectors are introduced with the help of (6.1). With their help
we pass to a new Gibbs system with a pair but complex potential described by the KS
equation having the unique solution in the Banach space Ẽξ with the norm given by (6.3).
For classical systems we have the following analog of the previous theorems.
Theorem 1.4. Let |φ′(x)|2 =
(∑d′
l=1
l2φ2
l (x)
)1
2
, ‖φ′‖ =
∫
|φ′(x)|2dx. If
‖φ′‖ < ∞ and φ0 is a stable potential then there exist functions ρn(x(m)), a positive
function ε(λ) decreasing at infinity, positive numbers η0, η1 such that the series
ρ(x(m)) =
∑
n≥0
zn+mρn(x(m)),
converges in the disc |z| < R
R =
η1
η0Cη
e−β(2B+
η1
η0
), Cη = 2
√
2βη1‖φ′‖ + η0‖e−βφ0 − 1‖1,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 981
and
|ρ(x(m)) − ρΛ(x(m))| ≤
(
η1
η0Cη
)m
ε(λ),
where λ = max
j
dist(xj , ∂Λ) and ∂Λ is a boundary of Λ. Moreover,
ρ(x(m)) ≤
(
|z|η0e
β(2B+
η1
η0
))mc, c = e−β(2B+
η1
η0
)(1 − |z|R−1
)−1
.
This theorem follows directly from Theorem 6.1 and Corollaries 6.1, 6.2, Lemma 6.1
and standard arguments from [1] (see Theorem 4.2.3, formula (2.48) and its proof with
the help of the inequality(2.39)) if one puts
ξ =
η1
η0Cη
.
This choice of ξ satisfies the condition of Theorem 6.1 and Lemma 6.1, i.e., 1−2
√
2β η0×
×‖φ′‖ξ > 0. From Lemma 6.1 it is easy to see that ε(λ) is proportional to the function
Cη,λ
(
η1
η0Cη
)
determined in this lemma. It is evident that the growth of R for small β is
proportional to
√
β−1. Our estimates allow to consider infinite d′.
We’ll employ the following abstract KS operator on the measure space (Ω, µ) describ-
ing classical and quantum systems with complex pair potential φ given by
(KF )(ω(m)) = exp
{
−βW (ω1|ω(m\1))
}
×
×
F (ω(m\1))(1 − δm,1) +
∑
n≥1
1
n!
∫
K(ω1|ω′
(n))F (ω(m\1), ω
′
(n))µ(dω′
(n))
, (1.12)
where the integration is performed over Ωn, (n\j) is a sequence (1, . . . , n) without the
integer j and δm,k is the Kronecker symbol, the function W determines the interaction of
one particle with others,
W (ω1|ω(m\1)) = U(ω(m)) − U(ω(m\1)),
K(ω|ω(n)) =
n∏
j=1
(
exp{−βφ(ω|ωj)} − 1
)
.
Now, its important to introduce the Ruelle’s symmetrization for the KS equation. The
following inequality ReW (ωj |ω(m\j)) ≥ −2B holds in some nonempty set of Ωm. Let’s
denote by χ(j,m) the characteristic function (it does not depend on the paths w∗
j ) of the
set and put
m∑
j=1
χ∗
(j,m) = 1, χ∗
(j,m) =
m∑
j=1
χ(j,m)
−1
χ(j,m). (1.13)
The first equality follows from the stability condition ReU(ω(m)) ≥ −Bm. After mul-
tiplying both sides of the KS equation by χ∗
(j,m) and summing over j from 1 to m, we
obtain the R-symmetrized KS equation
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
982 W. I. SKRYPNIK
(K̃F )(ω(m)) =
m∑
j=1
χ∗
(j,m)(ω(m)) exp
{
−βW (ωj |ω(m\j))
}
×
×
[
F (ω(m\j))(1 − δm,1) +
∑
n≥1
1
n!
∫
K(ωj |ω′
(n))F (ω(m\j), ω
′
(n))µ(dω′
(n))
]
, m ≥ 1.
(1.14)
Our paper is organized as follows. In the next section we reduce our Gibbs systems to
new Gibbs path systems with a complex pair potential and prove important bounds (2.7),
(2.8) in Lemma 2.1 connected with the functions C(β), CΛ
β (w). The first of them permits
to express the norm of the symmetrized KS equation in terms of C(β). In the third, fourth
and fifth sections we prove Theorems 1.2, 1.3 and 1.1, respectively. In the last sectio we
prove Theorem 1.4.
2. New Gibbs path systems. Our systems can be reduced to the Gibbs path system
with a complex pair potential with the help of the transformation
exp
−
β∫
0
n∑
k �=j,k=1
φl(wj(τ) − wk(τ))
2
dτ
=
=
∫
P0(dωl∗j) exp
−i
n∑
k �=j,k=1
β∫
0
φl(wj(τ) − wk(τ))dwl∗j
, (2.1)
where
∫
dwl∗j is the stochastic integral over the one-dimensional Wiener process
and P0(dwl∗j) is the Wiener measure concentrated on paths starting from the origin. As a
result
exp{−βU ′(w(n))} =
∫
exp
−β
d′∑
l=1
Ul(ω(n))
P0(dw∗(n)),
where ω = (w,w1
∗, . . . , w
l
∗) = (w,w∗), P0(dw∗(n)) =
∏n
j=1
P0(dw∗j), P0(dw∗) =
=
∏d′
l=1
P0(dwl∗),
Ul(ω(n)) =
∑
1≤k<j≤n
φl(ωj |ωk), l = 1, . . . , d′, (2.2)
φl(ω|ω′) = iβ
(
ϕl(w − w′|w∗) + ϕl(w′ − w|w′
∗)
)
, ϕl(w|w∗) =
β∫
0
dwl∗φl(w(τ)).
(2.3)
The stochastic integral ϕl(w|w∗) is the measurable function in L2(Ωd+1
0 , P0,x(dw)), x ∈
∈ R
d. Indeed, this function almost everywhere in w ∈ Ωd0 (w is a continuous function)
is defined as a limit in the topology of L2(Ω0, P0) of integral Riemannian sums, so it is a
measurable function in w∗. This function is, also, measurable in w, since it is defined as a
limit of almost everywhere convergent subsequence of measurable functions (a sequence
of functions, converging in the topology of L2(Ω0, P0) has a subsequence, converging
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 983
almost everywhere). These sums are cylinder functions in w,w∗. Hence the limit is a
measurable function.
So, we reduced the initial Gibbs path system to the Gibbs system with the complex
pair potential, described by the correlation functions and one-particle extended by the new
Wiener paths wl∗, l = 1, . . . , d′, phase space Ω = Ωd
′+d
0
ρΛ(ω(m)) =
= Ξ−1
Λ
∑
n≥0
zn+m
n!
∫
exp
{
−βU(ω(m), ω
′
(n))
}
χΛ(ω(m), ω
′
(n))dx
′
(n)P∗x(n)(dω
′
(n)),
(2.4)
where χΛ(ωn) = χΛ(wn), the integration is performed over Λn × Ωn, i.e., n-fold Carte-
sian product of Λ × Ω, and
U(ω(n)) =
d′∑
l=0
Ul(ω(n)), Ul(ω(n)) =
∑
1≤k �=j≤n
φl(ωk|ωj),
P∗x(dwdw∗) = P βx,x(dw)P0(dw∗), φ0(ω|ω′) = φ0(w − w′)
(2.5)
and ΞΛ coincides with the numerator for m = 0 in (2.4). The path correlation function
are expressed in terms of the new ones as follows:
ρΛ(w(m)) =
∫
ρΛ(ω(m))P0(dw∗(m)). (2.6)
So, we’ll deal with the KS operator in (1.12) and (1.14) for µ(dω) = dxP∗x(dwdw∗).
Our main estimates will concern the functions
K0
n,q = |Kn,q|0,
Kn,q(w) =
∫
dx(n)P
β
x(m),x(n)
(dw(n))Kq(w|w(n)), K0
0,q = 1,
Kq(w|w(n)) =
(∫
P0(dw∗)|Kq(ω|w(n))|q
) 1
q
=
=
(∫
P0(dw∗)P0(dw∗(n)|K(ω|ω(n))|q
)
)
1
q .
The first two important bounds are given in the following lemma.
Lemma 2.1. Let the condition (A) of Theorem 1.1 be satisfied. Then
K0
q,n ≤ (n!)
1
2 (4C0
q (β))n, q ∈ 2Z
+, (2.7)
where C0
q (β) = c(β) + 16
√
q
(∑d′
l=1
l−θ
)
c∗(β), and for Λ′ ⊆ Λ′′
∫
dxP βx,x(dw(n))Kq(w|w(n))[χΛ′′(w(n)) − χΛ′(w(n))] ≤
≤ 4nCΛ
β,q(w)((n− 1)!)
1
2 (4C0
q (β))n−1, (2.8)
where CΛ
β,q(w) = cΛβ (w) + 32
√
q
(∑d′
l=1
l−θ
)
cΛ∗β(w).
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984 W. I. SKRYPNIK
Proof. The generalized Helder inequality (3.7) yields
Kq(w|w(n)) ≤
n∏
j=1
[∫ ∫
P0(dw∗)P0(dw∗j)|K(ω|ωj)|qn
] 1
nq
. (2.9)
Integrating by dx(n)P
β
x(n),x(n)
(dw(n)) the left-hand side of the inequality we derive
Kn,q(w) ≤
[∫
dxP βx,x(dw
′)
(∫ ∫
P0(dw∗)P0(dw′
∗)|K(ω|ω′)|qn
) 1
qn
]n
.
From the inequality |eb+ia − 1| ≤ |eb − 1|+ 2|a| it follows that (by φ0, φ∗ we denote the
real and imaginary parts of the pair potential, respectively)
Kn,q(w) ≤ 2n
{∫
dxP βx,x(dw
′)
[
|e−βφ0(w−w′) − 1|+
+2
( ∫ ∫
P0(dw∗)P0(dw′
∗)|βφ∗(ω|ω′)|qn
)1
qn
]}n
.
Here we used the inequalities (a+b)n ≤ 2n(an+bn), (a+b)
1
m ≤ a
1
m +b
1
m . The Helder
inequality (∑
s
as
)m
≤
(∑
s
s−θ
m
m−1
)m−1 ∑
s
smθams , as ≥ 0,
for as = β|φs|, and m = qn implies
Kn,q(w) ≤ 2n
[
cβ(w) + 2
( d′∑
s=1
s−θ
) ∫
dxP βx,x(dw
′)×
×
(∫ ∫
P0(dw∗)P0(dw′
∗)
d′∑
s=1
snqθ|βφs(ω|ω′)|qn
) 1
qn
]n
.
Now, the inequality
[∑d′
s=1
as
] 1
nq
≤
∑d′
s=1
a
1
nq
s , yields
Kn,q(w) ≤ 2n
[
cβ(w) + 4
( d′∑
l=1
l−θ
) d′∑
s=1
sθ
∫
dxP βx,x(dw
′)×
×
( ∫
P0(dw∗)|ϕs(w − w′|w∗)|qn
) 1
qn
]n
.
Let q ∈ 2Z
+, then the function in the round brackets is equal to
∫
P0(dw∗)|ϕs(w|w∗)|qn =
(qn)!
qn
2
!
β∫
0
φ2
s(w(τ)dτ
qn
2
.
The inequality 4−nnn ≤ n! ≤ nn leads to
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ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 985
((qn
2
)
!
)−1
≤
(
8
q
)nq
2
n−nq
2 ,
(qn)!
(
qn
2
)!
1
q
≤
(√
8qn
)n
≤
(
4
√
2q
)n√
n!.
The inequality (2.7) is proved. The inequality (2.8) follows from (2.9) after repeating
the above arguments (instead of the Wiener measure its product with the characteristic
function is substituted in the bounds) and the inequality 0 ≤ χΛ′′(w′
(n)) − χΛ′(w′
(n)) ≤
≤
∑n
j=1
(1 − χΛ′(w′
j)).
3. Proof of Theorem 1.2. Let’s introduce the Banach space Eξ,q of sequences of
measurable functions F = {F (ω(n)), n ≥ 1} with the norm ‖F‖ξ,q
‖F‖ξ,q = max
n
ξ−n ess sup
w(n)
|F |q(w(n)),
|F |q(w(n)) =
[∫
|F (ω(n))|qP0(dw∗(n))
] 1
q
.
In this section we’ll prove that the norm ‖K̃‖ξ,q of the KS operator is bounded in the
Banach space for positive even integers q > 1 and a positive number ξ. This fact will
allow us to prove the convergence of the low activity cluster expansion for the correlation
functions ρΛ(ω(m)), m ≥ 1, which results from the expansion of the KS resolvent in the
series in powers of the KS operator, and represent them in the form
ρΛ(w(m)) =
∑
n≥0
zn+m
∫
P0(dw∗(m))(K̃n+m−1
Λ αΛ)(w(m), w∗(m)). (3.1)
Here we took into account that (K̃n
ΛαΛ)(ω(m)) = 0, n < m−1. One proves Theorem 1.2
with the aid of Theorem 3.1 and Corollary 3.1 putting q = 2 and taking into account that
C(β) = C0
2 (β). This gives
ρΛ
n(w(m)) =
∫
P0(dw∗(m))(K̃n+m−1
Λ α)(w(m), w∗(m)) (3.2)
and after applying the Schwartz inequality one derives (‖αΛ‖ξ,q = ξ−1)
|ρΛ
n(w(m))| ≤ ξm−1‖K̃‖n+m−1
ξ,q . (3.3)
This inequality and (3.4) imply the bound for ρn in Theorem 1.2.
Theorem 3.1. Let the condition (A) of Theorem 1.1 be satisfied. Then
‖K̃‖ξ,q ≤ ξ−1e2βBξ0
q , ξ0
q =
∑
n≥0
(4ξC0
q (β))n√
n!
, (3.4)
and for Λ ⊆ Λ′ ⊆ Λ′′
‖χ̂ΛK̃(χ̂Λ′′ − χ̂Λ′)‖ξ,q ≤ e2βB |χΛC
Λ′
β,q|0ξ0
q . (3.5)
If one puts ξ−1 = 4C0
q (β) then ‖K̃‖ξ,q ≤ e2βBe0C
0
q (β).
Corollary 3.1. If the condition |z|e0C0
q (β)e2βB < 1 is satisfied for even positive
integer q then there exists the unique solution of the symmetrized KS equations (1.5),
(1.6), in Eξ,q, ξ
−1 = 4C0
q (β) given by perturbation expansion convergent in the uniform
operator norm
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986 W. I. SKRYPNIK
ρΛ =
∑
n≥0
zn+1K̃n
ΛαΛ, ρ =
∑
n≥0
zn+1K̃nα (3.6)
and ‖ρΛ‖ξ,q ≤ |z|ξ−1(1 − |z‖|K̃‖ξ,q)−1. If the potential φ0 is positive then the same is
true for the KS equations (1.5), (1.6) and the expansion for the sequence of their solutions
in terms of powers of the operators KΛ and K instead of K̃Λ and K̃.
Proof of Theorem 3.1. Applying the Helder inequality for a probability measure
P0(dw∗) for the case when one of the function is the unity, we obtain for m > 1
∣∣(K̃F )(ω(m))
∣∣ ≤ m∑
j=1
χ∗
j,m(w(m))e2βB×
×
∑
n≥0
1
n!
∫
dx′(n)P
β
x′(n),x
′
(n)
(dw′
(n))K q
q−1
(ωj |w′
(n))|F |q(ω(m\j);w′
(n)),
|F |q(ω(m\j);w′
(n)) =
[∫
|F (ω(m\j), ω
′
(n))|qP0(dw′
∗(n))
] 1
q
.
For m = 1 the summation in n is restricted to n ≥ 1. Let χ∗
(j(q),m) =
∏q
l=1
χ∗
(jl,m).
Taking the both sides to the q-th power, we have∫
|(K̃F )|q(ω(m))P0(dw∗(m)) ≤
≤ e2βBq
∑
j1,...,jq
χ∗
(j(q),m)(w(m))
∑
n1,...,nq
(n1! . . . nq!)−1×
×
∫ q∏
s=1
dx′(ns)P
β
x′(ns),x
′
(ns)
(
dw
(s)′
(ns)
)
×
×
∫
P0
(
dw∗(m)
) q∏
r=1
K q
q−1
(
ωjr |w
(r)′
(nr)
)
|F |q
(
ω(m\jr);w
(r)′
(nr)
)
.
The integral over w∗(m) can be taken independently over w∗(m\js) and w∗js . Applying
the generalized Holder inequality
∫ n∏
j=1
|Fj(x)|µ(dx) ≤
n∏
j=1
(∫
|Fj(x)|nµ(dx)
) 1
n
(3.7)
for n = q and the two integrals, we see that the first integral coincides with the function
|F |q(w(m\js), w
′
(ns)) and the second with
∫
P0(dw∗js)
∣∣K q
q−1
(ωjs |w′
(n))
∣∣q ≤
≤ Kq(wjs |w′
(n)). Here we applied the ordinary Helder inequality for the integral in a
probability measure for the power q − 1. Taking ess sup and using (1.13) the following
bound is derived:
|K̃F |q(w(m)) ≤ ‖F‖ξ,qξm−1e2βB
∑
n≥0
ξn
n!
K0
n,q. (3.8)
The same is true for m = 1. The inequality (3.4) follows from Lemma 2.1. The similar
arguments yield
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ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 987
χ̂Λ(w(m))|K̃(χ̂Λ′′ − χ̂Λ′)F |q(w(m)) ≤
≤ ‖F‖ξ,qξm−1e2βB
∑
n≥0
ξn
n!
ess sup
w
χΛ(w)
∫
dx(n)P
β
x(n),x(n)
(dw(n))×
×Kq(w|w(n))[χΛ′′(w(n)) − χΛ′(w(n))] ≤
≤ ‖F‖ξ,qξm−1e2βB |χΛC
Λ′
β,q|0
∑
n≥0
ξn
n!
K0
n,q.
Here we applied (2.8). Now (3.5) is proven with the help of (2.7). Theorem 3.1 is proved.
4. Proof of Theorem 1.3. Application of inequalities (1.8), (3.5) yields for Λ = Λ′(δ)
(remark that ξ−1 = 4C(β))
‖χ̂ΛK̃(χ̂Λ′′ − χ̂Λ′)‖ξ,2 ≤ η(δ), η(δ) = δ−εe2βBC ′(β)e0.
Then the following estimate is proved in a standard way (see (2.39), (2.48) in [1] and
Section 4 in [6])
[∫
|ρ(ω(m)) − ρΛ(ω(m))|2P0(dw∗(m))
] 1
2
≤
≤ ξm‖χ̂Λ(ρ− ρΛ)‖ξ,2 ≤ ξmε(λ), w(m) ∈ Am.
The inequality (1.10) follows after an application of the Schwartz inequality. To prove
(1.11) one has to deal with the set Γβ(2−1δ) of paths that on the time interval [0, β]
depart at the distance 2−1δ, that is for all its paths |w(t′) − w(t)| ≥ δ, t′, t ∈ [0, β]. It
is well known (see Appendix 1 in [6]) that for the characteristic function χ(w|Γβ(R)) of
this set the inequality holds∫
P βx,y(dw)χ(w|Γβ(R)) ≤ γ(R, β) =
γ0
(4πβ)
d
2
∫
u≥(4
√
t)−1R
e−u
2
ud−1du, (4.1)
where γ0 is a constant. From the definition we derive for ξ−1 = 4C(β)
χA(x(m), y(m))|ρ(x(m)|y(m)) − ρΛ(x(m)|y(m))| ≤
≤ χA(x(m), y(m))
∫
P βx(m),y(m)
(dw(m))P0(w∗(m))×
×
[
(1 − χΛ(2−1λ))(w(m)) + χΛ(2−1λ))(w(m))
]
|ρ(ω(m)) − ρΛ(ω(m))| ≤
≤ χA(x(m), y(m))
∫
P βx(m),y(m)
(dw(m))×
×(1 − χΛ(2−1λ)(w(m)))(‖ρ‖ξ,2 + ‖ρΛ‖ξ,2)ξm+
+ξm(4πβ)−
md
2 ‖χ̂Λ(2−1λ)(ρ− ρΛ)‖ξ,2. (4.2)
Here we applied the Schwartz inequality for the integral by the measure P0(w∗
(m)) and
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988 W. I. SKRYPNIK
∫
P βx(m),y(m)
(dw(m)) =
m∏
j=1
P β(xj − yj) =
=
m∏
j=1
(
√
4πβ)−de−
|xj−yj |2
4β ≤ (
√
4πβ)−md. (4.2a)
For the second term in inequality (4.2) above the inequality (4.1) has to be applied. To
bound the first term one has to use Corollary 3.1 and the inequality 1−χΛ(2−1λ)(w(m)) ≤
≤
∑m
j=1
(1 − χΛ(2−1λ)(wj)). As a result
χA(x(m), y(m))
∫
P βx(m),y(m)
(dw(m))(1 − χΛ(2−1λ)(w(m))) ≤
≤ m(4πβ)
−m+1
2 d ess sup
x,y
χA(x, y)
∫
P βx,y(dw)(1 − χΛ(2−1λ)(w)) ≤
≤ m(4πβ)
−m+1
2 d ess sup
x,y
∫
P βx,y(dw)χ(w|Γβ(2−1λ)) ≤
≤ m(4πβ)
−m+1
2 dγ(2−1λ, β).
Here we employed (4.2a), took into account that 1−χΛ(2−1λ)(w) is concentrated on paths
that enter Λc(2−1λ) and {dist(A,Λc(2−1λ))} ≥ 2−1λ. So, inequality (1.11) holds with
ε′(λ) = ε(2−1λ) + 2(4πβ)
d
2 γ(2−1λ, β)|z|ξ−1(1 − |z|‖K̃‖ξ,2)−1. (4.3)
5. Proof of Theorem 1.1. From the bound |e−βφ0 − 1| ≤ e2βBβ|φ0| (see for-
mula (3.16) in [6]), it follows that cβ(w) ≤ e2βB
∫
dxPx,x(dw′)
∫ β
0
|φ0(w(τ) −
− w′(τ))|dτ}. After changing the order of integrations (the Fubini theorem is used) one
easily derives the bound taking into account translation invariance of Px,x(dw) in x it
follows that c(β) ≤ e2βBβ‖φ0‖1. The first step in the proof of the second bound for c∗β
is the following bound:
c∗β(w) ≤ (4πβ)−
d
2
d′∑
l=1
l−θ
h∗(w), (5.1)
where
h∗(w) =
∫
dx′
[∫ β
0
dτh0(|w(τ) − x′|)
]1
2
,
h0(|x|
)
= h2
( |x|
2
)
+ exp
{
−|x|2
16β
}
2
d
2 |h2|0.
The second step is the proof of the bound∫
(h∗(w))nP0(dw) ≤
(
β2d+1[1 +
√
2d] ‖
√
h0‖2
1
)n
2
. (5.2)
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ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 989
The following proposition shows that h∗ is bounded by the square root of the term in the
round brackets in the right-hand side of this inequality and (5.1), (5.2) lead to the needed
inequality for c∗β with c =
(
22d+3(1 +
√
2d)
) 1
2 ‖
√
h0‖1.
Proposition 5.1. Let f(x), x ∈ X , be a positive function on the probability space
(X,µ). Then |f |0 ≤ a iff
∫
X
fn(x)µ(dx) ≤ an, n ∈ Z
+.
Proof. The first condition implies the second one. Now, suppose that the last condi-
tion holds. Then for arbitrary ε > 0 we have∫
X
((a+ ε)−1f)n(x)µ(dx) ≤ ((a+ ε)−1a)n, n ∈ Z
+.
The right-hand side is less than arbitrary small number δ for sufficiently large n. This
means that f(x) ≥ a + ε, x ∈ A, holds for the set A such that µ(A) ≤ δ. Tending δ to
zero, i.e., n to infinity, we prove the proposition.
Proof of (5.1). Applying the Schwartz inequality for the integral in the measure
Px,x(dw′) we obtain, taking into account the condition for φl from Theorem 1.1,
c∗β(w) ≤
d′∑
l=1
l−θ
∫
dx
[∫
P βx,x(dw
′)
] 1
2
β∫
0
dτG(w(τ);x)
1
2
, (5.2a)
G(w(τ);x) =
∫
P βx,x(dw
′)h2(|w(τ) − w′(τ)|
)
.
We’ll prove now that
G(w(τ);x′) ≤ (4πβ)−
d
2 h0(|w(τ) − x′|
)
. (5.2b)
It is not difficult to derive the following equalities G(w(τ);x) = G(w(τ) − x), G(x) =
=
∫
dyP τ (y−x)P β−τ (y−x)h2(|y|
)
.Here we used the definition of the Wiener measure
(tj−1 ≤ tj ≤ t) for n = 1 and f(y1) = h2(y1 − x)∫
P tx,y(dw)f(w(t1), . . . , w(tn)) =
=
∫
dy(n)P
t1(x− y1)
n∏
j=2
P tj−tj−1(yj − yj−1)P t−tn(yn − y)f(y(n)).
In all our following estimates we ’ll use that for monotone integrable bounded positive
functions f, g and positive ψ the following bound is valid (monotonicity inequality)
(fψ ∗ g)(x) =
∫
f(|x− y|ψ(y − x))g(|y|
)
dy ≤
≤ g
( |x|
2
)
‖ψf‖1 + f
( |x|
2
)
(ψ ∗ g)(x) ≤ g
( |x|
2
)
‖ψf‖1 + f
( |x|
2
)
|g|0‖ψ‖1.
(5.3)
The bounds are obtained, representing the integral into the sum of two integrals over two
domains: |y| ≥ |x|
2
, |y| ≤ |x|
2
, using the monotonicity of the functions, the fact that in
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990 W. I. SKRYPNIK
these domains either |y| ≥ |x|
2
or |x − y| ≥ |x|
2
, and after that enlarging the domains
to the whole space. Now, let’s put P [2]t(x) = P t
(x
2
)
, ψ(x) = P [2]τ (x)P [2](β−τ)(x),
g = h2, f = exp
{
−|x|2
8τ
− |x|2
8(β − τ)
}
in (5.3). Then
∫
dyP τ (y − x)P β−τ (y − x)h2(|y|
)
≤ h2
( |x|
2
)
‖P τP β−τ‖1+
exp
{
−|x|2
16τ
− |x|2
16(β − τ)
} ∥∥∥P [2]τP [2](β−τ)
∥∥∥
1
|h2|0.
The equalities ‖P τP β−τ‖1 = P β(0) = (4πβ)−
d
2 , ‖P [2]τP [2](β−τ)‖1 =
= 2
d
2 ‖P τP β−τ‖1 = 2
d
2 (4πβ)−
d
2 and inequalities τ ≤ β, β − τ ≤ β yield (5.2b). It
and (5.2a) lead to (5.1).
Proof of (5.2). We have
|h∗|0 = ess sup
w∈Ωd
0 ,w(0)=0
h∗(w). (5.4)
This equality follows from translation invariance of h∗: h∗(w + x) = h∗(w). With
the help of the Schwartz inequality for the probability measure and the definition of the
Wiener integral
∫
Px(dw)f(w(t1), . . . , w(tn)) =
=
∫
dy(n)P
t1(x− y1)
n∏
j=2
P tj−tj−1(yj − yj−1)f(y(n))
we derive
∫
P0(dw)(h∗(w))n =
∫
dx(n)
∫
P0(dw)
n∏
j=1
β∫
0
dτh0(|xj − w(τ)|
)
1
2
≤
≤
∫
dx(n)
∫
P0(dw)
n∏
j=1
β∫
0
dτh0(|xj − w(τ)|
)
1
2
=
=
∫
dx(n)
β∫
0
dτ(n)
∫
P0(dw)
n∏
j=1
h0(|xj − w(τj)|
)
1
2
=
∫
dx(n)I(x(n)) = In,
where
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ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 991
I2(x(n)) = n!
∫
[β](n)
dτ(n)
∫
P τ1(|y1|
)
h0(|y1 − x1|
)
×
×
n∏
j=2
P τj−τj−1(|yj − yj−1|
)
h0(|yj − xj |
)
dy(n)
and [t](n) = 0 ≤ τ1 ≤ τ2 . . . ≤ τn ≤ t ∈ R
n. From inequality (5.3) with ψ =
= p[2]τ , g(x) = h0
( |x|
2l
)
, f(x) = e−
|x|2
8τ and the elementary inequality
exp
{
−|x|2
a
}
≤ exp
{
−|x|2
b
}
, b > a, it follows that
∫
P τ (|x− y|
)
h0
( |y|
2l
)
dy ≤ 2
(
1 +
√
2d
)
h0
( |x|
2l+1
)
, l ∈ Z
+. (5.5)
Here we took into account that |h0|0 ≤ 2
(
1 +
√
2d
)
|h2|0. Applying this inequality for
the integral in yn, translating yn by xn at first, one gets (x stands for xn − yn−1)
I2(x(n)) ≤ n!2
(
1 +
√
2d
) ∫
[β](n)
dτ(n)
∫
P τ1
(
|y1|
)
h0
(
|y1 − x1|
)
×
×
n−1∏
j=2
P τj−τj−1
(
|yj − yj−1|
)
h0
(
|yj − xj |
)
h0(|yn−1 − xn|
)
dy(n−1).
From the first inequality in (5.3) with ψ = P τ and (5.5) we obtain for τ ≤ β translating
the variable yn−1 by xn and then, on the second step, by −xn∫
P τ
(
|yn−1 − yn−2|
)
h0
(
|yn−1 − xn−1|
)
h0
( |yn−1 − xn|
2l
)
dyn−1 ≤
≤ h0
( |xn − xn−1|
2l+1
) ∫
P τ
(
|yn−1 − yn−2|
)
h0
(
|yn−1 − xn−1|
)
dyn−1+
+h0
( |xn − xn−1|
2
) ∫
P τ
(
|yn−1 − yn−2|
)
h0
( |yn−1 − xn|
2l
)
dyn−1 ≤
≤ 2
(
1 +
√
2d
) [
h0
( |xn − xn−1|
2l+1
)
h0
( |yn−2 − xn−1|
2
)
+
+h0
( |xn − xn−1|
2
)
h0
( |yn−2 − xn|
2l+1
) ]
.
Let’s substitute this inequality into the expression for In for l = 1 (here τj − τj−1 ≤ β)
then
In ≤
(
2
(
1 +
√
2d
))1
2
[
I
(1)
n−1(n)
∫
dx
(
h0
( |x|
4
))1
2
+ I
(2)
n−1(n)
∫
dx
(
h0
( |x|
2
))1
2
]
,
I
(l)
n−1(n) =
√
n!
∫
dx(n−1)
[ ∫
[β](n)
dτ(n)
∫
dy(n−1)P
τ1
(
|y1|
)
h0
(
|y1 − x1|
)
×
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
992 W. I. SKRYPNIK
×
n−2∏
j=1
P τj−τj−1
(
|yj − yj−1|
)
h0
(
|yj − xj |
)
h0
( |yn−2 − xn−1|
2l
)]1
2
, l = 1, 2.
Here we used the fact that the square root of a finite sum is less than the sum of square
roots of its elements. Iterating this bound n− 1 times we’ll obtain
In ≤
√
n!
(
2
(
1 +
√
2d
))n
2
( ∫
[β]n
dτ(n)
)1
2
2(n−1)(d+1)
∥∥√
h0
∥∥
1
.
The integral in τ(n) is equal to (n!)−1βn. That is, In ≤
(√
βa ‖
√
h0‖1
)n
, a =
= 22d+3(1 +
√
2d ) and (5.2) is proved.
Proof of (1.8). Let’s insert the equality under the sign of the integral in the expression
for CΛ
β (w) the equality
1 =
(
1 − χΛc(2−1δ)(w′)
)
+ χΛc(2−1δ)(w′). (5.6)
By c
Λ−(+)
β (w), cΛ−(+)
∗β (w) we’ll denote the functions which are obtained from the ex-
pressions for the corresponding functions without the marks −(+) by inserting the first
and and the second terms in this equality. The function 1 − χΛc(w′) is concentrated
on paths that intersect Λ at least once. We have
(
1 − χΛc(2−1δ)(w′)
)(
1 − χΛ(w′)
)
≤
≤ χ
(
w′|Γβ(2−1δ)
)
, since the left-hand side is non-zero on the set Γβ(2−1δ). Now, ap-
plying the Schwartz inequality for the integral in the measure P βx,x(dw
′) we obtain,taking
into account the condition for A in Theorem 1.1 and (4.1),
cΛ−
∗β;σ(w) ≤ γ
1
2
(
2−1δ, β
)
d′∑
l=1
l−θ
∫
dx
β∫
0
G (w(τ);x) dτ
1
2
.
The inequality (5.2b) gives
cΛ−
∗β;σ(w) ≤ γ
1
2 (2−1δ, β)c−∗ , c−∗ = (4πβ)−
d
4
d′∑
l=1
l−θ
|h∗|0. (5.7)
c−∗ is finite since h∗ is a bounded function as it is proved at the beginning of this sec-
tion. Applying (4.1) and the Schwartz inequality for the measure P βx,x(dw
′) we derive
cΛ−
β (w) ≤ γ
1
2 (2−1δ, β)
∫ β
0
dτ
∫
dx[G(w(τ);x)]
1
2 . The inequality (5.2b) leads to
cΛ−
β (w) ≤ γ
1
2 (2−1δ, β)c−, c− = β(4πβ)−
d
4 ‖
√
h0‖1. (5.8)
From the the inequality |w(τ) − w′(τ)| ≥ 2−1δ we derive for w ∈ Λ(δ) employing once
more the Schwartz inequality for the measure P βx,x(dw
′)
cΛ+
β (w) ≤ δ−ε
β∫
0
dτ
∫
dx
[∫
P βx,x(dw
′)
] 1
2
[G′(w(τ);x)]
1
2 ,
where G′ is derived from h′ in the same way as G from h, h′(x) = h(x), |x| ≤ δ
2
;
h′(x) = h̄|x|−d−ε, |x| ≥ δ
2
, δ ≥ 2R. As a result with the help of (5.2b) we derive
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 993
cΛ+
β (w) ≤ δ−εc+, c+ = β(4πβ)−
d
4 ‖
√
h′0‖1, w ∈ Λ(δ), (5.9)
where h′0 is determined in the same way as h0. Repeating the above arguments we ob-
tain for w ∈ Λ(δ) the bound cΛ+
∗β;σ(w) ≤ δ−ε
(∑d′
l=1
l−θ
) ∫
dx
[∫
P βx,x(dw
′)
]1
2
×
×
[∫ β
0
G′(w(τ);x)dτ
] 1
2
. As a result
cΛ+
∗β (w) ≤ δ−εc+∗ , c+∗ = (4πβ)−
d
4
d′∑
l=1
l−θ
|h′∗|0, w ∈ Λ(δ), (5.10)
where h′∗ is determined in the same way as h∗. Hence c+∗ is finite. Let r(ε) = max
{
δ−ε×
×γ 1
2 (2−1δ, β)
}
. Then combining (5.7) – (5.10) we conclude that (1.8) holds withC ′(β) =
= c++r(ε)c−+16
√
2
(∑d′
l=1
l−θ
)
(c+∗ +r(ε)c−∗ ). This ends the proof of Theorem 1.1.
6. Classical correlation functions. For classical systems we have the following
simple analog of (2.1):
e−βU
′(x(n)) =
∫
e−i
√
βU ′(x(n);s(n))µ0(ds(n)), (6.1)
where the integration is performed over R
d′n,
µ0(ds) = (
√
2π)−d
′
e−2−1‖s‖2
ds, ‖s‖2 = (s, s) =
d′∑
l=1
s2l ,
U ′(x(n); s(n)) =
√
2
n∑
j1=1
sj1 ,
∑
0≤j2≤n,j2 �=j1
φ′(xj2 − xj1)
=
=
√
2
∑
1≤k<j≤n
(φ′(xj − xk), sj + sk). (6.2)
In our formulas d′ may be infinite and we’ll omit it in our formulas. Details about ex-
istence of the Gaussian measure on R
∞ and finite character of ηj a reader may find
in [8]. The equality (6.1) reduces the classical Gibbs system with the potential en-
ergy U to the Gibbs particle system with with an additional R
ν-valued degree of free-
dom, smeared by the measure µ0, and a pair complex interaction potential φ(x; s, s′) =
= φ0(x) + i
√
2β−1(φ′(x), s+ s′). The function (s, φ(x)) is a limit of finite sums in the
topology of L2(R∞, µ0) if d′ is infinite.
Now we have to deal with the KS equations (1.5), (1.6) determined by (1.12), (1.14)
with Ω = R
d′+d, ω = (x, s), µ(dω) = µ0(ds)dx. The most natural space for a descrip-
tion of the KS operator is the Banach space Ẽξ of sequences F =
{
F (x(n); s(n))
}
n≥1
of
measurable functions with the following norm:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
994 W. I. SKRYPNIK
‖F‖ξ = max
n≥1
ξ−n ess sup
x(n),s(n)
exp
−
n∑
j=1
‖s‖1
∣∣F (x(n), s(n))
∣∣,
‖s‖2
1 =
d′∑
l=1
l−2|sl|2.
(6.3)
Our main result for the classical systems is formulated as follows.
Theorem 6.1. If the conditions of Theorem 1.4 are satisfied, ηj =
∫
‖s‖j1e‖s‖1 ×
×µ0(ds), then the KS operator K̃ is bounded in Ẽξ if 1 − 2
√
2βη0‖φ′‖ξ ≥ 0 and its
norm ‖K̃‖ξ is given by ‖K̃‖ξ ≤ ξ−1e2βB+ξCη .
Proof. From the definition of the R-symmetrized KS operator and (1.12) – (1.14) we
obtain
‖K̃F‖ξ ≤ ess sup
x,s
ξ−1e−‖s‖1+2βB
∑
n≥0
ξn
n!
∫ ∣∣K(x; s|(x′(n), s
′
(n))
∣∣dx′(n)µ1(ds′(n)) ‖F‖ξ,
where
µ1(ds) = e‖s‖1µ0(ds),
K(x; s|x′(n), s
′
(n)) =
n∏
j=1
(
e−βφ0(x−x′j)−i
√
2β(φ(x−xj),s+s
′
j) − 1
)
.
We’ll need further to employ the following relation |eb+ia − 1| ≤ |eb − 1| + 2|a|.
Putting b = −βφ0(x− x′j), a = −
√
2β (φ′(x− x′j), s+ s′j) one derives with the help of
the Schwartz inequality∫ ∣∣K(x; s|x′j , s′j)
∣∣µ1(ds′j)dx
′
j ≤ η0‖e−βφ0 − 1‖1+
+ 2
√
2β
[∫ ∣∣(φ′(x′j), s)
∣∣dx′jµ1(ds′j) +
∫ ∣∣(φ′(x′j), s
′
j)
∣∣dx′jµ1(ds′j)
]
≤
≤ η0‖e−βφ0 − 1‖1 + 2
√
2β ‖φ′‖(η0‖s‖1 + η1).
This results in∫ ∣∣K(x; s|x′(n), s
′
(n))
∣∣dx′(n)µ1(ds′(n)) ≤
(
Cη + 2
√
2βη0 ‖φ′‖ ‖s‖1
)n
,
where ‖K̃‖ξ = ξ−1 max
s
e2βB+ξCη−‖s‖1(1−2
√
2βη0‖φ′‖ξ). The theorem is proved.
Hence we came to the following conclusion.
Corollary 6.1. If the conditions of Theorem 1.4 are satisfied and 1 − 2
√
2βη0 ×
×‖φ′‖ξ ≥ 0 then the sequence ρ =
∑
n≥0
zn+1K̃nα belongs to the Banach space Ẽξ
if |z| ≤ ξe−β(2B+Cηξ) and is the unique solution of the R-symmetrized KS equation in
the space. If φ0 is nonnegative then the sequence
ρ =
∑
n≥0
zn+1Knα
is the unique solution of the KS equation (1.6) in the same Banach space and the same
values of z with B = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
ON GIBBS QUANTUM AND CLASSICAL PARTICLE SYSTEMS WITH THREE-BODY FORCES 995
Taking into account that (K̃nα)(x(m)) = 0, n < m− 1 it is seen that
ρ(x(m); s(m)) =
∑
n≥m−1
zn+1(K̃nα)(x(m); s(m)).
Thus the following conclusion holds if one takes into account that ‖α‖ξ = ξ−1.
Corollary 6.2. Functions ρn from Theorem 1.1 are given by
ρn(x(m)) =
∫
(K̃m+n−1α)(x(m); s(m))µ0(ds(m))
and satisfy the inequality
∣∣ρn(x(m))
∣∣ ≤ ξ−1‖K̃‖m+n−1
ξ (η0ξ)m ≤ ηm0 (eβ(2B+Cηξ))n+m−1ξ−n.
If φ0 is nonnegative then the equality and inequality hold after substituting K instead of
K̃ and putting B = 0, respectively, in them.
Lemma 6.1. Let the conditions of Theorem 1.4 hold, 1 − 2
√
2βη0‖φ′‖ξ > 0 and
Λ ⊂ Λ′ ⊂ Λ′′ and δ be the distance of Λ to the boundary of Λ′ then
∥∥χΛK̃(χΛ′′ − χΛ′)
∥∥
ξ
≤ Cη,δ(ξ)
where 0 < Cη,δ(ξ) tends to zero if δ tends to infinity.
Proof. Applying the following relations:
χΛ(x(m))(χΛ′′(x(m), x
′
(n)) − χΛ′(x(m), x
′
(n))) = χΛ(x(m))(χΛ′′(x′(n)) − χΛ′(x′(n))),
0 ≤ χΛ′′(x′(n)) − χΛ′(x′(n)) ≤
n∑
j=1
(1 − χΛ′(x′j))
one obtains, taking into account that the considered functions are symmetric, the inequal-
ities
χΛ(x)χΛ(x(m−1))
∫ ∣∣K(x; s|x′(n), s
′
(n))
∣∣(χΛ′′(x(n)) − χΛ′(x(n))
)
×
×
∣∣F (x(m−1), x
′
(n); s(m−1), s
′
(n))
∣∣dx′(n)µ0(ds′(n)) ≤
≤ nξm+n−1‖F‖ξe
∑ m−1
j=1 ‖sj‖1
(∫
dx′µ1(ds′)
∣∣K(x, s|x′s′)
∣∣)n−1
×
×ess sup
x
χΛ(x)
∫
dx′
(
1 − χΛ′(x′)
)
µ1(ds′)
∣∣K(x, s|x′s′)
∣∣ ≤
≤ nξm+n−1‖F‖ξe
∑ m−1
j=1 ‖sj‖1
(
Cη + 2
√
2βη0‖φ′‖ ‖s‖1
)n−1×
×
η0
∫
|x|≥δ
∣∣∣e−βφ0(x) − 1
∣∣∣ dx+ 2
√
2β(η0‖s‖1 + η1)
∫
|x|≥δ
∣∣φ′(x)
∣∣
2
dx
.
As a result, we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
996 W. I. SKRYPNIK
∥∥χΛK̃(χΛ′′ − χΛ′)
∥∥
ξ
≤ eβB+ξCηe−‖s‖1(1−2
√
2βη0‖φ′‖ξ)×
×
η0
∫
|x|≥δ
|e−βφ0(x) − 1|dx+ 2
√
2β
(
η0‖s‖1 + η1
) ∫
|x|≥δ
∣∣φ′(x)
∣∣
2
dx
.
That is
Cη,δ(ξ) =
= e2βB+ξCη
η0
∫
|x|≥δ
|e−βφ0(x) − 1|dx+ 2
√
2β(η0θξ + η1)
∫
|x|≥δ
|φ′(x)|2dx
,
where θξ =
(
max
q≥0
qe−q
) (
1 − 2
√
2βη0‖φ′‖ξ
)−1
. The lemma is proved.
Remark 6.1. A general potential energy is represented as
U(x(n)) =
∑
k
∑
1≤j1 �=j2... �=jk≤n
φk(xj2 − xj1 , xj3 − xj1 , . . . xjk − xj1),
where φk are k-particle potentials.
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Received 30.08.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
|
| id | umjimathkievua-article-3508 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:51Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/02/17cf5010396e33d05c6d206897dac502.pdf |
| spelling | umjimathkievua-article-35082020-03-18T19:56:18Z On Gibbs quantum and classical particle systems with three-body forces Про гібсівсьські квантові та класичні системи частинок з тричастинковими силами Skrypnik, W. I. Скрипник, В. І. For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit. Для рівноважних квантових та класичних систем частинок, що взаємодіють завдяки тернарному i парному (непозитивним) далекосяжним потенціалам, побудовано кластерний розклад для їх редукованих матриць щільності та кореляційних функцій великого канонічного ансамблю, збіжний при низьких активностях у термодинамічній границі. Institute of Mathematics, NAS of Ukraine 2006-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3508 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 7 (2006); 976–996 Український математичний журнал; Том 58 № 7 (2006); 976–996 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3508/3753 https://umj.imath.kiev.ua/index.php/umj/article/view/3508/3754 Copyright (c) 2006 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. On Gibbs quantum and classical particle systems with three-body forces |
| title | On Gibbs quantum and classical particle systems with three-body forces |
| title_alt | Про гібсівсьські квантові та класичні системи частинок з тричастинковими силами |
| title_full | On Gibbs quantum and classical particle systems with three-body forces |
| title_fullStr | On Gibbs quantum and classical particle systems with three-body forces |
| title_full_unstemmed | On Gibbs quantum and classical particle systems with three-body forces |
| title_short | On Gibbs quantum and classical particle systems with three-body forces |
| title_sort | on gibbs quantum and classical particle systems with three-body forces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3508 |
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