On lattice oscillator equilibrium equation with positive infinite-range many-body potentials
The symmetrized lattice Kirkwood-Salsburg (KS) equation for the Gibbs grand canonical correlation functions of the lattice oscillators, interacting via positive infinite-range manybody potentials, is solved. The symmetrization is based on the superstability condition for the potentials. Розв'яз...
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| Cite this: | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials / W.I. Skrypnik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43702:1-6. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials / W.I. Skrypnik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43702:1-6. — Бібліогр.: 10 назв. — англ. |
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| description | The symmetrized lattice Kirkwood-Salsburg (KS) equation for the Gibbs grand canonical correlation functions of the lattice oscillators, interacting via positive infinite-range manybody potentials, is solved. The symmetrization is based on the superstability condition for the potentials.
Розв'язано симетризоване граткове рівняння Кірквуда - Зальзбурга для гіббсівських великоканонічних кореляційних функцій граткових осциляторів, що взаємодіють через позитивні нескінченносяжні багаточастинкові потенціали. Симетризація базується на умові суперстійкості для потенціалів.
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Condensed Matter Physics 2010, Vol. 13, No 4, 43702: 1–6
http://www.icmp.lviv.ua/journal
On lattice oscillator equilibrium equation with positive
infinite-range many-body potentials
W.I. Skrypnik∗
The Institute of Mathematics, Tereshchenkivska Str. 3, Kyiv, Ukraine
Received October 1, 2010
The symmetrized lattice Kirkwood-Salsburg (KS) equation for the Gibbs grand canonical correlation functions
of the lattice oscillators, interacting via positive infinite-range manybody potentials, is solved. The symmetriza-
tion is based on the superstability condition for the potentials.
Key words: lattice oscillators, Gibbs grand canonical ensemble, superstability, Kirkwood-Salsburg equation
PACS: 71.38.-k, 71.38.Fp
We consider Gibbsian (equilibrium) systems of oscillators, whose one-dimensional coordinates
qx ∈ R are indexed by sites x of the hyper-cubic lattice Z
d with the potential energy (see also [1])
Uc(qΛ) =
∑
x∈Λ
u(qx) + U(qΛ), U(qΛ) =
∑
|X|>1,X⊆Λ
φX(qX),
where the summation is performed over the one-point sets and sets with greater then 1 number of
sites in the first and second sums, respectively, qX = (qx, x ∈ X ⊆ Λ), u is an external potential,
φX is a |X |-body positive interaction potential, Λ ⊂ Z
d and the number of sites in Λ is finite, that
is |Λ| < ∞. States of these systems are described in the thermodynamic limit by the sequence ρ
of grand canonical correlation functions ρ = {ρ(qX), X ⊂ Z
d, 1 6 |X | < ∞} satisfying the lattice
oscillator Kirkwood-Salsburg (KS) equation (we derive it in the Appendix)
ρ = zKρ+ zα,
where z is the activity(a thermodynamic parameter), α(qX) = δ|X|,1 = 0, |X | 6= 1, δ|X|,1 = 1, |X | =
1. The linear KS operator K is defined on sequences of measurable functions F = {F (qX), X ⊂
Z
d, 1 6 |X | < ∞}as follows
(KF )(qX) =
∑
Y⊆Xc
∫
K(qx|qX\x; qY )
[
F (qX\x∪Y )−
∫
ν(dqx)F (qX∪Y )
]
ν(dqY ), 1 < |X | < ∞,
where the integrations are performed over R and R
|Y | in the integral within the square brackets
and the other, respectively,
ν(dqY ) =
∏
y∈Y
ν(dqy), ν(dq) = e−βu(q)
(∫
e−βu(q)dq
)−1
dq, Xc = Z
d\X,
and β > 0 is the inverse temperature. If X = x, then the first term in the square bracket is missing.
The KS kernels are defined as follows
K(qx|qX\x; qY )=e−βW (qx|qX\x)
|Y |
∑
n=1
∑
∪Yj=Y,Yj 6=∅
n
∏
j=1
(
e−βW (qX ;qYj
|x) − 1
)
=e−βW (qx|qX\x)Kx(qX ; qY ),
∗E-mail: wolodymyr.skypnyk@ukr.net
c© W.I. Skrypnik 43702-1
http://www.icmp.lviv.ua/journal
W.I. Skrypnik
W (qx|qY ) = U(qx, qY )− U(qY ) > 0, W (qX ; qY |x) =
∑
x∈Z⊆X
uZ∪Y (qZ∪Y ).
Where the summation in the expression for the KS kernel is performed over n subsets whose union
is Y . We will also demand that the superstability condition [2] for positive potentials should hold
uY (qY ) 6 JY
∑
y∈Y
v(qy), N0 =
∫
eβγv(q)ν(dq) < ∞, γ > 0,
||J ||1 = max
x
∑
Y,x∈Y
JY < ∞ and the summation is performed over subsets of Zd containing a site x.
Usually the considered systems are described by canonical correlation functions [3, 4] which
are determined in the high-temperature regime by convergent cluster expansions. One hopes to
show that they satisfy the KS equation by analogy with correlation functions of particle systems
[5]. It is well known that the canonical ensemble correlation functions generate vacuum averages
in the lattice boson Euclidean quantum field theory for the case of the nearest neighbor bilinear
pair potential [6, 7].
The KS equation for an integer valued spin Ising model with a pair potential, whose solutions
describe vacuum field averages in the two-dimensional lattice Higgs-Villain model, appeared earlier
in [8]. The case of unbounded spins (oscillator variables sometimes are treated as unbounded
continuous spins) is more complicated than the case of bounded spins or lattice gas and the results
concerning solutions of the KS operator for the latter, exposed in [9], cannot be easily generalized
to unbounded spins either for infinite-range manybody or non-positive potentials (this conclusion
follows from [8]).
In [1] we considered the case of finite-range positive manybody potentials and super-stable
pair potentials with a different representation of the KS kernels and showed that for positive
pair potentials the KS equation is easily solved. For a non-positive pair potential we proposed to
symmetrize the KS operator (with respect to the super-stability condition) and established that
the symmetrized KS equation (the KS equation with the symmetrized KS operator) can be solved
proving that the symmetrized KS operator is bounded in a natural Banach space Ef,ξ which will
be used by us in this paper. But our method failed for infinite-range manybody potentials. Starting
with the above representation for the KS kernels we are able to derive in this paper the basic bound
K̄x(qX) 6 exp {ξβ(c1 + c2v(qx) +W ′(x|qX))} , (1)
where c1, c2 do not depend on oscillator variables and lattice sites and
K̄x(qX) =
∑
Y⊆Xc
ξ|Y |
∫
|K(qx|qX\x; qY )|ν
′(dqY ), ν′(dqY ) = exp
{
∑
x∈Y
f(qx)
}
ν(dqY ),
W ′(x|qX) =
∑
y∈X\x
v(qy)
∑
Y ⊆(x∪y)c
Jx∪y∪Y (1 +N0)
|Y |.
Note that positivity of W permits to substitute Kx(qX ; qY ) into the expression for the kernel
K̄x(qX) instead of Kx(qX ; qY ). W
′ satisfies the following remarkable inequality
∑
x∈X
W ′(x|qX) =
∑
x∈X
v(qx)
∑
y∈X\x
∑
Y⊆(x∪y)c
Jy∪x∪Y (1 +N0)
|Y |
6
∑
x∈X
v(qx)
∑
y∈(x)c
∑
Y⊆(x∪y)c
Jy∪x∪Y (1 +N0)
|Y |
6 |J |1
∑
x∈X
v(qx), (2)
where
|J |l = max
x
|J |l(x), |J |l(x) =
∑
Z⊆(x)c
Jx∪Z(1 +N0)
|Z|(|Z|+ 1)l−1, l > 1.
43702-2
On lattice oscillator equilibrium equation
The basic bound (1) permits to symmetrize the KS operator and prove that the symmetrized oper-
ator is bounded in the Banach space Ef,ξ which is the linear space of sequences F = {FX(qX), X ⊂
Z
d, 1 6 |X | < ∞} of measurable functions with the norm
||F ||f,ξ = max
X
ξ−|X|ess sup
qX
exp
{
−
∑
x∈X
f(qx)
}
|FX(qX)|, f(q) = γβv(q).
Let χx(qX) be the characteristic(indication) function of the set Dx where the inequality
W ′(x|qX) 6 |J |1v(qx) (2′)
holds. Then (2) implies that ∪x∈XDx = R
|X| or
∑
x∈X
χx(qX) > 1
since Dx may intersect for different x. It is more convenient to deal with χ∗
x
χ∗
x(qX) =
∑
y∈X
χy(qX)
−1
χx(qX),
∑
x∈X
χ∗
x(qX) = 1. (3)
The symmetrized KS operator K̃ is given by
(K̃F )(qX) =
∑
x∈X
χ∗
x(qX)
∑
Z⊆Xc
∫
K(qx|X\x; qZ)
[
F (qX\x∪Z)−
∫
ν(dqx)F (qX∪Z)
]
ν(dqZ),
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 (4)
is derived after multiplying both sides of the KS equation by the characteristic function χ∗
x(qX)
and applying (3). Our main result is formulated in the following theorem.
Theorem. Let γ > 2ξ|J |1, G(ξ) = ξN ′
0|J |2, where N ′
0 = N−1
0
∫
v(q)ν′(dq). Then the norm
of the symmetrized KS operator in the Banach space Ef,ξ satisfies the following bound ||K̃||f,ξ 6
(ξ−1 +N0)e
βG(ξ) and the vector ρ from the space Ef,ξ
ρ =
∑
n>0
zn+1K̃nα
determines the unique solution of the symmetrized KS equations in Ef,ξ if |z| < ||K̃||−1
f,ξ.
Proof. If the basic bound (1) holds then the proof is almost trivial since for the norm of the
KS operator we have the following inequality
||K̃||ξ,f 6 (ξ−1 +N0)max
X
ess sup
qX
K̄(qX |f), K̄(qX |f) =
∑
x∈X
χ∗
x(qX)e−f(qx)K̄x(qX).
As a result of (2′) and (3) one obtains for γ > ξ(c2 + |J |1)
K̄(qX |f) 6 ec1ξβess sup
q
exp {−γβv(q) + βξ(c2 + |J |1)v(q)} = ec1ξβ .
The most simple choice is ξ = (βc1)
−1. The theorem will be proved if we prove the basic bound
(1) and show that c1 = N ′
0|J |2, c2 = |J |1.
43702-3
W.I. Skrypnik
Proof of the basic bound (1). We have the following inequalities which are analogues of the
inequalities for the KS kernels for the lattice gas systems from [9]
K̄x(qX) 6
∑
Y⊆Xc
ξ|Y |
|Y |
∑
l=1
∑
∪Yj=Y
l
∏
j=1
∫
|e−βW (qX ;qYj
|x) − 1|ν′(dqYj
)
=
∑
n>0
ξn
∑
|Y |=n,Y⊆Xc
n
∑
l=1
∑
∪Yj=Y
l
∏
j=1
∫
|e−βW (qX ;qYj
|x) − 1|ν′(dqYj
)
6
∑
n>0
ξn
n!
∑
Y⊆Xc
∫
|e−βW (qX ;qY |x) − 1|ν′(dqY )
n
= exp
ξ
∑
Y⊆Xc
∫
|e−βW (qX ;qY |x) − 1|ν′(dqY )
.
Hence, for positive potentials uY > 0 one derives the following estimate
K̄x(qX) 6 exp
ξβ
∑
Y⊆Xc
∫
|W (qX ; qY |x)|ν
′(dqY )
. (5)
Moreover,
|W (qX ; qY |x)| 6
∑
x∈Z⊆X
|uZ∪Y (qZ∪Y )| 6
∑
x∈Z⊆X
JZ∪Y
∑
y⊆Z∪Y
v(qy)
=
∑
x∈Z⊆X
JZ∪Y
∑
y∈Y
v(qy) +
∑
y∈Z\x
v(qy) + v(qx)
.
Then the last inequality yields
∫
|W (qX ; qY |x)|ν
′(dqY ) 6 N
|Y |
0
∑
Z⊆X\x
Jx∪Z∪Y
N ′
0|Y |+
∑
y∈Z\x
v(qy) + v(qx)
= N
|Y |
0
∑
Z⊆X\x
Jx∪Z∪Y [N ′
0|Y |+ v(qx)] +N
|Y |
0
∑
y∈X\x
v(qy)
∑
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).
As a result
∑
Y⊆Xc
∫
|W (qX ; qY |x)|ν
′(dqY ) 6 N ′
0|J |2 + |J |1v(qx) +W ′(x|qX).
The last inequality and (5) prove the basic bound. The theorem is proven.
The analogue of the theorem can be easily proven for the quantum lattice oscillator KS equation
generalizing the result of [10] where only finite-range manybody potentials (special non-positive)
were considered. The theorem can be also easily generalized to the classical case of special non-
positive infinite-range manybody potentials from [10].
43702-4
On lattice oscillator equilibrium equation
1. Appendix
To derive the KS equation one has to start from the following expression for the grand canonical
correlation functions in a compact set Λ
ρΛ(qX) = χΛ(X)Ξ−1
Λ
∑
Y⊆Λ\X
z|Y∪X|
∫
ν(dqY )e
−βU(qX∪Y ),
where χΛ is the characteristic function of Λ, the grand partition function ΞΛ coincides with the nu-
merator of the right-hand side of the empty setX . The usual Gibbs correlation functions are derived
by multiplying the righthand side of the last equality by the exp{−β
∑
x∈Λ
u(qx)}(
∫
e−βu(q)dq)−|X|
and renormalizing the activity by the multiplier
∫
e−βu(q)dq. We have the equality
U(qX∪Y ) = U(qX∪Y \x) +W (qx|qX\x∪Y ). (6)
In order to derive the finite-volume KS equation one has to represent the exponent of
−βW (qx|qX\x∪Y ) in terms of the KS kernels. Let
W̃ (qX ; qY |x) =
∑
x∈Z⊆X
∑
∅6=S⊆Y
uZ∪S(qZ∪S) =
∑
∅6=S⊆Y
W (qX ; qS |x).
Then
W (qx|qX\x, qY ) = W (qx|qX\x) + W̃ (qX ; qY |x),
and
e−βW̃(qX ;qY |x) =
∏
∅6=S⊆Y
(
1 +
(
e−βW (qX ;qS |x) − 1
))
=
∑
S⊆Y
Kx(qX ; qS), Kx(qX ; q∅) = 1.
Then, using (6) and substituting this equality into the expression of the finite volume grand canon-
ical correlation functions one obtains
ρΛ(qX) = Ξ−1
Λ χΛ(X)
∑
Y⊆Λ\X
z|Y∪X|
∫
ν(dqY )e
−βU(qX∪Y \x)
∑
S⊆Y
K(qx|qX\x; qS)
= Ξ−1
Λ χΛ(X)
∑
Y⊆Λ\X
z|Y∪X|
∑
S⊆Y
∫
ν(dqY )K(qx|qX\x; qS)e
−βU(qX∪Y \x)
= z
∑
Z⊆Λ\X
∫
ν(dqZ)K(qx|qX\x; qZ)Ξ
−1
Λ χΛ(X ∪ Z)
×
∑
Y ⊆Λ\(Z∪X)
z|Y∪X∪Z|−1
∫
ν(dqY )e
−βU(qX\x,qY ).
The equality
ρΛ(qX\x) = Ξ−1
Λ χΛ(X\x)
∑
Y⊆(Λ\X)∪x
z|Y ∪X|−1
∫
ν(dqY )e
−βU(qX\x,qY )
leads to
Ξ−1
Λ χΛ(X ∪ Z)
∑
Y⊆Λ\(Z∪X)
z|Y∪X∪Z|−1
∫
ν(dqY )e
−βU(qX\x,qY )
= χΛ(x)(ρ
Λ(qX\x∪Z)−
∫
ν(dqx)ρ
Λ(qX∪Z)).
43702-5
W.I. Skrypnik
It is 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. That is,
the KS equation is given for x ∈ X, |X | > 1 by
ρΛ(qX) = zχΛ(x)
∑
Z⊆Λ\X
∫
K(qx|qX\x; qZ)
[
ρΛ(qX\x∪Z)−
∫
ν(dqx)ρ
Λ(qX∪Z)
]
ν(dqZ )
and for X = x by
ρΛ(qx) = zχΛ(x)
{
1−
∫
ρΛ(qx)ν(dqx)
+
∑
|Z|>1,Z⊆Λ\x
∫
K(qx|qZ)
[
ρΛ(qZ)−
∫
ν(dqx)ρ
Λ(qZ∪x)
]
ν(dqZ)
}
.
The infinite volume KS equation is derived if one puts Λ = Z
d.
References
1. Skrypnik W., Ukrainian Math. Journ., 2008, 60, No. 10, 1427–1433.
2. Ruelle D., Commun. Math. Phys., 1976, 50, 189–194.
3. Kunz H., Commun. Math. Phys., 1978, 59, No. 1, 53–69.
4. Park Y.M., Yoo H.J., J. Stat.Phys., 1995, 80, No. 1/2, 223–271.
5. Bogolubov N.N., Petrina D.Ya., Khatset B.I., TMF, 1969, 1, 251–274.
6. Simon B., Euclidean P (φ)2 Quantum Field Theory. Princeton University Press, Princeton, New Jersey,
1974.
7. Glimm J., Jaffe A., Quantum Physics. A Functional Integral Point of View. Springer-Verlag, New
York, Heidelberg, Berlin, 1981.
8. Israel R., Nappi C., Commun. Math. Phys., 1979, 64, No. 2, 177–189.
9. Ruelle D., Statistical Mechanics. Rigorous Results. W.A. Benjamin Inc., New York, Amsterdam, 1969.
10. Skrypnik W., Ukrainian Math. Journ., 2009, 61, No. 5, 689–700.
До рiвняння рiвноваги ґраткового осцилятора з позитивними
нескiнченносяжними багаточастинковими потенцiалами
В.I. Скрипнiк
Iнститут математики, Київ, Україна
Розв’язано симетризоване ґраткове рiвняння Кiрквуда-Зальзбурга для гiббсiвських великоканонiч-
них кореляцiйних функцiй ґраткових осциляторiв, що взаємодiють через позитивнi нескiнченносяжнi
багаточастинковi потенцiали. Симетризацiя ґрунтується на умовi суперстiйкостi для потенцiалiв.
Ключовi слова: ґратковi осцилятори, великий канонiчний ансамбль Гiббса, суперстiйкiсть,
рiвняння Кiрквуда-Зальзбурга
43702-6
Appendix
|
| id | nasplib_isofts_kiev_ua-123456789-32131 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-28T04:28:16Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Skrypnik, W.I. 2012-04-09T21:28:15Z 2012-04-09T21:28:15Z 2010 On lattice oscillator equilibrium equation with positive infinite-range many-body potentials / W.I. Skrypnik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43702:1-6. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 71.38.-k, 71.38.Fp https://nasplib.isofts.kiev.ua/handle/123456789/32131 The symmetrized lattice Kirkwood-Salsburg (KS) equation for the Gibbs grand canonical correlation functions of the lattice oscillators, interacting via positive infinite-range manybody potentials, is solved. The symmetrization is based on the superstability condition for the potentials. Розв'язано симетризоване граткове рівняння Кірквуда - Зальзбурга для гіббсівських великоканонічних кореляційних функцій граткових осциляторів, що взаємодіють через позитивні нескінченносяжні багаточастинкові потенціали. Симетризація базується на умові суперстійкості для потенціалів. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On lattice oscillator equilibrium equation with positive infinite-range many-body potentials До рівняння рівноваги граткового осцилятора з позитивними нескінченносяжними багаточастинковими потенціалами Article published earlier |
| spellingShingle | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials Skrypnik, W.I. |
| title | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| title_alt | До рівняння рівноваги граткового осцилятора з позитивними нескінченносяжними багаточастинковими потенціалами |
| title_full | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| title_fullStr | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| title_full_unstemmed | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| title_short | On lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| title_sort | on lattice oscillator equilibrium equation with positive infinite-range many-body potentials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32131 |
| work_keys_str_mv | AT skrypnikwi onlatticeoscillatorequilibriumequationwithpositiveinfiniterangemanybodypotentials AT skrypnikwi dorívnânnârívnovagigratkovogooscilâtorazpozitivnimineskínčennosâžnimibagatočastinkovimipotencíalami |