Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions
The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operato...
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Institute of Mathematics, NAS of Ukraine
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| author | Caraffini, G. L. Petrina, D. Ya. Цараффіні, Г. Л. Петрина, Д. Я. |
| author_facet | Caraffini, G. L. Petrina, D. Ya. Цараффіні, Г. Л. Петрина, Д. Я. |
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| datestamp_date | 2020-03-18T19:55:07Z |
| description | The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operators. The infinitesimal generator of the semigroup coincides on a certain everywhere dense set with the operator on the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data, they are generalized ones. |
| first_indexed | 2026-03-24T02:42:56Z |
| format | Article |
| fulltext |
UDC 517.9+531.19
D. Ya. Petrina (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
G. L. Caraffini (Univ. Parma, Italy)
SOLUTIONS OF THE BBGKY HIERARCHY
FOR A SYSTEM OF HARD SPHERES
WITH INELASTIC COLLISIONS∗
ROZV’QZKY I{RARXI} BBHKI DLQ SYSTEMY
TVERDYX KUL\ IZ NEPRUÛNYM ROZSIQNNQM
The problem of existence of solutions of the hierarchy for the sequence of correlation functions is investigated
in direct sum of spaces of summable functions. It is proved existence and uniqueness of solutions which are
represented through a semigroup of bounded strongly continuous operators.
Infinitesimal generator of the semigroup coincides, on certain everywhere dense set, with the operator in
the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data,
they are generalized ones.
DoslidΩeno problemu isnuvannq rozv’qzkiv i[rarxi] dlq poslidovnosti korelqcijnyx funkcij pry
poçatkovyx danyx z prqmo] sumy prostoriv intehrovnyx funkcij. Dovedeno isnuvannq ta [dynist\
rozv’qzkiv, podanyx çerez pivhrupu obmeΩenyx syl\no neperervnyx operatoriv.
Infinitezymal\nyj operator pivhrupy zbiha[t\sq na pevnij skriz\ wil\nij mnoΩyni z operatorom,
wo vyznaça[ pravu çastynu i[rarxi]. Dlq poçatkovyx danyx z ci[] mnoΩyny rozv’qzky [ strohymy, dlq
zahal\nyx poçatkovyx danyx — uzahal\nenymy.
Introduction. We consider the analogue of the BBGKY hierarchy for systems of hard
spheres with inelastic collisions. It is commonly accepted that these systems are proper
models of granular flow. We continue the investigation of these systems that have been
begun in our paper [1]. We use the same denotation.
In paper [1] the Liouville equation for distribution functions of systems of finite num-
bers N of hard spheres with inelastic collisions has been investigated. It has been proved
that the distribution function is defined as follows:
DN (t, (x)N ) =
[
∂X(−t, (x)N )
∂(x)N
]2
SN (−t, (x)N )D(0, (x)N ), N ≥ 1, (1)
where D(0, (x)N ) is the initial distribution function, SN (−t, (x)N ) is the operator of shift
along the trajectory X(−t, (x)N ) of N spheres with initial data (x)N ,
∂X(−t, (x)N )
∂(x)N
is
the Jacobian. It has been shown that only distribution functions (1), with squared Jaco-
bian, satisfy the law of conservation of full probability∫
DN (t, (x)N )d(x)N =
∫
DN (0, (x)N )dxN , (2)
the Liouville equation
∂
∂t
DN (t, (x)N ) = −
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ), (3)
and specific boundary conditions according to which at qi − qj = aη, 〈η, (pi − pj)〉 >
> 0, (|η| = 1, a — diameter of hard spheres) momenta pi, pj should be replaced by the
following ones:
p∗i = pi +
ε
1 − 2ε
η〈η, (pi − pj)〉, p∗j = pj −
ε
1 − 2ε
η〈η, (pi − pj)〉, (4)
∗ This paper has been completed during stay of D. Ya. Petrina in November 2004 in Dipartimento di
Matematica, Università di Parma.
c© D. YA. PETRINA, G. L. CARAFFINI, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3 371
372 D. YA. PETRINA, G. L. CARAFFINI
and
DN (−t, x1, . . . , qi, pi, . . . , qi − aη, pj , . . . , xN ) =
=
1
(1 − 2ε)2
DN (−t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN ). (5)
The parameter ε, characterizes inelasticity,
1
2
< ε < 1. The corresponding sequence of
correlation functions satisfies the following analogue of the BBGKY hierarchy
∂
∂t
ρs(t, (x)N ) = −
s∑
i=1
pi
∂
∂qi
ρs(t, (x)s) + a2
s∑
i=1
∫
dps+1
∫
S+
2
dη〈η, (pi − ps+1)〉×
×
[
1
(1 − 2ε)2
ρs+1(t, x1, . . . , qi, p
∗
i , . . . , xs, qi − aη, p∗s+1)−
−ρs+1(t, x1, . . . , qi, pi, . . . , xs, qi + aη, ps+1)
]
, s ≥ 1, (6)
with the same boundary condition in the first term on the right-hand side of (6) as those
(4), (5) for DN (t, (x)N ) and with the initial conditions
ρs(t, (x)s)|t=0 = ρs(0, (x)s), s ≥ 1. (7)
In given paper we consider hierarchy (6) with initial data (7), i.e., the Cauchy problem for
the hierarchy (6) in the Banach space L1 of sequences of integrable symmetric functions
f = (f1(x1), . . . , fs((x)s), . . .) (8)
equal to zero on forbidden configurations where |qi−qj | < a at least for one pair (i, j) ⊂
⊂ (1, . . . , s), with norm
‖f‖ =
∞∑
s=1
‖fs‖, ‖fs‖ =
∫
|fs((x)s)|d(x)s. (9)
The corresponding group of bounded and strongly continuous in L1 evolution operators
U(t), t ≥ 0, has been constructed
U(t) = e
∫
dxJ(−t)S(−t)e−
∫
dx, (10)
where ( ∫
dxf
)
s
(
(x)s
)
=
∫
dxs+1fs+1(x1, . . . , xs, xs+1) (11)
is a bounded operator in L1, ‖
∫
dx‖ ≤ 1, J(−t) is direct sum of the operators of multi-
plication by squared Jacobian
(J(−t)f)s((x)s) =
[
∂X(−t, (x)s)
∂(x)s
]2
fs((x)s)
and S(−t) is direct sum of the operators Ss(−t, (x)s) of shift along the trajectory
X(−t, (x)s)(
S(−t)f
)
s
(
(x)s
)
=
(
Ss(−t, (x)s)fs
)(
(x)s
)
= fs
(
X(−t, (x)s)
)
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
SOLUTIONS OF THE BBGKY HIERARCHY FOR A SYSTEM OF HARD SPHERES ... 373
It is proved that group U(t) is strongly differentiable on everywhere dense set L0
1 ⊂
⊂ L1 consisting of finite sequences f ∈ L1 of differentiable functions equal to zero in
some neighborhood of forbidden configurations and with compact support. The infinites-
imal generator B of the group U(t) coincides on L0
1 with the operator in the right-hand
side of the hierarchy (6).
Denoting by ρ(t) the sequence of correlation functions
(
ρ1(t, (x)1), . . . , ρs(t,
(x)s), . . .)
)
, we consider the hierarchy (6) as an abstract evolution equation in L1 for
the sequence ρ(t)
dρ(t)
dt
= Bρ(t), ρ(t)|t=0 = ρ(0), (12)
and show that the Cauchy problem (12) for the hierarchy (6) has the unique solution
ρ(t) = U(t)ρ(0), ρ(0) ∈ L1, (13)
that is strong for ρ(0) ∈ L0
1 and the generalized one for arbitrary ρ(0) ∈ L1.
1. Solution of hierarchy for correlation functions. 1.1. Formulae that express
ρ(t) through ρ(0). As known [1] the correlation functions defined by formulae
ρs(t, x1, . . . , xs) =
1
Ξ
∞∑
n=0
1
n!
∫ [
∂X(−t, x1, . . . , xs, xs+1, . . . , xs+n)
∂(x1, . . . , xs, xs+1, . . . , xs+n)
]2
×
×
[
Ss+n(−t, x1, . . . , xs, xs+1, . . . , xs+n)×
×Ds+n(0, x1, . . . , xs, xs+1, . . . , xs+n)
]
dxs+1 . . . dxs+n =
=
1
Ξ
∞∑
n=0
1
n!
∫ [
∂X(−t, (x)s+n)
∂(x)s+n
]2
×
×
[
Ss+n(−t, (x)s+n)Ds+n(0, (x)s+n)
]
d(x)s
s+n, s ≥ 1,
(1.1)
Ξ =
∞∑
n=0
1
n!
∫ [
∂X(−t, (x)n)
∂(x)n
]2
[Sn(−t, (x)n)Dn(0, (x)n)]d(x)n =
=
∞∑
n=0
1
n!
∫
Dn(0, (x)n)d(x)n
are formal solutions of the hierarchy
∂ρs(t, (x)s)
∂t
=
= −
s∑
i=1
pi
∂
∂qi
ρs(t, (x)s) + a2
s∑
i=1
∫
dps+1
∫
S2
+
dη〈η, (pi − ps+1)〉×
×
[
1
(1 − 2ε)2
ρs+1(t, q1, p1, . . . , qi, p
∗
i , . . . , qs, ps, qi − aη, p∗s+1)−
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
374 D. YA. PETRINA, G. L. CARAFFINI
−ρs+1(t, q1, p1, . . . , qi, pi, . . . , qs, ps, qi + aη, ps+1)
]
, s ≥ 1, (1.2)
with corresponding boundary and initial conditions. Recall that we use the same denota-
tion as in our previous paper [1]
(x)s+n = (x1, . . . , xs, xs+1, . . . , xs+n),
d(x)s
s+n = dxs+1 . . . dxs+n, |η| = 1, S+
2 (η|〈η, (pi − pj)〉 > 0),
Ss+n(−t, (x)s+n) is the operator of shift along the trajectory Xs+n(−t, (x)s+n),
Ds+n(0, (x)s) is the initial distribution function,
∂X(−t, (x)s+n)
∂(x)s+n
is the Jacobian, Ξ
is the grand partition function.
Correlation function satisfies the following boundary condition: at qi − qj = aη,
〈η, (pi − pj)〉 > 0 momenta pi and pj in the first term of the right-hand side of (1.2)
should be replaced by
p∗i = pi +
ε
1 − 2ε
η〈ηi, (pi − pj)〉, p∗j = pj −
ε
1 − 2ε
η〈ηi, (pi − pj)〉
in −
∑s
i=1
pi
∂
∂qi
and ρs(t, (x)s) should be replaced by
1
(1 − 2ε)2
ρs(t, x1, . . . , qi, p
∗
i , . . .
. . . , qi − aη, p∗i , . . . , xN ).
In (1.1) the correlation functions ρs(t, (x)s) are expressed by the initial distribu-
tion functions Ds+n(0, (x)s+n), n ≥ 0. We transform formulae (1.1) in such a way
that ρs(t, (x)s) will be expressed by the initial correlation functions ρs(0, (x)s+n), n ≥
≥ 0. To do it, we use (1.1) with t = 0, when S(0, (x)s+n) = I, I — unit operator,
∂X(0, (x)s+n)
∂(x)s+n
= 1. One obtains
ρs(0, (x)s) =
1
Ξ
∞∑
n=0
1
n!
∫
Ds+n(0, (x)s+n)d(x)s
s+n. (1.3)
Denote by ρ(0), D(0) and f the following sequences:
ρ(0) = (ρ1(0, x1), . . . , ρs(0, (x)s), . . .),
D(0) = (D1(0, x1), . . . , Ds(0, (x)s), . . .),
f = (f1(x1), . . . , fs((x)s), . . .),
(1.4)
and by
∫
dx the following operator:( ∫
dxf
)
s
((x)s) =
∫
fs+1((x)s, xs+1)dxs+1. (1.5)
Formulae (1.3) can be represented as follows:
ρ(0) =
1
Ξ
e
∫
dxD(0), ρs(0, (x)s) =
1
Ξ
(e
∫
dxD(0))s(x)s,
D(0) = Ξe−
∫
dxρ(0). (1.6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
SOLUTIONS OF THE BBGKY HIERARCHY FOR A SYSTEM OF HARD SPHERES ... 375
Denote by J(t), as in introduction, the direct sum of operators of multiplication of se-
quences (1.4) by
[
∂X(−t, (x)s)
∂(x)s
]2
(J(−t)f)s((x)s) =
[
∂X(−t, (x)s)
∂(x)s
]2
fs((x)s)
and by S(−t) the direct sum of operator Ss(−t, (x)s)
(S(−t)f)s((x)s) = (Ss(−t, (x)s)fs)(xs) = fs(X(−t, (x)s)).
In terms of these operators formulae (1.1) can be represented as follows:
ρ(t) =
1
Ξ
e
∫
dxJ(−t)S(−t)D(0),
ρs(t, (x)s) =
1
Ξ
(e
∫
dxJ(−t)S(−t)D(0))s(x)s.
(1.7)
Finally, expressing D(0) in (1.7) through ρ(0), according to (1.6), we have
ρ(t) = e
∫
dxJ(−t)S(−t)e−
∫
dxρ(0) = U(t)ρ(0),
U(t) = e
∫
dxJ(−t)S(−t)e−
∫
dx
(1.8)
or, component-wise,
ρs(t, (x)s) =
∞∑
n=0
n∑
k=0
(−1)k
(n− k)!k!
∫ [
∂X(−t, (x)s+n−k)
∂(x)s+n−k
]2
×
×Ss+n−k(−t, (x)s+n−k)ρs+n(0, (x)s+n)d(x)s
s+n.
U(t) is the evolution operator of hierarchy (1.2). These formulae have been obtained
on formal level. In the next subsection the justification of these formulae will be pre-
sented.
1.2. Convergence of series (1.8). Suppose that sequence f (1.4) consists of integrable
symmetric functions
∫ ∣∣fs((x)s)
∣∣ = ‖fs‖ < ∞ equal to zero on forbidden configurations
and with norm
‖f‖ =
∞∑
s=1
‖fs‖ < ∞. (1.9)
This means that f belongs to the Banach space L1 consisting of sequences of inte-
grable symmetric functions equal to zero on forbidden configurations with norm ‖f‖ and
component-wise linear operations.
In previous paper [1] we proved that∫ [
∂X(−t, (x)s)
∂(x)s
]2
|S(−t, (x)s)fs((x)s)|d(x)s =
∫ ∣∣∣fs((x)s)
∣∣∣d(x)s.
It means that ∥∥J(−t)S(−t)f
∥∥ = ‖f‖. (1.10)
If is obvious that
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
376 D. YA. PETRINA, G. L. CARAFFINI
∥∥∥∥
∫
dxf
∥∥∥∥ ≤ ‖f‖
so that ∥∥∥∥
∫
dx
∥∥∥∥ ≤ 1, ‖e±
∫
dx‖ ≤ e, e±
∫
dxe∓
∫
dx = 1. (1.11)
Taking into account (1.9) – (1.11) one obtains from (1.8) that
‖U(t)f‖ = ‖e
∫
dxJ(−t)S(−t)e−
∫
dxf‖ ≤ e2‖f‖
for arbitrary f ⊂ L1 and it means that the operator of evolution U(t) is a bounded one in
space L1, i.e.,
‖U(t)‖ ≤ e2. (1.12)
1.3. Group property of U(t). We have proved in [1] that the operator S(−t) has the
group property
S(t1 + t2) = S(−t1)S(−t2) = S(−t2)S(−t1) (1.13)
for arbitrary t1 > 0, t2 > 0.
It has also been proved in [1] that the operator J(−t) has the following property:[
∂X(−t1 − t2, (x)s)
∂(x)s
]2
=
[
∂X(−t1, X(−t2, (x)s))
∂X(−t2, (x)s)
∂X(−t2, (x)s)
∂(x)s
]2
=
=
[
∂X(−t2, X(−t1, (x)s))
∂X(−t1, (x)s)
∂X(−t1, (x)s)
∂(x)s
]2
. (1.14)
This equality follows from the fact that the Jacobian
∂X(−t1 − t2, (x)s)
∂(x)s
is equal to
product of the Jacobians that correspond to consecutive time intervals [0, t2], [t2, t2 + t1]
or [0, t1], [t1, t1 + t2].
Now we show that product of the operators J(−t) and S(−t) satisfies group property
J(−t1 − t2)S(−t1 − t2) = J(−t1)S(−t1)J(−t2)S(−t2) =
= J(−t2)S(−t2)J(−t1)S(−t1). (1.15)
Prove (1.15) in s-particle subspace. One has[
∂X(−t1, (x)s)
∂(x)s
]2
Ss(−t1, (x)s)
{[
∂X(−t2, (x)s)
∂(x)s
]2
Ss(−t2, (x)s)
}
=
=
[
∂X(−t1, (x)s)
∂(x)s
]2 [
∂X(−t2, X(−t1, (x)s))
∂X(−t1, (x)s)
]2
Ss(−t1 − t2, (x)s) =
=
[
∂X(−t1 − t2, (x)s)
∂(x)s
]2
Ss(−t1 − t2, (x)s).
We used that
Ss(−t, (x)s)
[
fs((x)s)gs((x)s)
]
=
[
Ss(−t, (x)s)fs((x)s)
][
Ss(−t, (x)s)gs((x)s)
]
.
These equalities are equivalent to the following ones:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
SOLUTIONS OF THE BBGKY HIERARCHY FOR A SYSTEM OF HARD SPHERES ... 377
J(−t1)S(−t1)J(−t2)S(−t2) = J(−t1 − t2)S(−t1 − t2).
One can prove likewise that
J(−t2)S(−t2)J(−t1)S(−t1) = J(−t1 − t2)S(−t1 − t2).
Thus equality (1.15) is proved.
Using (1.13) – (1.15) we can prove the group property of the operator U(t)
U(t1 + t2) = U(t1)U(t2) = U(t2)U(t1). (1.16)
Indeed
U(t1 + t2) = e
∫
dxJ(−t1 − t2)S(−t1 − t2)e−
∫
dx =
= e
∫
dxJ(−t1)S(−t1)J(−t2)S(−t2)e−
∫
dx =
= e
∫
dxJ(−t1)S(−t1)e−
∫
dxe
∫
dxJ(−t2)S(−t2)e−
∫
dx = U(t1)U(t2)
because e−
∫
dxe
∫
dx = I .
By similar calculation one obtains
U(t1 + t2) = U(t2)U(t1)
and (1.16) is proved.
1.4. Strong continuity of the group U(t). We have proved that evolution operators
U(t), t ≥ 0, are bounded in L1, according to (1.12), and possess the group property
according to (1.16).
Now we prove that evolution operator U(t) is strongly continuous, i.e., that
lim
∆t→0
‖U(t + ∆t)f − U(t)f‖ = 0, f ∈ L1. (1.17)
We will follow books [2, 3] with some modification.
From boundness of U(t) and its group property it is sufficient to prove that
lim
∆t→0
‖U(∆t)f − f‖ = 0. (1.18)
The operator U(∆t) is product of the operators e
∫
dxJ(−∆t)S(−∆t)e−
∫
dx, where oper-
ators e±
∫
dx are bounded in L1. It means that in order to prove (1.18) it is sufficient to
prove strong continuity of group J(−t)S(−t) for f ∈ L0
1. It follows from the fact that
e−
∫
dxJ(−∆t)S(−∆t)e−
∫
dxf − f = e
∫
dx
[
J(−∆t)S(−∆t)e−
∫
dxf − e−
∫
dxf
]
and that e−
∫
dxf ∈ L0
1 when f ∈ L0
1. Recall that subspace L0
1 consists of finite sequences
of functions fs((x)s) ∈ L0
s. Functions fs((x)s) ∈ L0
s are continuously differentiable with
compact support and equal to zero in some neighborhood of the forbidden configuration.
Subspace L0
s is everywhere dense in Ls and L0
1 in L1 [1].
Note that strong continuity of J(−t)S(−t) has already been proved in [1] (see (2.1) –
(2.3)). Indeed, it has been proved that functions [J(−t)S(−t)f ]s((x)s), fs((x)s) ∈ L0
s
are continuous functions with respect to t, t ≥ 0, uniformly with respect to (x)s on
compacts. Therefore
lim
∆t→0
∫ ∣∣∣J(−∆t)S(−∆t)fs((x)s) − fs((x)s)
∣∣∣d(x)s = 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
378 D. YA. PETRINA, G. L. CARAFFINI
because integrand has compact support and tends to zero as ∆t → 0 uniformly on com-
pacts with respect to (x)s.
Taking into account that f ∈ L0
1 is finite sequence, we proved that
lim
∆t→0
∥∥U(∆t)f − f
∥∥ = 0, f ∈ L0
1.
It follows from boundness of U(t) and taking into account that L0
1 is everywhere dense
in L1 that
lim
∆t→0
∥∥U(∆t)f − f
∥∥ = 0
for arbitrary f ∈ L1. Thus strong continuity of the evolution operator U(t) (1.17) is
proved.
We summarize obtained above results in the following theorem.
Theorem I. The evolution operators U(t), t ≥ 0, are a group of bounded strongly
continuous operators in L1.
2. Infinitesimal generator of group U(t) and solution of the BBGKY hierarchy
in L1. 2.1. Infinitesimal operator of group U(t). As known, the group of bounded
strongly continuous operators U(t) in L1 is strongly differentiable and its infinitesimal
generator is defined on everywhere dense set in L1. Now we proceed to determine this
infinitesimal generator. We will follow the books [2, 3] with some modification.
Theorem II. The infinitesimal generator B of the group U(t) is closed; its spectrum is
concentrated on the imaginary axis. On the set L0
1, everywhere dense in L1, B coincides
with the operator B = −H +
[∫
dx,H
]
, or, componentwise,
(Bf)s(x1, . . . , xs) =
= −Hfs(x1, . . . , xs) + a2
s∑
i=1
∫
dps+1
∫
S2
+
dη〈η, (pi − ps+1)〉×
×
[
1
(1 − 2ε)2
fs+1(x1, . . . , qi, p
∗
i , . . . , xs, qi − aη, p∗s+1)−
−fs+1(x1, . . . , qi, pi, . . . , xs, qi + aη, ps+1)
]
,
(2.1)
Hfs(x1, . . . , xs) = −
s∑
i=1
pi
∂
∂qi
fs(x1, . . . , xs),
with the following boundary condition:
If qi−qj = aη, 〈η, (pi−ps+1)〉 > 0, then momenta pi, pj should be replaced by p∗i , p
∗
j
and fs+2(x1, . . . , xs, qs+1, ps+1, qs+1−aη, ps+2) by
1
(1 − 2ε)2
fs+2(x1, . . . , xs, qs+1,
p∗s+1, qs+1−aη, p∗s+2). The operator H with this boundary condition is the infinitesimal
generator of the group J(−t)S(−t).
On set L0
1 the operators B and U(t) commute,
BU(t) = U(t)B
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
SOLUTIONS OF THE BBGKY HIERARCHY FOR A SYSTEM OF HARD SPHERES ... 379
and
d
dt
U(t) = BU(t) = U(t)B.
Proof. The proof of Theorem II coincides completely with the corresponding proof
for a system of hard spheres with elastic collisions (see [2, 3]). As for the systems of hard
spheres with elastic collisions the crucial point is identity[∫
dx,
[∫
dx,H
]]
f = 0, f ∈ L0
1.
In our case of systems of hard spheres with inelastic collisions, projection of this identity
on s-particle subspace has the following form:∫
dxs+2a
2
∫
dps+1
∫
S2
+
dη〈η, (ps+1 − ps+2)〉×
×
[
1
(1 − 2ε)2
fs+2(x1, . . . , xs, qs+1, p
∗
s+1, qs+1 − aη, p∗s+2)−
−fs+2(x1, . . . , xs, qs+1, ps+1, qs+1 + aη, ps+2
]
and it is equal to zero, as follows from [1] (formulae (3.5)).
Note that functions (J(−t)S(−t)fs+1)(x1, . . . , xs+1) are (possibly) different from
zero in a neighborhood of forbidden configurations, where the functions fs+1(x1, . . .
. . . , xs+1) ∈ L0
s+1 vanish. From this it follows that the functions (U(t)f)s+1(x1, . . .
. . . , xs+1), f ∈ L0
1 are (possibly) different from zero in a neighborhood of forbidden
configurations and on the hypersurfaces |qi − qs+1| = a, i = 1, . . . , s. This means that
the second term in formula (see (2.1))(
d
dt
U(t)f
)
s
(x1, . . . , xs) = (BU(t)f)s(x1, . . . , xs)
is different from zero.
2.2. Existence of solutions of the BBGKY hierarchy. The BBGKY hierarchy (1.2)
is the evolution equation for the infinite sequence ρ(t) of correlation functions ρ(t) =
= (ρ1(t, x1), . . . , ρs(t, x1, . . . , xs), . . .). This equation reads
dρ(t)
dt
= Bρ(t) = −Hρ(t) +
[
H,
∫
dx
]
ρ(t) (2.2)
with initial condition
ρ(t)|t=0 = ρ(0). (2.3)
The operator B is the infinitesimal generator of the group U(t) (1.7).
One can consider the BBGKY hierarchy as the abstract evolution equation (2.2) in the
Banach space L1 with initial data (2.3), ρ(0) ∈ L1. Then in follows from Theorem II that
ρ(t) = U(t)ρ(0). (2.4)
is the strong solution of the Cauchy problem for the BBGKY hierarchy (2.2) with initial
data ρ(0) ∈ L0
1. For general initial data ρ(0) ∈ L0
1, ρ(t) = U(t)ρ(0) is a generalized solu-
tion in the following sense. The strong solutions exist for ρ(0) ∈ L0
1 and are represented
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
380 D. YA. PETRINA, G. L. CARAFFINI
by formulae (2.4). The set L0
1 is everywhere dense in L1 and for arbitrary ρ(0) ∈ L0
1 there
exists a sequence ρ(0)i ∈ L0
1 which strongly converges to ρ(0). From the boundedness of
U(t) it follows that the sequence U(t)ρ(0)i also converges to U(t)ρ(0) and in this sense
ρ(t) = U(t)ρ(0) is a generalized solution of the BBGKY hierarchy.
The above results can be summarized in the following theorem.
Theorem III. The Cauchy problem for the BBGKY hierarchy (2.2) has a solution in
L1, given by formula (2.4). For initial data ρ(0) ∈ L0
1 ⊂ L1 this solution is strong, for
arbitrary initial data ρ(0) ∈ L1 it is a generalized solution.
2.3. States of infinite systems. Solutions of the hierarchy (1.2) or (2.2) constructed
above describe states of finite systems, because the average number N̄ of particles corre-
sponding to the state ρ(t) ∈ L1, ρ(0) ∈ L1,
N̄ =
∫
ρ1(t, x1)dx1 < ∞ (2.5)
is finite. This means that solutions of the hierarchy (1.2) or (2.2) for initial data ρ(0) ∈
∈ L1 can not describe states of infinite systems, i.e., system consisting of infinite average
number of particles situated in the entire phase space on admissible configurations.
Usually, in the case of elastic collisions, perturbations of equilibrium states of infinite
systems — i.e., Gibbs states for given temperature and density are considered, as initial
data for an infinite system. We hope to realize this approach in the case of an infinite
system of hard spheres with inelastic collisions.
As it has been pointed in paper [1] hierarchy (1.2) or (2.2), has the stationary solution
ρs((x)s) =
s∏
i=1
δ(pi − p)
s∏
i<j=1
Θ(|qi − qj | − a), s ≥ 1, (2.6)
where p is arbitrary momentum. It would be naturally to consider as initial data for infinite
system some perturbation of sequence (2.6). For example
ρs((x)s) =
s∏
i=1
1
(2πβ)
3
2
e−β(pi−p)2
s∏
i<j=1
Θ(|qi − qj | − a), s ≥ 1, (2.7)
where β is some positive parameter, β > 0. Sequence (2.7) can be considered as a pertur-
bation of sequence (2.6), because it converges as β → 0 to it, in the sense of generalized
functions.
1. Caraffini G. L., Petrina D. Ya. Analogue of Liouville equation and BBGKY hierarchy for a system of hard
spheres with inelastic collisions // Ukr. Math. J. – 2005. – 57, # 6. – P. 818 – 839.
2. Cercignani C., Gerasimenko V. I., Petrina D. Ya. Many-particle dynamics and kinetic equations. – Dor-
drecht: Kluwer Acad./Plenum Publ., 1997. – 252 p.
3. Petrina D. Ya., Gerasimenko V. I., Malyshev P. V. Mathematical foundations of classical statistical me-
chanics. Continuous systems. – Second ed. – London; New York: Taylor and Francis Sci. Publ., 2002. –
352 p.
Received 08.02.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, 58, # 3
|
| id | umjimathkievua-article-3460 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:56Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e9/861c333c1bd6868c70cef4e6bce092e9.pdf |
| spelling | umjimathkievua-article-34602020-03-18T19:55:07Z Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions Розвязки ієрархії ББГКІ для ситем твердих куль із непружним розсіянням Caraffini, G. L. Petrina, D. Ya. Цараффіні, Г. Л. Петрина, Д. Я. The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operators. The infinitesimal generator of the semigroup coincides on a certain everywhere dense set with the operator on the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data, they are generalized ones. Досліджено проблему існування розв'язків ієрархії для послідовності кореляційних функцій при початкових даних з прямої суми просторів інтегровних функцій. Доведено існування та єдиність розв'язків, поданих через півгрупу обмежених сильно неперервних операторів. Інфінітезимальний оператор півгрупи збігається на певній скрізь щільній множині з оператором, що визначає праву частину ієрархії. Для початкових даних з цієї множини розв'язки є строгими, для загальних початкових даних — узагальненими. Institute of Mathematics, NAS of Ukraine 2006-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3460 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 3 (2006); 371–380 Український математичний журнал; Том 58 № 3 (2006); 371–380 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3460/3657 https://umj.imath.kiev.ua/index.php/umj/article/view/3460/3658 Copyright (c) 2006 Caraffini G. L.; Petrina D. Ya. |
| spellingShingle | Caraffini, G. L. Petrina, D. Ya. Цараффіні, Г. Л. Петрина, Д. Я. Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title | Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title_alt | Розвязки ієрархії ББГКІ для ситем твердих куль із непружним розсіянням |
| title_full | Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title_fullStr | Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title_full_unstemmed | Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title_short | Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions |
| title_sort | solutions of the bbgky hierarchy for a system of hard spheres with inelastic collisions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3460 |
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