Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions
Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution func...
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nasplib_isofts_kiev_ua-123456789-1657502025-02-09T14:22:22Z Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions Аналог рівняння Ліувілля та ББГКІ ієрархії для системи твердих сфер з непружним розсіянням Petrina, D.Ya. Caraffini, G.L. Статті Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions. Досліджується динаміка твердих сфер з непружним розсіянням. Така система є моделлю для гранульованих потоків. Відображення, індуковане зсувом уздовж траєкторій, не зберігає об'єм фазового простору, а відповідний якобіан є відмінним від одиниці. Визначено спеціальну функцію розподілу як добуток звичайної функції розподілу та квадрата якобіана. Для цієї функції розподілу виведено рівняння Ліувілля з граничними умовами. Послідовність кореляційних функцій визначено для канонічного та великого канонічного ансамблів. Для кореляційних функцій виведено узагальнену ієрархію ББГКІ та відповідні граничні умови. This paper has been completed during stay of D.Ya. Petrina in November – December of 2003 in Dipartimento di Matematica of Politecnico di Milano and Universita di Parma. 2005 Article Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions / D.Ya. Petrina, G.L. Caraffini // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 818–839. — Бібліогр.: 4 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165750 517.9 + 531.19 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті |
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Статті Статті Petrina, D.Ya. Caraffini, G.L. Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions Український математичний журнал |
| description |
Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions. |
| format |
Article |
| author |
Petrina, D.Ya. Caraffini, G.L. |
| author_facet |
Petrina, D.Ya. Caraffini, G.L. |
| author_sort |
Petrina, D.Ya. |
| title |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions |
| title_short |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions |
| title_full |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions |
| title_fullStr |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions |
| title_full_unstemmed |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions |
| title_sort |
analog of the liouville equation and bbgky hierarchy for a system of hard spheres with inelastic collisions |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| topic_facet |
Статті |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165750 |
| citation_txt |
Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions / D.Ya. Petrina, G.L. Caraffini // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 818–839. — Бібліогр.: 4 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T19:20:51Z |
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2025-11-26T19:20:51Z |
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| fulltext |
UDC 517.9+531.19
D. Ya. Petrina (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
G. L. Caraffini (Univ. Parma, Italy)
ANALOGUE OF LIOUVILLE EQUATION
AND BBGKY HIERARCHY FOR A SYSTEM
OF HARD SPHERES WITH INELASTIC COLLISIONS∗
ANALOH RIVNQNNQ LIUVILLQ
TA BBHKI I{RARXI} DLQ SYSTEMY TVERDYX SFER
Z NEPRUÛNYM ROZSIQNNQM
Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for gran-
ular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and
the corresponding Jacobian is different from one.
A special distribution function is defined as the product of the usual distribution function and the squared
Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence
of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hier-
archy and boundary condition are derived for correlation functions.
DoslidΩu[t\sq dynamika tverdyx sfer z nepruΩnym rozsiqnnqm. Taka systema [ modellg dlq hra-
nul\ovanyx potokiv. VidobraΩennq, indukovane zsuvom uzdovΩ tra[ktorij, ne zberiha[ ob’[mfazovoho
prostoru, a vidpovidnyj qkobian [ vidminnym vid odynyci.
Vyznaçeno special\nu funkcig rozpodilu qk dobutok zvyçajno] funkci] rozpodilu ta kvadrata
qkobiana. Dlq ci[] funkci] rozpodilu vyvedeno rivnqnnq Liuvillq z hranyçnymy umovamy. Posli-
dovnist\ korelqcijnyx funkcij vyznaçeno dlq kanoniçnoho ta velykoho kanoniçnoho ansambliv. Dlq
korelqcijnyx funkcij vyvedeno uzahal\nenu i[rarxig BBHKI ta vidpovidni hranyçni umovy.
Introduction. It is commonly accepted that systems of hard spheres with inelastic colli-
sion are proper model of granular flow. Statistical mechanics of systems of hard spheres
should be a theoretical basis of the theory of granular flow. In attempts to adapt classical
statistical mechanics to systems of hard spheres with inelastic collisions, one is faced with
new very difficult problems connected with inelasticity.
First of all it is necessary to define density of probability (distribution function) on
phase space, because the Jacobian of the transformation induced by shift along trajectories
of hard spheres with inelastic collision is different from one and is singular (its derivative
with respect to time contains δ-functions). It is necessary to derive the Liouville equation
for defined distribution function and correctly formulate boundary conditions associated
with inelasticity. And at last one should derive the analogue of BBGKY hierarchy for
corresponding correlation functions.
Above mentioned problems are solved in given paper. The distribution function is
defined as follows:
DN (t, (x)N ) = DN (0, X(−t, (x)N ))
[
∂X(−t, (x)N )
∂(x)N
]2
(1)
where DN (0, (x)N ) is the initial distribution function, X(−t, (x)N ) the trajectory of
N hard spheres at time −t with initial data (x)N at initial time t = 0,
∂X(−t, (x)N )
∂(x)N
the corresponding singular Jacobian different from one. Distribution function satisfies the
law of conservation of full probability
∗ This paper has been completed during stay of D.Ya. Petrina in November – December of 2003 in Dipar-
timento di Matematica of Politecnico di Milano and Universita di Parma.
c© D. Ya. PETRINA, G. L. CARAFFINI, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6 818
ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 819
∫
DN (t, (x)N )d(x)N =
∫
DN (0, (x)N )d(x)N (2)
and the Liouville equation
∂
∂t
DN (t, (x)N ) = −
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ), DN (t, (x)N ) |t=0 = DN (0, (x)N )
(3)
with boundary condition according to which for qi − qj − aη = 0 (where |η| = 1 and
a is the diameter of the sphere), and 〈η, (pi − pj)〉 > 0, momenta pi, pj should be
replaced by
p∗i = pi +
ε
1 − 2ε
η〈η, (pi − pj)〉, p∗j = pj −
ε
1 − 2ε
η〈η, (pi − pj)〉
in the operator −
∑N
i=1
pi
∂
∂qi
and in DN (t, (x)N ); moreover the identity is valid
DN (t, q1, p1, . . . , qi, pi, . . . , qj , pj , . . . , qN , pN ) =
=
1
(1 − 2ε)2
DN (t, q1, p1, . . . , qi, p
∗
i , . . . , qj , p
∗
j , . . . , qN , pN )
for above mentioned phase points; ε is a parameter associated with inelastic collisions(
1
2
< ε ≤ 1
)
. Momenta do not change if 〈η, (pi − pj)〉 < 0, i.e., p∗i = pi, p∗j = pj .
The following BBGKY hierarchy is derived for a sequence of correlation functions
ρ(N)
s (t, (x)s) =
= N(N − 1) . . . (N − s + 1)
∫
DN (t, x1, . . . , xs, xs+1, . . . , xN ) dxs+1 . . . xN :
(4)
∂ρ
(N)
s (t, (x)s)
∂t
=
= −
s∑
i=1
pi
∂
∂qi
ρ(N)
s (t, (x)s) + a2
s∑
i=1
∫
dps+1
∫
S2
+
dη
〈
η, (pi − ps+1)
〉
×
×
[
1
(1 − 2ε)2
ρ
(N)
s+1
(
t, q1, p1, . . . , qi, p
∗
i , . . . , qs, ps, qi − aη, p∗s+1
)
−
−ρ
(N)
s+1 (t, q1, p1, . . . , qi, pi, . . . , qs, ps, qi + aη, ps+1)
]
,
S+
2
(
η
∣∣〈η, (pi − ps+1)〉 ≥ 0
)
, ρ(N)
s (t, (x)s)
∣∣
t=0
= ρ(N)
s ((x)s), 1 ≤ s ≤ N.
One should add the same boundary condition as for DN (t, (x)N ) in the Liouville equa-
tion.
In given paper we did not touch the problem of existence of solution of hierarchy (4)
and existence of the thermodynamic limit.
1. Trajectories of system of hard spheres with inelastic collisions.
1.1. Dynamics. Consider in three-dimensional space R3 particles with mass m
that are hard spheres with diameter a. Particles move freely until they touch each other
and the distance between their centers is equal to a. Then they inelastically collide.
Denote position of the center of sphere by q ∈ R3 and its momentum by p ∈ R3.
Let N be the number of particles of the considered system. Particles with numbers i
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
820 D. YA. PETRINA, G. L. CARAFFINI
and j collide if qi − qj = aη. If their momenta before collisions are pi and pj and
〈η, (pi − pj)〉 < 0, then after inelastic collision they become
p∗i = pi − εη〈η, (pi − pj)〉,
p∗j = pj + εη〈η, (pi − pj)〉
(1.1)
where parameter
1
2
< ε ≤ 1 characterizes inelastic collision, unit vector η is di-
rected from the center of sphere with number i to the center of sphere with number j,
〈η, (pi − pj)〉 is scalar product of vectors η and pi − pj . Formulae (1.1) define a linear
transformation of momenta pi, pj . If 〈η, (pi − pj)〉 > 0 then after collision p∗i = pi,
p∗j = pj .
Note that this law of inelastic collision (1.1) is true for real evolution of system with
increasing time t (for dynamics forward in time). In statistical mechanics we also need
an imaginary evolution of system with decreasing time (backward in time dynamics). We
define the law of inelastic collision with decreasing time as inverse to (1.1) transformation.
To obtain this desired transformation we consider (1.1) as an equation with respect to pi,
pj for given p∗i , p∗j .
Calculating scalar product
〈η, (p∗i − p∗j )〉 = (1 − 2ε)〈η, (pi − pj)〉
one obtains from (1.1) desired inverse to (1.1) linear transformation
pi = p∗i +
ε
1 − 2ε
η〈η, (p∗i − p∗j )〉,
pj = p∗j −
ε
1 − 2ε
η〈η, (p∗i − p∗j )〉, 〈η, (p∗i − p∗j )〉 > 0.
(1.2)
In what follows we will need only backward in time dynamics and it will be useful to
change in (1.2) denotation and write instead of momenta (pi, pj) momenta (p∗i , p
∗
j ) and
vice versa. Then transformation (1.2) looks like the following one
p∗i = pi +
ε
1 − 2ε
η〈η, (pi − pj)〉,
p∗j = pj −
ε
1 − 2ε
η〈η, (pi − pj)〉, 〈η, (pi − pj)〉 > 0.
(1.3)
It follows from (1.3) that components of vectors p∗i , p
∗
j perpendicular to vector η do
not change, and components parallel to vector η change according to (1.3). It is obvious
that the Jacobian of transformation (1.3) J can be easily calculated:
J =
1
1 − 2ε
. (1.4)
If 〈η, (pi − pj)〉 < 0 momenta do not change, i.e., p∗i = pi, p∗j = pj . Let us
calculate kinetic energy after collision in backward motion of particles with number i
and j. We have according to (1.3)
p∗i
2 + p∗j
2 = p2
i + p2
j + 2
ε− ε2
(1 − 2ε)2
(〈η, (pi − pj)〉)2 ≥ p2
i + p2
j (1.5)
because ε− ε2 > 0 for
1
2
< ε < 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 821
Now calculate kinetic energy after collision in forward motion. According to (1.1)
one obtains
p∗i
2 + p∗j
2 = p2
i + p2
j + 2(−ε + ε2) (〈η, (pi − pj)〉)2 ≤ p2
i + p2
j (1.6)
because −ε + ε2 < 0 for
1
2
< ε < 1. From (1.5), (1.6) one can see that p∗i
2 + p∗j
2 is
greater than p2
i + p2
j for backward motion (t < 0), and p∗i
2 + p∗j
2 is less than p2
i + p2
j
for forward motion:
p∗i
2 + p∗j
2 ≥ p2
i + p2
j , t < 0,
p∗i
2 + p∗j
2 ≤ p2
i + p2
j , t > 0.
(1.7)
Thus in defined above dynamics with inelastic collisions kinetic energy increases for
t < 0 and decreases for t > 0. Only in the case 〈η, (pi−pj)〉 =0, one has p∗i
2 +p∗j
2 =
= p2
i + p2
j and kinetic energy is preserved, even for
1
2
< ε < 1.
1.2. Trajectory. Denote by Q1(−t), . . . , QN (−t) positions of hard spheres at time
−t, t > 0, by P1(−t), . . . , PN (−t) their momenta, and by q1, . . . , qN their initial po-
sitions, by p1, . . . , pN their initial momenta at time t = 0, by (x)N = (q1, p1, . . . , qN ,
pN ) the initial phase point. Obviously we will consider only admissible configurations,
i.e., |qi−qj | ≥ a for all (i, j) ⊂ (1, . . . , N). As it was mentioned above, particles move
freely until they touch each other and then collide and their momenta change according
to (1.3).
We will neglect instantaneous collisions of three or more particles because the set
of such initial positions and momenta has Lebesgue measure equal to zero. Denote
by tij((x)N ) the time of collision of particles with number i and j. Considered as a
function of (x)N , tij((x)N ) is continuously differentiable outside of a certain set with
Lebesgue measure equal to zero.
The trajectory X(−t, (x)N ) =
(
Q1(−t, (x)N ), P1(−t, (x)N ), . . . , QN (−t, (x)N ),
PN (−t, (x)N )
)
, Qi(−t) ≡ Qi(−t, (x)N ), Pi(−t) = Pi(−t, (x)N ), i = 1, . . . , N,
is constructed as follows. Until the first collision
X(−t, (x)N ) = (q1 − p1t, p1, . . . , qN − pN t, pN ). (1.8)
If at time tij((x)N ) particles with numbers i and j collide, then for t > tij(x)N
the trajectory X(−t, (x)N ) is again given by formula (1.8), but positions and momenta
of i-th and j-th particles are given by
qi − pitij(x) − p∗i (t− tij(x)), p∗i , qj − pjtij(x) − p∗j (t− tij(x)), p∗j , (1.9)
where p∗i , p∗j are expressed in terms of pi, pj according to (1.3).
One can continue the trajectory according to (1.9) after all collisions if infinitely many
collisions on finite time interval are absent. Then momenta of all particles involved in
these infinite number of collisions coincide and their spheres touch each other. The cor-
responding set of initial phase points lie on the hyperplanes of lower dimension and has
Lebesgue measure equal to zero.
It is obvious that the trajectory has the group property
X(−t1 − t2, (x)N ) = X(−t1, X(−t2, (x)N )) = X(−t2, X(−t1, (x)N ))
and it satisfies the following boundary condition:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
822 D. YA. PETRINA, G. L. CARAFFINI
for qi − qj = aη, 〈η, (pi − pj)〉 > 0, (i, j) ⊂ (1, . . . , N),
X(−t, q1, p1, . . . , qi, pi, . . . , qj , pj , . . . , qN , pN ) =
= X(−t, q1, p1, . . . , qi, p
∗
i , . . . , qj , p
∗
j , . . . , qN , pN ); (1.10)
if qi − qj = aη, but 〈η, (pi − pj)〉 < 0 momenta pi, pj do not change. This boundary
condition means that after collision particles depart and distance between them increases.
The trajectory X(−t, (x)N ) is a continuously differentiable function almost every-
where with respect to its initial data (x)N and time on every time interval between col-
lision. The detailed proof of above mentioned properties of the trajectory X(−t, (x)N )
can be found in books [1, 2], after some modification connected with the inelastic charac-
ter of collisions.
We summarize above formulated results in the following theorem.
Theorem I. The trajectory X(−t, (x)N ) of N hard spheres that inelastically col-
lide exists for arbitrary time t > 0, is continuously differentiable with respect to initial
phase points (x)N and time t on intervals between collisions, and has group property
for almost all initial (x)N , that belong to certain domain outside of hypersurface with
Lebesgue measure equal to zero.
Theorem I asserts that trajectories X(−t, (x)N ) are well defined between times of
collisions almost everywhere (a.e.) with respect to (x)N . In many respects, trajectories of
our system of hard spheres with inelastic collisions have the same properties as a system of
hard spheres with elastic collisions. These properties were formulated in Theorem I. But
trajectories of hard spheres with inelastic collisions have also certain specific properties
different from those of the case of elastic collisions. One of these specific properties is
that the map of the phase space induced by the shift along trajectories does not preserve
the volume.
According to definition of trajectories (1.8), (1.9), the Jacobian
∂(X1(−t, (x)N ), . . . , XN (−t, (x)N ))
(∂x1, . . . , ∂xN )
=
∂(X(−t, (x)N ))
∂(x)N
(1.11)
is equal to one if for initial point (x)N there are no collisions until time −t, and is
equal to
∂(P1(−t, (x)N ), . . . , PN (−t, (x)N ))
(∂p1, . . . , ∂pN )
=
( 1
1 − 2ε
)n
(1.12)
if there are n pair collisions for initial point (x)N . The Jacobian of transformation (1.3)
is equal to
∂(p∗i , p
∗
j )
∂(pi, pj)
=
1
1 − 2ε
.
2. Evolution operator.
2.1. Definition of evolution operator. Let fN (x1, . . . , xN ) = fN ((x)N ) be a con-
tinuous symmetric (permutation invariant) function defined on phase space R6N of N
particles and equal to zero on the set of forbidden configurations. Define at first formally
operator SN (−t) as the operator of shift along the trajectory X(−t, (x)N ) as follows:
(SN (−t)fN )(x1, . . . , xN ) =
= fN (X1(−t, (x)N ), . . . , XN (−t, (x)N )) = fN (X(−t, (x)N )) (2.1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 823
on admissible configurations, and
(SN (−t)fN )(x1, . . . , xN ) = 0
on the set of forbidden configurations.
According to definition of the trajectory X(−t, (x)N ), function fN (X(−t, (x)N ))
has jumps of momenta at the time of collision tij((x)N ), because momenta after colli-
sions are different from momenta before collisions, and is again a symmetric function.
In classical statistical mechanics of systems of particles with elastic collisions function
fN (X(−t, (x)N )) is proportional to the probability density of the considered system
at time t in phase space. It should satisfy the law of conservation of full probability,
i.e., full probability has to be independent of time. We also need to derive equation for
fN (X(−t, (x)N )) and equation for the sequence of correlation functions. Therefore we
impose some condition on function fN ((x)N ).
We suppose that fN ((x)N ) belongs to the Banach space LN of functions equal to
zero on the set of forbidden configurations, for which |qi − qj | < a at least for one pair
(i, j) ⊂ (1, . . . , N), and Lebesgue integrable with norm
‖fN‖ =
∫
|fN (x1, . . . , xN )|dx1 . . . dxN =
∫
|fN ((x)N )|d(x)N . (2.2)
Denote by L0
N the subspace of LN consisting of continuously differentiable func-
tions with compact support and equal to zero in some neighbourhood of the forbidden
configuration. Subspace L0
N is everywhere dense in LN .
If fN ∈ L0
N , then fN (X(−t, (x)N )) is a continuously differentiable function with
respect to t and (x)N almost everywhere. Indeed, the trajectory X(−t, (x)N ) is a con-
tinuously differentiable function with respect to time t and initial points (x)N a.e. on
time intervals between collisions. Collisions happen if |Qi(t, (x)N ) −Qj(−t, (x)N )| =
= a for some (i, j) ⊂ (1, . . . , N), but function fN (X(−t, (x)N )) is equal to zero
in some neighborhood of these hypersurfaces. Outside of these hypersurfaces trajecto-
ries are continuously differentiable with respect to time t and initial points (x)N a.e.,
and therefore functions fN (X(−t, (x)N )) have the same property because fN ((x)N ) ∈
∈ L0
N . According to definition (2.1), fN (X(−t, (x)N )) it is equal to zero on the forbid-
den configuration together with fN ((x)N ) ∈ L0
N . (For more details see [1, 2].)
It is obvious that operator SN (−t) has the group property
SN (−t1 − t2) = SN (−t1)SN (−t2) = SN (−t2)SN (−t1) (2.3)
2.2. Properties of operator SN (−t). Consider again fN (X(−t, (x)N )) with
fN ((x)N ) ∈ L0
N and show that it is Lebesgue integrable. Indeed it is continuous with
respect to (x)N a.e., has compact support and therefore∫
|fN (X(−t, (x)N )|d(x)N < ∞.
We need only to prove that fN (X(−t, (x)N )) has compact support with respect to
(x)N if fN ((x)N ) has compact support. If fN ((x)N ) has compact support,
say
∑N
i=1
(q2
i + p2
i ) ≤ R, R > 0, then fN (X(−t, (x)N )) has compact support∑N
i=1
[
Q2
i (−t, (x)N ) + P 2
i (−t, (x)N
]
≤ R, with respect to Qi, Pi, i = 1, . . . , N.
If
∑N
i=1
P 2
i (−t, (x)N ) ≤ R, then
∑N
i=1
p2
i ≤ R, because at each collision of
i-th and j-th particles at time 0 ≤ τ ≤ t one has P ∗2
i (−τ, (x)N ) + P ∗2
j (−τ, (x)N ) ≥
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
824 D. YA. PETRINA, G. L. CARAFFINI
≥ P 2
i (−τ, (x)N ) + P 2
j (−τ, (x)N ), and therefore R ≥
∑N
i=1
P 2
i (−t, (x)N ) ≥
≥
∑N
i=1
p2
i .
One has
∑N
i=1
Q2
i (−t, (x)N ) ≤ R2 and therefore
∑N
i=1
q2
i < r, where r > 0 is
finite, because Qi(−t, (x)N ) is shifted from qi at finite distance by finite Pi(−τ, (x)N ),
0 ≤ τ ≤ t.
Thus fN (X(−t, (x)N )) has compact support with respect to (x)N , together with
fN ((x)N ).
It the case of elastic collision, operator SN (−t) is isometric, because the Jacobian
(1.11) is equal to one. In our case of inelastic collision the Jacobian (1.11) is different
from one for such initial (x)N that collisions occur.
If D is some domain in R6N and D−t is the image of D induced by shift along
trajectories X(−t, (x)N ), then∫
D
d(x)N �=
∫
D−t
d(X(−t, (x)N )) =
∫
D
∂X(−t, (x)N )
∂(x)N
d(x)N ,
because
∂(X(−τ, (x)N )
∂(x)N
�= 1 and
∂
∂τ
∂(X(−τ, (x)N ))
∂(x)N
is proportional to δ(τ − τl),
for those (x)N that collisions occur at time τ = τl, 0 ≤ τl ≤ t, and the Jacobian has a
jump at time τl.
Denote by V (D) and V (D−t) the volumes of domains D and D−t respectively.
Then one has
V (D−t) =
∫
D
∂X(−t, (x)N )
∂(x)N
d(x)N =
=
∫
D
∂X(0, (x)N )
∂(x)N
d(x)N +
∫
D
t∫
0
∂
∂τ
∂X(−τ, (x)N )
∂(x)N
dτ
d(x)N =
= V (D) +
∫
D
t∫
0
∂
∂τ
∂X(−τ, (x)N )
∂(x)N
dτ
d(x)N .
It follows from these formulae that contributions in V (D−t) from hypersurfaces
|Qi(−τl, (x)N ) −Qj(−τl, (x)N )| = a, (i, j) ⊂ (1, . . . , N),
are finite (see for more details in Appendix A and B).
Nevertheless operator S(−t) is “isometric” on L0
N in the following sense.
Consider function fN (X(−t, (x)N ))
(
∂(X(−t, (x)N ))
∂(x)N
)2
. Later we will show that
∫
fN (X(−t, (x)N ))
(∂(X(−t, (x)N ))
∂(x)N
)2
d(x)N =
∫
fN ((x)N )d(x)N ,
∫
|fN (X(−t, (x)N ))|
(∂(X(−t, (x)N ))
∂(x)N
)2
d(x)N =
∫
|fN ((x)N )|d(x)N .
(2.4)
It is obvious that
∂(X(−t, (x)N ))
∂(x)N
∣∣∣∣
t=0
= 1 and it follows from (2.4) that function
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 825
DN (t, (x)N ) = fN (X(−t, (x)N ))
(
∂(X(−t, (x)N ))
∂(x)N
)2
, (2.5)
that is equal to fN ((x)N ) at t = 0, may be considered as a probability density in the
phase space of systems of hard spheres with inelastic collisions.
Function (2.5) has the following “group” property:
fN (X(−t1 − t2, (x)N ))
(
∂(X(−t1 − t2, (x)N ))
∂(x)N
)2
=
= fN (X(−t1, X(−t2, (x)N ))
(
∂(X(−t1, X(−t2, (x)N )))
∂(x)N
)2
=
= fN (X(−t2, X(−t1, (x)N ))
(
∂(X(−t2, X(−t1, (x)N )))
∂(x)N
)2
,
∂(X(−t1, X(−t2, (x)N )))
∂(x)N
=
∂(X(−t1, X(−t2, (x)N )))
∂X(−t2, (x)N )
∂(X(−t2, (x)N ))
∂(x)N
,
∂(X(−t2, X(−t1, (x)N )))
∂(x)N
=
∂(X(−t2, X(−t1, (x)N )))
∂X(−t1, (x)N )
∂(X(−t1, (x)N ))
∂(x)N
.
(2.6)
Note that function (2.5) is continuously differentiable together with functions
fN (X(−t, (x)N )), fN ((x)N ) ∈ L0
N . Indeed function fN (X(−t, (x)N )) is contin-
uously differentiable with respect to time t and initial data (x)N a.e. The Jacobian
∂(X(−t, (x)N ))
∂(x)N
is a constant function of time on time intervals between collisions and
has a jump at time of collisions, but function fN (X(−t, (x)N )) is equal to zero in a
neighbourhood of time of collisions and, therefore, function DN (t, (x)N ) is continu-
ously differentiable as well as fN (X(−t, (x)N )). The proof is presented below.
2.3. Differential equation for DN(t, (x)N). Let us show that the function
DN (t, (x)N ) is differentiable with respect to time. It is product of two functions
fN (X(−t, (x)N )) and
∂X(−t, (x)N )
∂(x)N
. One obtains
∂
∂t
DN (t, (x)N ) =
=
[
∂
∂t
fN (X(−t, (x)N ))
](∂X(−t, (x)N )
∂(x)N
)2
+
+fN (X(−t, (x)N ))
[
∂
∂t
(∂X(−t, (x)N )
∂(x)N
)2
]
. (2.7)
Now calculate derivatives of fN (X(−t, (x)N )) for fN ((x)N ) ∈ L0
N . Using group
property of SN (−t) (2.3) one obtains (see details in [1, 2])
∂
∂t
fN (X(−t, (x)N )) =
= lim
∆t→0
[
SN (−t)
(SN (−∆t) − I)fN ((x)N )
∆t
]
=
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826 D. YA. PETRINA, G. L. CARAFFINI
= SN (−t)
[
−
N∑
i=1
pi
∂
∂qi
fN ((x)N )
]
=
= −
N∑
i=1
Pi(−t, (x)N )
∂
∂Qi(−t, (x)N )
fN (X(−t, (x)N )),
(2.8)
∂
∂t
fN (X(−t, (x)N )) = lim
∆t→0
SN (−∆t) − I
∆t
SN (−t)fN ((x)N ) =
= −
N∑
i=1
pi
∂
∂qi
fN (X(−t, (x)N )).
Now explain derivation of formulas (2.8).
One has
lim
∆t→0
1
∆t
(SN (−∆t) − I)fN ((x)N ) = −
N∑
i=1
pi
∂
∂qi
fN ((x)N )
on set |qi − qj | > a, (i, j) ⊂ (1, . . . , N) because fN ∈ L0
N and it is equal to zero on
some neighbourhood of forbidden configuration.
The trajectory X(−t, (x)N ) at qi−qj = aη, 〈η, (pi−pj)〉 > 0 has the jump at t =
= +0, X(−0, (x)N )−X(0, (x)N ) = (x)∗N−(x)N , (x)∗N = (q1, p1, . . . , qi, p
∗
i , . . . , qj ,
p∗j , . . . , qN , pN ), but function fN ∈ L0
N is equal to zero in some neighbourhood of such
points, therefore fN ((x)∗N ) = fN ((x)N ) = 0, i.e., function fN (X(−t, x)) has not
jump at t = +0. At time t > 0X(−t, (x)N ) = X(−t, (x)∗N ) and fN (X(−t, (x)N )) =
= fN (X(−t, (x)∗N )) for qi − qj = aη, 〈η, (pi − pj)〉 > 0.
Note that fN (X(−t, (x)N )) may be different from zero with respect to (x)N on this
neighbourhood of forbidden configuration where fN ((x)N ) is equal to zero. Therefore
lim
∆t→0
1
∆t
(SN (−∆t) − I)fN (X(−t, (x)N )) = −
N∑
i=1
pi
∂
∂qi
fN (X(−t, (x)N ))
with boundary condition according to which at qi − qj = aη, 〈η, pi − pj〉 > 0, (i, j) ⊂
⊂ (1, . . . , N) in expression −
∑N
i=1
pi
∂
∂qi
fN (X(−t, (x)N )) momenta pi, pj should
be replaced by p∗i , p∗j .
Thus we obtain two expressions for
∂
∂t
fN (X(−t, (x)N )), namely
∂
∂t
fN (X(−t, (x)N )) = −
N∑
i=1
Pi(−t, (x)N )
∂
∂Qi(−t, (x)N )
fN (X(−t, (x)N )),
(2.8a)
and
∂
∂t
fN (X(−t, (x)N )) = −
N∑
i=1
pi
∂
∂qi
fN (X(−t, (x)N )) =
= −
N∑
j=1
[
∂fN (X(−t, (x)N ))
∂Qj(−t, (x)N )
N∑
i=1
pi
∂
∂qi
Qj(−t, (x)N ) +
+
∂fN (X(−t, (x)N ))
∂Pj(−t, (x)N )
N∑
i=1
pi
∂
∂qi
Pj(−t, (x)N )
]
. (2.8b)
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 827
The right-hand side of (2.8a) is a continuous function with respect to (x)N a.e.,
because fN ((x)N ) ∈ L0
N and Pi(−t, (x)N ), i = 1, . . . , N, are continuous functions
of time and (x)N a.e. on time intervals between collisions and have jumps only at time
of collisions, but function fN (X(−t, (x)N )) is equal to zero in some neighbourhood of
times of collisions. Qi(−t, (x)N ), i = 1, . . . , N, are continuous functions of time and
(x)N a.e. on time intervals between collisions.
The right-hand side of (2.8b) is also a continuous function with respect to (x)N a.e.,
because (fN (x)N ) ∈ L0
N , and Qj(−t, (x)N ), Pj(−t, (x)N ), 1 ≤ j ≤ N, are con-
tinuously differentiable functions with respect to (x)N a.e. on time intervals between
collisions, but functions
∂fN (X(−t, (x)N ))
∂Qj(−t, (x)N )
,
∂fN (X(−t, (x)N ))
∂Pi(−t, (x)N )
are equal to zero in
some neighbourhood of times of collisions.
Note that in the right-hand side of (2.8b) we have the following boundary conditions:
at qi − qj = aη, |η| = 1, 〈η, (pi − pj)〉 > 0, (i, j) ⊂ (1, . . . , N),
the expressions −
∑N
i=1
pi
∂
∂qi
fN (X(−t, (x)N )), fN (X(−t, (x)N )) should be repla-
ced by:
−
N∑
i=1
pi
∂
∂qi
fN (X(−t, (x)N ))|pi=p∗
i ,pj=p∗
j
, fN (X(−t, (x)N ))|pi=p∗
i ,pj=p∗
j
. (2.9)
At 〈η, (pi − pj)〉 < 0, on the contrary, momenta do not change. These boundary
conditions follow from the definition of the trajectory at qi − qj = aη, 〈η, (pi − pj)〉 >
> 0 (1.10), namely X(−t, (x)N ) = X(−t, (x)N ) |pi=p∗
i , pj=p∗
j
, and the fact that
fN ((x)N ) ∈ L0
N (see [1, 2]).
In the right-hand side of (2.8a) the analogous boundary conditions are absent, because
function fN (X(−t, (x)N )) is equal to zero when |Qi(−t, (x)N ) − Qj(−t, (x)N )| =
= a, (i, j) ⊂ (1, . . . , N), and the term
∂
∂t
Pi(−t, (x)N )
∂fN (X(−t, (x)N ))
∂Pi(−t, (x)N )
is equal
to zero, because fN (X(−t, (x)N )) is equal to zero where Pi(−t, (x)N ) have jumps,
i = 1, . . . , N.
At first sight, according to the boundary condition (2.9) at qi − qj = aη, 〈η, (pi −
−pj)〉 > 0, there are jumps in the right-hand side of (2.8b), because momenta (pi, pj)
are replaced by (p∗i , p
∗
j ). We show that it is not true.
As it was mentioned above, the right-hand side of (2.8a) is a continuous function of
(x)N a.e. on the entire phase space of admissible configurations, i.e., |qi − qj | ≥ a for
all pairs (i, j) ⊂ (1, . . . , N). The right-hand side of (2.8b) together with boundary con-
dition (2.9) identically coincide with the right-hand side of (2.8a) and therefore it is also a
continuous function of (x)N a.e. on the entire phase space of admissible configurations.
Note that only the sum −
∑N
i=1
pi
∂
∂qi
fN (X(−t, (x)N )) is continuous on admissi-
ble configuration. The each term −pi
∂
∂qi
fN (X(−t, (x)N )), −pj
∂
∂qj
fN (X(−t, (x)N ))
has jumps for qi − qj = aη, 〈η, (pi − pj)〉 > 0 because pi, pj should be replaced by
p∗i , p∗j and
∂
∂qi
fN (X(−t, (x)N )),
∂
∂qj
fN (X(−t, (x)N )) are continuous with respect
to (x)N on admissible configuration (see (2.8b)).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
828 D. YA. PETRINA, G. L. CARAFFINI
Now consider the second term in the right-hand side of (2.7) and show that it is equal
to zero. Indeed, the Jacobian
∂X(−t, (x)N )
∂(x)N
for given fixed (x)N is a constant function
of t and at time of collisions has a jumps. Therefore
∂
∂t
∂X(−t, (x)N )
∂(x)N
is equal to zero
on time intervals between collisions, but the function f(X(−t, (x)N )) is equal to zero
in neighbourhoods of times of collisions and, as result,
fN (X(−t, (x)N ))
[
∂
∂t
(∂X(−t, (x)N )
∂(x)N
)2
]
≡ 0,
i.e., the second term in the right-hand side of (2.7) is equal to zero.
Taking into account above obtained results we have
∂
∂t
DN (t, (x)N ) =
[
∂
∂t
fN (X(−t, (x)N ))
](∂X(−t, (x)N )
∂(x)N
)2
=
=
[
−
N∑
i=1
Pi(−t, (x)N )
∂
∂Qi(−t, (x)N )
fN (X(−t, (x)N ))
](∂X(−t, (x)N )
∂(x)N
)2
=
=
[
−
N∑
i=1
pi
∂
∂qi
fN (X(−t, (x)N ))
](∂X(−t, (x)N )
∂(x)N
)2
=
= −
N∑
i=1
pi
∂
∂qi
[
fN (X(−t, (x)N ))
(∂X(−t, (x)N )
∂(x)N
)2
]
=
= −
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ). (2.10)
The last equality in (2.10) follows from the fact that the Jacobian
∂X(−t, (x)N )
∂(x)N
is
constant (piece-wise constant) everywhere with respect to (x)N excluding points (x)N
at which there are collisions at time t, but the function fN (X(−t, (x)N )) is equal to
zero in neighbourhoods of such points and, therefore,
fN (X(−t, (x)N ))
(
−
N∑
i=1
pi
∂
∂qi
∂X(−t, (x)N )
∂(x)N
)
is equal to zero at such points. Remembering that
DN (t, (x)N ) = fN (X(−t, (x)N ))
(∂X(−t, (x)N )
∂(x)N
)2
one obtains for η ∈ S2
+(〈η, (pi − pj)〉 > 0)
DN (−t, x1, . . . , qi, pi, . . . , qi − aη, pj , . . . , xN ) =
= fN (X(−t, x1, . . . , qi, pi, . . . , qi − aη, pj , . . . , xN )) ×
×
(
∂X(−t, x1, . . . , qi, pi, . . . , qi − aη, pj , . . . , xN )
∂(x)N
)2
=
= fN (X(−t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )) ×
×
(
∂X(−t− 0, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )
∂(x)∗N
×
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 829
×
∂X(−0, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )
∂(x)N
)2
=
= fN (X(−t− 0, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )) ×
×
(
∂X(−t− 0, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )
∂(x)∗N
)2 1
(1 − 2ε)2
=
= DN (−t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN )
1
(1 − 2ε)2
. (2.11)
In deriving (2.11) we took into account that fN (X(−t, (x)N )) = fN (X(−t, (x)∗N ))
for qi − qj = aη, 〈η, (pi − pj)〉 > 0 and used that the Jacobian
∂X(−t, (x)N )
∂(x)N
can be
calculated as product of Jacobians on consecutive time intervals between collisions and
extract Jacobian that corresponds to collisions of i-th and j-th particles at time t = +0.
The last Jacobian is equal to
1
1 − 2ε
.
We must add to (2.10) the following boundary condition: at
qi − qj = aη, | η |= 1, 〈η, (pi − pj)〉 > 0, (i, j) ⊂ (1, . . . , N),
expressions
−
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ), DN (t, (x)N )
should be replaced by
− 1
(1 − 2ε)2
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ) |pi=p∗
i ,pj=p∗
j
,
1
(1 − 2ε)2
DN (t, (x)N ) |pi=p∗
i ,pj=p∗
j
,
(2.12)
and at 〈η, (pi − pj)〉 < 0 momenta (pi, pj) do not change.
The boundary condition (2.12) for DN (t, (x)N ) follows directly from the boundary
condition (2.9) for fN (X(−t, (x)N )), fN ((x)N ) ∈ L0
N , and from equality (2.11).
Obtained results can be summarized in the following fundamental theorem.
Theorem II. The probability density on phase space of system of hard spheres with
inelastic collisions DN (t, (x)N ) is a differentiable function with respect to time t and
(x)N a.e., and satisfies the Liouville equation
∂
∂t
DN (t, (x)N ) = −
N∑
i=1
pi
∂
∂qi
DN (t, (x)N ) (2.13)
with boundary condition (2.12) and initial condition DN (t, (x)N )|t=0 = DN (0, (x)N ) =
= fN ((x)N )
(
∂X(−t, (x)N )
∂(x)N
∣∣∣∣
t=0
= 1, because X(−t, (x)N )|t=0 = (x)N
)
.
Note that the right-hand side of (2.13) together with boundary conditions (2.12) is a
continuous function on phase space of admissible configurations a.e., as it follows from
the second expression in the right-hand side of (2.10). Indeed it was already shown that
−
N∑
i=1
Pi(−t, (x)N )
∂
∂Qi(−t, (x)N )
fN (X(−t, (x)N ))
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
830 D. YA. PETRINA, G. L. CARAFFINI
is a continuous function of (x)N on admissible configuration of phase space a.e. The
Jacobian
∂X(−t, (x)N )
∂(x)N
has jumps only at such points (x)N that there are collisions at
time t, but above written multiplier is equal to zero in neighbourhoods of these points,
and therefore expression
−
N∑
i=1
Pi(−t, (x)N )
∂
∂Qi(−t, (x)N )
fN (X(−t, (x)N ))
(
∂X(−t, (x)N )
∂(x)N
)2
has desired property of continuity. Expression −
∑N
i=1
pi
∂
∂qi
DN (t, (x)N ), together
with boundary condition (2.12), coincide with this continuous function and has the same
property of continuity a.e. on the entire phase space of admissible configurations.
Remark. One can impose some additional conditions on functions fN ((x)N ) ∈ L0
N
in order to make function DN (t, (x)N ) continuous everywhere on admissible configura-
tions in phase space.
Namely, we restrict ourselves with functions fN ((x)N ) ∈ L0
N also equal to zero in
neighbourhoods of the hyperplanes where three or more particles collide instantaneously,
times of collisions become infinite and the number of collisions on finite time interval
is infinite. (The trajectories after instantaneous collisions of three or more particles are
defined as the same as before collisions but in opposite directions.) Obviously this set of
functions is again everywhere dense in LN . We continue denoting it by L0
N . Functions
DN (t, (x)N ) that correspond to such fN ((x)N ) are continuous with respect to (x)N
everywhere on phase space and for each time t.
3. Equation for sequence of correlation functions.
3.1. Definition of correlation functions. We will use commonly accepted definition
of correlation function. Namely, correlation functions ρ
(N)
s (t, x1, . . . , xs) in the frame-
work of the canonical ensemble is defined through the probability density DN (t, x1, . . .
. . . , xN ) as follows:
ρ(N)
s (t, x1, . . . , xs) =
= N(N − 1) . . . (N − s + 1)
∫
DN (t, x1, . . . , xs, xs+1, . . . , xN )dxs+1 . . . dxN ,
(3.1)
1 ≤ s ≤ N.
We integrate in (3.1) over entire phase space of particles with numbers s + 1, . . . , N,
but function DN (t, (x)N ) is equal to zero on forbidden configurations, and actually in-
tegration in (3.1) is carried out over the admissible configuration |qi − qj | ≥ a, (i, j) ⊂
⊂ (1, . . . , N). It is supposed that initial probability density DN (0, (x)N ) = fN ((x)N )
is normalized to unity:
∫
DN (0, (x)N )d(x)N =
∫
fN ((x)N )d(x)N = 1.
3.2. Equation for correlation function. In order to derive equations for correlation
functions we differentiate both sides of (3.1) with respect to time and use in the right-hand
side of (3.1) the Liouville equation (2.12). One obtains
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 831
∂
∂t
ρ(N)
s (t, x1, . . . , xs) =
= N(N − 1) · · · (N − s + 1) ×
×
∫ (
−
N∑
i=1
pi
∂
∂qi
DN (t, x1, . . . , xs, xs+1, . . . , xN )
)
dxs+1 . . . dxN . (3.2)
Now we are in the same situation as for system of N hard spheres with elastic colli-
sions, and we obtain the following hierarchy of equations [1 – 3]:
∂
∂t
ρ(N)
s (t, x1, . . . , xs) = −
s∑
i=1
pi
∂
∂qi
ρ(N)
s (t, x1, . . . , xs) +
+a2
s∑
i=1
∫
dps+1
∫
S2
dη〈η, (pi − ps+1)〉ρ(N)
s+1(t, x1, . . . , xs, qi − aη, ps+1) +
+
1
2
a2
∫
dxs+1
∫
dps+2
∫
S2
dη〈η, (ps+1 − ps+2)〉 ×
×ρ
(N)
s+2(t, x1, . . . , xs+1, qs+1 − aη, ps+2), 1 ≤ s ≤ N, (3.3)
where η is unit vector and S2 is unit sphere.
Now split spheres S2 in the second and third terms in the right-hand side of (3.3) into
two parts, S2 = S2
+ ∪ S2
−, where
S2
+(η|〈η, (pi − ps+1)〉 > 0), S2
−(η|〈η, (pi − ps+1)〉 < 0), i = 1, . . . , s,
and
S2
+(η|〈η, (ps+1 − ps+2)〉 > 0), S2
−(η|〈η, (ps+1 − ps+2)〉 < 0).
It follows from (2.11) that correlation functions satisfy the following boundary condi-
tions:
ρ
(N)
s+1(t, x1, . . . , qi, pi, . . . , xs, qi − aη, ps+1) =
= ρ
(N)
s+1(t, x1, . . . , qi, p
∗
i , . . . , xs, qi − aη, p∗s+1)
1
(1 − 2ε)2
,
(3.4)
ρ
(N)
s+2(t, x1, . . . , xs, qs+1, ps+1, qs+1 − aη, ps+2) =
= ρ
(N)
s+2(t, x1, . . . , xs, qs+1, p
∗
s+1, qs+1 − aη, p∗s+2)
1
(1 − 2ε)2
for 〈η, (pi − ps+1)〉 > 0 and 〈η, (ps+1 − ps+2)〉 > 0 correspondingly.
Show that the third term in the right-hand side of (3.3) is equal to zero. To this aim,
represent it as follows using (3.4):
a2
2
∫
dqs+1
∫
dps+1
∫
dps+2
[ ∫
S2
+
dη
∣∣〈η, (ps+1 − ps+2)〉
∣∣ ×
×ρ
(N)
s+2(t, x1, . . . , qs+1, p
∗
s+1, qs+1 − aη, p∗s+2)
1
(1 − 2ε)2
−
−
∫
S2
−
dη|〈η, (ps+1 − ps+2)〉|ρ(N)
s+2(t, x1, . . . , q+1, ps+1, qs+1 − aη, ps+2)
]
. (3.5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
832 D. YA. PETRINA, G. L. CARAFFINI
In the first term we use as new variables of integration momenta p∗s+1, p
∗
s+2. Taking
into account that 〈η, (p∗s+1 − p∗s+2)〉 =
1
1 − 2ε
〈η, (ps+1 − ps+2)〉 < 0 for 〈η, (ps+1 −
− ps+2)〉 > 0,
1
2
< ε < 1, we have η ∈ S2
− with respect to the variables p∗s+1, p∗s+2.
We have also dps+1dps+2
∣∣∣∣ 1
1 − 2ε
∣∣∣∣ = dp∗s+1dp
∗
s+2. (Note that we used the constant Ja-
cobian in momentum space equal to
∣∣∣∣ 1
1 − 2ε
∣∣∣∣ and take into account that linear transfor-
mation (1.3) maps domain 〈η, (ps+1−ps+2)〉 > 0 into domain 〈η, (p∗s+1−p∗s+2)〉 < 0. )
Therefore the first term is equal to
a2
2
∫
dqs+1
∫
dp∗s+1
∫
dp∗s+2
∫
S2
−
dη|〈η, (p∗s+1 − p∗s+2)〉| ×
×ρ
(N)
s+2(t, x1, . . . , qs+1, p
∗
s+1, qs+1 − aη, p∗s+2)
and it cancels with the second term.
Now split spheres S2 into two parts, S2
+(η | 〈η, (pi − ps+1)〉 > 0), S2
−(η | 〈η, (pi −
− ps+1)〉 < 0), change the vector η ∈ S2
− to vector −η ∈ S2
+, and use for η ∈ S2
+ the
boundary conditions (3.4)
ρ
(N)
s+1(t, x1, . . . , qi, pi, . . . , qi − aη, ps+1) =
= ρ
(N)
s+1(t, x1, . . . , qi, p
∗
i , . . . , qi − aη, ps+1)
1
(1 − 2ε)2
in the second term of the right-hand side of (3.3).
Finally, taking this into account, hierarchy (3.3) takes the following form:
∂
∂t
ρ(N)
s (t, x1, . . . , xs) = −
s∑
i=1
pi
∂
∂qi
ρ(N)
s (t, x1, . . . , xs) +
+ a2
s∑
i=1
∫
dps+1
∫
S2
+
dη〈η, (pi − ps+1)〉 ×
×
[
1
(1 − 2ε)2
ρ
(N)
s+1(t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗s+1) −
−ρ
(N)
s+1(t, x1, . . . , qi, pi, . . . , qi + aη, ps+1)
]
, N ≥ s ≥ 1, (3.6)
with the same boundary condition at qi−qj = aη for −
∑s
i=1
pi
∂
∂qi
ρ(N)
s (t, x1, . . . , xs),
ρ
(N)
s (t, x1, . . . , xs) as for the Liouville equation (2.12) for Ds(t, x1, . . . , xs).
(In the term ρs+1(t, x1, . . . , qi, pi, . . . , qi − aη, ps+1), with η ∈ S2
−, one uses a
new η′ = −η, η′ ∈ S2
+.) We have also initial condition ρ
(N)
s (t, x1, . . . , xs)|t=0 =
= ρ
(N)
s (0, x1, . . . , xs), s ≥ 1.
Consider equation for ρ
(N)
1 (t, x1)
∂
∂t
ρ
(N)
1 (t, x1) = −p1
∂
∂q1
ρ
(N)
1 (t, x1) + a2
∫
dp2
∫
S2
+
dη〈η, (p1 − p2)〉 ×
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 833
×
[
1
(1 − 2ε)2
ρ
(N)
2 (t, q1, p∗1, q1 − aη, p∗2) − ρ
(N)
2 (t, q1, p1, q1 + aη, p2)
]
and integrate it with respect to x1 over the entire phase space. Using the same tricks as
in proof that the term with ρs+2 is zero and supposing that lim
|q1|→∞
ρ
(N)
1 (t, q1, p1) = 0,
one obtains
∂
∂t
∫
ρ
(N)
1 (t, x1)dx1 = 0. (3.7)
This means that
∫
ρ
(N)
1 (t, x1)dx1 does not depend on t, i.e.,
∫
ρ
(N)
1 (t, x1)dx1 =
∫
ρ
(N)
1 (0, x1)dx1.
Taking into account that, according to definition (3.1),
ρ
(N)
1 (t, (x)1) = N
∫
DN (t, x1, x2, . . . , xN )dx2 . . . dxN
one obtains the law of conservation of full probability∫
DN (t, x1, . . . , xN )dx1 . . . dxN =
∫
DN (0, x1, . . . , xN )dx1 . . . dxN . (3.8)
Summarize the obtained above results in the following theorem.
Theorem III. The sequence of correlation functions ρ
(N)
s (t, x1, . . . , xs) (3.1), 1 ≤
≤ s ≤ N, satisfies the hierarchy of equations (3.6) with boundary and initial condition
and the probability density DN (t, x1, . . . , xN ) (2.5) satisfies the law of conservation of
full probability (3.8).
Remark. If one introduces the probability density by formula
DN (t, (x)N ) = fN (X(−t, (x)N )
∣∣∣∣
[
∂X(−t, (x)N )
∂(x)N
]∣∣∣∣
n
(3.9)
for n ≥ 1, n �= 2, then the sequence of correlation functions (3.1) satisfies the following
hierarchy:
∂
∂t
ρ(N)
s (t, x1, . . . , xs) = −
s∑
i=1
pi
∂
∂qi
ρ(N)
s (t, x1, . . . , xs) +
+a2
s∑
i=1
∫
dps+1
∫
S2
+
dη〈η, (pi − ps+1)〉 ×
×
[∣∣∣∣ 1
(1 − 2ε)n
∣∣∣∣ρ(N)
s+1(t, x1, . . . , qi, p
∗
i , qi − aη, p∗s+1) −
−ρ
(N)
s+1(t, x1, . . . , qi, pi, . . . , qi + aη, ps+1)
]
+
+
1
2
a2
∫
dxs+1
∫
dps+1
∫
S2
+
dη〈η, (ps+1 − ps+2)〉 ×
×
[∣∣∣∣ 1
(1 − 2ε)n
∣∣∣∣ρ(N)
s+2(t, x1, . . . , qs+1, p
∗
s+1, qs+1 − aη, p∗s+2) −
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
834 D. YA. PETRINA, G. L. CARAFFINI
−ρ
(N)
s+2(t, x1, . . . , qs+1, ps+1, qs+1 + aη, ps+2)
]
. (3.10)
The third term in the right-hand side of (3.10) is different from zero, because the
calculation used in the case n = 2 is not true for n �= 2. After the change of integration
variables from (ps+1, ps+2) to (p∗s+1, p
∗
s+2) in the first term with ρs+2 there will be
left the multiplier
∣∣∣∣ 1
(1 − 2ε)n−2
∣∣∣∣ , and this term does not cancel with the second term.
For the probability density DN (t, (x)N ) (3.9) with n �= 2 the law of conserva-
tion of full probability (3.8) is not true. This means that for the system of hard spheres
with inelastic collisions the unique “candidate” for the probability density is function
DN (t, (x)N ) (3.9) with n = 2, i.e., DN (t, (x)N ) defined according to (2.5).
3.3. Boundary conditions for correlation functions. According to boundary con-
ditions for function DN (t, (x)N ) and for Liouville equation (2.11), we have boundary
conditions (3.4) for correlation functions and the following boundary conditions for the
BBGKY hierarchy (3.6): in expression
−
S∑
i=1
pi
∂
∂qi
ρ(N)
s (t, x1, . . . , xs)
for qi − qj − aη = 0, η ∈ S2
+, (i, j) ⊂ (1, . . . , s), momenta pi and pj should be
replaced by p∗i , p∗j (1.3) and ρ
(N)
s by
1
(1 − 2ε)2
ρ(N)
s .
At first sight momenta pi, ps+1, i = 1, . . . , s, in 〈η, (pi − ps+1)〉 in the first term
of the integral in the right-hand side of equation (3.6) should be also replaced by p∗i ,
p∗s+1. It is not true. The reason is that under integral sign in (3.2) behavior of integrand
on hypersurfaces of lower dimension can be neglected. But we prefer to explain this
assertion on a very simple example of system of two spheres (rods) in one-dimensional
case.
We have Liouville equation
∂D2(t, x1, x2)
∂t
= −
(
p1
∂
∂q1
+ p2
∂
∂q2
)
D2(t, x1, x2)
with boundary condition: for q1 − q2 − aη = 0, 〈η, (p1 − p2)〉 > 0
D2(t, q1, p1, q2, p2) =
1
(1 − 2ε)2
D2(t, q1, p∗1, q2, p
∗
2),(
−p1
∂
∂q1
− p2
∂
∂q2
)
D2(t, q1, p1, q2, p2) =
=
1
(1 − 2ε)2
(
−p1
∗ ∂
∂q1
− p∗2
∂
∂q2
)
D2(t, q1, p∗1, q2, p
∗
2).
We have, following [3] and taking into account that
∂
∂t
D2(t, x1, x2) is different from
zero on admissible configurations |q1−q2| ≥ a and continuous with respect to (x1, x2),∫
∂
∂t
D2(t, x1, x2)dq2dp2 =
=
∫ (
−p1
∂
∂q1
− p2
∂
∂q2
)
D2(t, q1, p1, q2, p2)dq2dp2 =
= lim
ε→0
∫ {( q1−a−ε∫
−∞
dq2 +
∞∫
q1+a+ε
dq2
)
×
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 835
×
(
−p1
∂
∂q1
− p2
∂
∂q2
)
D2(t, q1, p1, q2, p2)
}
dp2. (3.11)
Now calculate the following integrals:( q1−a−ε∫
−∞
dq2 +
∞∫
q1+a+ε
dq2
)(
−p2
∂
∂q2
D2(t, q1, p1, q2, p2)
)
=
= −p2D2(t, q1, p1, q1 − a− ε, p2) + p2D2(t, q1, p1, q1 + a + ε, p2),( q1−a−ε∫
−∞
dq2 +
∞∫
q1+a+ε
dq2
)(
−p1
∂
∂q1
D2(t, q1, p1, q2, p2)
)
=
= −p1
∂
∂q1
( q1−a−ε∫
−∞
dq2 +
∞∫
q1+a+ε
dq2
)
D2(t, q1, p1, q2, p2) +
+p1D2(t, q1, p1, q1 − a− ε, p2) − p1D2(t, q1, p1, q1 + a + ε, p2).
Tend ε → 0 in (3.11) using above obtained formulas, continuity of function
D2(t, x1, x2) on admissible configurations and take into account that p1, p2 are fixed
and independent on ε. One obtains
∂F1(t, q1, p1)
∂t
=
= lim
ε→0
∫ {( q1−a−ε∫
−∞
dq2 +
∞∫
q1+a+ε
dq2
)
×
×
(
−p1
∂
∂q1
− p2
∂
∂q2
)
D2(t, q1, p1, q2, p2)
}
dp2 =
= −p1
∂
∂q1
F1(t, q1, p1) +
+
∫
dp2
{
(p1 − p2)
[
D2(t, q1, p1, q1 − a, p2) −D2(t, q1, p1, q1 + a, p2)
]}
.
Consider the following two cases: 1) p1 − p2 > 0; 2) p1 − p2 < 0. In the first case
one has, according to the boundary condition,
D2(t, q1, p1, q1 − a, p2) =
1
(1 − 2ε)2
D2(t, q1, p∗1, q1 − a, p∗2),
D2(t, q1, p1, q1 + a, p2) = D2(t, q1, p1, q1 + a, p2).
In the second case one has
D2(t, q1, p1, q1 − a, p2) = D2(t, q1, p1, q1 − a, p2),
D2(t, q1, p1, q1 + a, p2) =
1
(1 − 2ε)2
D2(t, q1, p∗1, q1 + a, p∗2).
Denote by η unit inner vector of sphere (rod) |q2−q1| = a with center in q1, η = +1
in point q2 = q1 − a, η = −1 in point q2 = q1 + a.
We have in the first case (p1 − p2) = 〈η, (p1 − p2)〉 > 0, η = +1
(p1 − p2)
[
D2(t, q1, p1, q1 − a, p2) −D2(t, q1, p1, q1 + a, p2)
]
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
836 D. YA. PETRINA, G. L. CARAFFINI
= η(p1 − p2)
[
1
(1 − 2ε)2
D2(t, q1, p∗1, q1 − aη, p∗2) −D2(t, q1, p1, q1 + aη, p2)
]∣∣∣∣
η=1
and in the second case −(p1 − p2) = (p1 − p2)η, 〈η, (p1 − p2)〉 > 0, η = −1
(p1 − p2)
[
D2(t, q1, p1, q1 − a, p2) −D2(t, q1, p1, q1 + a, p2)
]
=
= (p1 − p2)
[
D2(t, q1, p1, q1 + aη, p2) −
1
(1 − 2ε)2
D2(t, q1, p∗1, q1 − aη, p∗2)
]
=
= η(p1 − p2)
[
1
(1 − 2ε)2
D2(t, q1, p∗1, q1 − aη, p∗2) −D2(t, q1, p1, q1 + aη, p2)
]∣∣∣∣
η=−1
.
Denote by S1
+ vector η for which 〈η, (p1 − p2)〉 > 0. For (p1 − p2) > 0, S1
+
consists of vector η = +1, for (p1 − p2) < 0, S1
+ consists of vector η = −1. Denote
D2(t, q1, p1, q2, p2) = F2(t, q1, p1, q2, p2). Finally one obtains equations
∂F1(t, q1, p1)
∂t
= −p1
∂
∂q1
F1(t, q1, p1) +
∫
dp2
∑
η⊂S1
+
〈η, (p1 − p2)〉 ×
×
[
1
(1 − 2ε)2
F2(t, q1, p∗1, q1 − aη, p∗2) − F2(t, q1, p1, q1 + aη, p2)
]
,
(3.12)
∂F2(t, q1, p1, q2, p2)
∂t
=
(
− p1
∂
∂q1
− p2
∂
∂q2
)
F2(t, q1, p1, q2, p2)
and boundary condition for the second equation is the same as for D2(t, q1, p1, q2, p2).
Equations (3.12) is the hierarchy (3.6) for N = 2.
Analogous calculation has been performed for one-dimensional point-wise particles
in [4] on formal level.
3.4. Grand canonical ensemble. As known [1, 2], in grand canonical ensemble one
has a sequence of nonnormalized distribution functions DN (t, (x)N ), N ≥ 0, D0 = 1,
that satisfy Liouville equation (2.13) with boundary condition (2.12). The sequence of
correlation functions is defined as follows:
ρs(t, (x)s) =
1
Ξ
∞∑
n=0
∫
1
n!
Ds+n(t, x1, . . . , xs, xs+1, . . . , xs+n)dxs+1 . . . dxs+n, s ≥ 1,
(3.13)
where Ξ is the grand partition function
Ξ = 1 +
∞∑
n=1
∫
Dn(t, x1, . . . , xn)dx1 . . . dxn =
= 1 +
∞∑
n=1
∫
1
n!
Dn(0, x1, . . . , xn)dx1 . . . dxn. (3.14)
In (3.14) we used the law of conservation of full probability (3.8).
By repeating the derivation of hierarchy (3.6) for canonical ensemble [1, 2], one ob-
tains the hierarchy for grand canonical ensemble
∂ρs(t, x1, . . . , xs)
∂t
= −
s∑
i=1
pi
∂
∂qi
ρs(t, x1, . . . , xs) +
+ a2
s∑
i=1
∫
dps+1
∫
S2
+
dη〈η, (pi − ps+1)〉 ×
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 837
×
[
1
(1 − 2ε)2
ρs+1(t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗s+1) −
−ρs+1(t, x1, . . . , qi, pi, . . . , qi − aη, ps+1)
]
, s ≥ 1, (3.15)
with the same boundary conditions as for canonical ensemble.
Appendix A. In this appendix we present two very simple examples that explains the
reason why ∫
f(X(−t, x))
∂(X(−t, x))
∂x
dx �=
∫
f(x)dx.
Consider interval [0,∞) and on this interval define the following map T
T (x) = x, if 0 ≤ x ≤ 1,
(A.1)
T (x) = 2x, if x > 1.
Show that the Jacobian
dT (x)
dx
is defined as follows:
dT (x)
dx
= 1, if 0 ≤ x < 1,
dT (x)
dx
= δ(x− 1), if x = 1, (A.2)
dT (x)
dx
= 2, if 1 < x < ∞.
Calculate
dT (x)
dx
as distribution (generalized function). Let ϕ(x) be a test function,
then
∞∫
0
dT (x)
dx
ϕ(x)dx = −
∞∫
0
T (x)ϕ′(x)dx = −
1∫
0
xϕ′(x)dx−
∞∫
1
2xϕ′(x)dx =
= −ϕ(1) +
1∫
0
ϕ(x)dx + 2ϕ(1) + 2
∞∫
1
ϕ(x)dx = ϕ(1) +
1∫
0
ϕ(x)dx + 2
∞∫
1
ϕ(x)dx =
=
∞∫
0
δ(x− 1)ϕ(x)dx +
1∫
0
1 · ϕ(x)dx +
∞∫
1
2 · ϕ(x)dx. (A.3)
This formula gives us
dT (x)
dx
as stated before in (A.2).
Now consider the following integral with arbitrary smooth function f(x) defined on
[0,∞):
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838 D. YA. PETRINA, G. L. CARAFFINI
∞∫
0
f(T (x))
dT (x)
dx
dx =
1∫
0
1 · f(x)dx + f(1) +
∞∫
1
2 · f(2x)dx =
= f(1) +
1∫
0
f(x)dx +
∞∫
2
f(x)dx. (A.4)
From (A.4) one can see that there is finite contribution from ”hypersurface” x = 1
where the map T (x) is discontinuous and the interval 1 ≤ x ≤ 2 is absent, i.e., is lost
in the map T (x). Let us suppose that f(1) = 0. Then (A.4) is reduced to the following
final formula:
∞∫
0
f(T (x))
dT (x)
dx
dx =
1∫
0
f(x)dx +
∞∫
2
f(x)dx =
∞∫
0
f(x)dx−
2∫
1
f(x)dx. (A.5)
It follows from (A.5) that
∞∫
0
f(T (x))
dT (x)
dx
dx <
∞∫
0
f(x)dx (A.6)
for positive “distribution” f(x) ≥ 0, different from zero on interval (1, 2].
Consider second example with map
T (x) = x, 0 ≤ x ≤ 1, T (x) =
1
2
x, x > 1.
For T (x) one obtains
dT (x)
dx
= 1, 0 ≤ x ≤ 1,
dT (x)
dx
= −1
2
δ(x − 1), x = 1,
dT (x)
dx
=
1
2
, x > 1. If is easy to check that
∞∫
0
f(T (x))
dT (x)
dx
dx = −1
2
f(1) +
1∫
0
f(x)dx +
∞∫
1
2
f(x)dx.
If f(x) ≥ 0 and f(1) = 0 then
∞∫
1
f(T (x))
dT (x)
dx
dx =
∞∫
0
f(x)dx +
1∫
1
2
f(x)dx >
∞∫
0
f(x)dx. (A.7)
This two examples show that for discontinuous map there may be “loss” or “gain” of
domains.
This simple examples can help us to understand why
∫
fN (X(−t, (x)N ))
∂(X(−t, (x)N ))
∂(x)N
d(x)N �=
∫
f((x)N )d(x)N . (A.8)
It is because the map X(−t, (x)N ) is discontinuous and after collisions
∂(X(−t, (x)N ))
∂(x)N
�= 1, in the left hand side contributions from the hypersurfaces, where
collisions occur, may be finite, and some domains in the phase space may be “lost” in the
map induced by shift along trajectories X(−t, (x)N ), or may be “gained” in the map
induced by shift along trajectories X(t, (x)N ).
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ANALOGUE OF LIOUVILLE EQUATION AND BBGKY HIERARCHY . . . 839
We have shown in Section III that∫
fN (X(−t, (x)N ))
(
∂(X(−t, (x)N ))
∂(x)N
)2
=
∫
fN ((x)N )d(x)N
for fN ((x)N ) ⊂ L0
1 and it means that additional multiplier
∂(X(−t, (x)N ))
∂(x)N
, different
from 1 after collisions compensates “loss” of domains in the phase space. Note that
fN ((x)N ) ⊂ L0
1 is equal to zero on hypersurfaces where collisions occur and, therefore,
contributions from these hypersurfaces are equal to zero.
Appendix B. In deriving formulae (2.11) we did not take into account that for some
pi, pj momenta after collisions p∗i , p
∗
j are equal to pi, pj and
∂X(+0, (x)N )
∂(x)N
= 1.
For example, if 〈η, (pi − pj)〉 = 0. These momenta belong to hypersurfaces of lower
dimension and one can neglect them because DN (t, (x)N ) ⊂ LN for fN ((x)N ) ⊂ L0
N .
If one considers generalized functions fN ((x)N ) concentrated, for example, on hy-
persurfaces p1 = . . . = pN = p and with compact support with respect to (q)N then
∂X(−t, (x)N )
∂(x)N
= 1 and in DN (t, x1, . . . , qi, p
∗
i , . . . , qi − aη, p∗j , . . . , xN ), 〈η, (pi −
−pj)〉 > 0 momenta p∗i = pi, p
∗
j = pj .
For such initial distribution functions fN ((x)N ) hierarchy for correlation functions
(3.15) is reduced to the following one
∂
∂t
ρs(t, x1, . . . , xs) = −
s∑
i=1
pi
∂
∂qi
ρs(t, x1, . . . , xs), s ≥ 1. (B.1)
The second and third term in the right-hand side of (3.3) is equal to zero because pi −
− ps+1 = 0, ps+1 − ps+2 = 0.
Hierarchy (B.1) has stationary solution
ρs(t, x1, . . . , xs) =
s∏
i=1
(pi − p)
s∏
i<j=1
Θ(|qi − qj | = a).
1. Petrina D. Ya., Gerasimenko V. I., Malyshev P. V. Mathematical foundation of classical statistical mechan-
ics. Continuous systems. – Second edition. – London: Taylor and Francis, 2002. – 338 p.
2. Cercignani C., Gerasimenko V. I., Petrina D. Ya. Many-particle dynamics and kinetic equations. – Dor-
drecht: Kluwer, 1998. – 244 p.
3. Cercignani C. Theory and application of the Boltzmann equation. – Edinburgh: Scottich Acad. Press,
1975. – 496 p.
4. Benedetto D., Caglioti E., Pulvirenti M. A kinetic equation for granular media // Math. Model. and Numer.
Anal. – 1999. – 33, # 2. – P. 439 – 441.
Received 09.03.2004
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