Advances in theory of evolution equations of many colliding particles
The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations.
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Інститут математики НАН України
2023
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| author | Gerasimenko, Viktor Gapyak, Igor Герасименко, Віктор Гап'як, Ігор |
| author_facet | Gerasimenko, Viktor Gapyak, Igor Герасименко, Віктор Гап'як, Ігор |
| author_institution_txt_mv | [
{
"author": "Віктор Герасименко",
"institution": "Інтистут математики НАН України, м. Київ"
},
{
"author": "Ігор Гап'як",
"institution": "Київський національний університет імені Тараса Шевченка"
}
] |
| author_sort | Gerasimenko, Viktor |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2023-08-17T09:06:54Z |
| description | The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations. |
| doi_str_mv | 10.3842/trim.v20n1.528 |
| first_indexed | 2026-08-04T01:08:55Z |
| format | Article |
| fulltext |
Збірник праць Ін-ту математики НАН України (2023) т. 20, №1, 729–804
Advances in theory of evolution
equations of many colliding particles
V. I. Gerasimenko, I. V. Gapyak
Abstract. The review presents rigorous results of the theory of fundamen-
tal equations of evolution of many-particle systems with collisions and also
considers their connection with nonlinear kinetic equations describing the
collective behavior of particles in scaling approximations.
Анотація. В огляді подано строгі результати теорії фундаментальних
еволюційних рівнянь систем багатьох частинок із зіткненнями, а також
розглянуто їх зв’язок із нелінійними кінетичними рівняннями, які опи-
сують колективну поведінку частинок у скейлінгових наближеннях.
CONTENTS
1. Preface 730
1.1. A chronological overview of the theory of evolution
equations for many colliding particles 732
1.2. Evolution equations of finitely many hard spheres 734
2. Hierarchies of evolution equations for colliding particles 741
2.1. Hierarchy of evolution equations for reduced observables 741
2.2. The BBGKY hierarchy for reduced distribution functions 748
2.3. The Liouville hierarchy for correlation functions 756
2.4. The hierarchy of nonlinear evolution equations for reduced
correlation functions 767
3. Nonlinear kinetic equations for many hard spheres 772
3.1. On the Boltzmann–Grad scaling approximation 773
3.2. The Boltzmann–Grad limit of reduced observables 774
3.3. The Boltzmann kinetic equation 777
3.4. The Boltzmann kinetic equation with initial correlations 780
4. Origin of kinetic equations 783
4.1. The generalised Enskog kinetic equation 784
4.2. Dynamics of correlations governed by kinetic equations 788
2020 Mathematics Subject Classification: 35C10; 35Q20; 82C05; 82D05; 70F45
Keywords: hierarchy of evolution equations; kinetic equation; cumulant; semigroup of
operators; colliding particles
DOI : http://dx.doi.org/10.3842/trim.v20n1.528
729
730
5. Conclusion 792
5.1. On the dynamics of inelastic collisions 792
5.2. Some bibliographic notes on collisional dynamics 796
5.3. Outlook 797
References 799
1. PREFACE
This review presents the modern theory of evolution equations for sys-
tems of many particles with collisions. The traditional approach to describ-
ing the evolution of both finitely and infinitely many classical particles is
based on the description of the evolution of all possible states by means
of the reduced distribution functions governed by the BBGKY hierarchy
(Bogolyubov–Born–Green–Kirkwood–Yvon), which for finitely many parti-
cles is an equivalent to the Liouville equation for the probability distribution
function [14,82,83].
As is known, in the basis of the description of many-particle systems,
there are notions of the state and the observable. The functional of the
mean value of observables defines a duality between observables and states,
and as a consequence, there exist two approaches to the description of
evolution. Thus, an equivalent approach to the description of evolution is
to describe the evolution by means of observables governed by the so-called
dual BBGKY hierarchy for reduced observables [10,51].
In certain situations [14], the collective behavior of many-particle systems
can be adequately described by the kinetic equations. The conventional
philosophy of the description of kinetic evolution consists of the following:
if the initial state is specified by a one-particle reduced distribution function,
then the evolution of the state can be effectively described by means of a
one-particle reduced distribution function governed by the nonlinear kinetic
equation in a suitable scaling limit. A well-known historical example of a
kinetic equation is the Boltzmann equation, which describes the process of
particle collisions in rarefied gases [13].
Nowadays, a number of papers have appeared that discuss possible ap-
proaches to the rigorous description of the evolution of many colliding par-
ticles [25, 29, 52, 88, 89, 92, 94]. In particular, this is related to the prob-
lem of the rigorous derivation of the Boltzmann kinetic equation from the
underlying hierarchies of fundamental evolution equations. The conven-
tional method of deriving the Boltzmann equation consists of constructing
the Boltzmann–Grad scaling asymptotics of the BBGKY hierarchy solu-
tion represented by series expansions within the framework of perturbation
theory. The most advanced and rigorous results to date have been ob-
tained for systems of colliding particles, which is why the motivation for
Evolution equations of colliding particles 731
writing this work is to discuss the theory of evolution equations for such
many-particle systems [4, 52,98].
In what follows, we mainly consider three challenges are left open until
recently [52].
One of them is related to the construction of solutions to the Cauchy
problem for hierarchies of fundamental evolution equations for systems of
many particles with collisions, using the example of hard spheres with elas-
tic collisions. It is established that the cluster expansions of groups of
operators for the Liouville equations for observables and a state of many
hard spheres underlie the classification of possible non-perturbative solu-
tion representations of the Cauchy problem for the dual BBGKY hierarchy
and the BBGKY hierarchy, respectively. As a consequence, these solutions
are represented in the form of series expansions whose generating operators
are the cumulants of the groups of operators for the Liouville equations.
In a particular case, the non-perturbative solutions of these hierarchies are
represented in the form of the perturbation (iteration) series as a result of
applying analogs of the Duhamel equation to their generating operators.
The paper also formulated the Liouville hierarchy of evolution equations
for correlation functions of the state and established that the dynamics of
correlations underlie the description of the evolution of an infinitely many
hard spheres that are governed by the BBGKY hierarchy for the reduced
distribution functions or the hierarchy of nonlinear evolution equations for
the reduced correlation functions [53].
Another challenge considered below is an approach to the description of
the kinetic evolution within the framework of the evolution of the observ-
ables of many colliding particles [51]. The problem of a rigorous description
of the kinetic evolution of hard sphere observables is considered by giving
the example of the Boltzmann–Grad asymptotics of the non-perturbative
solution of the dual BBGKY hierarchy. One of the advances of this appro-
ach is the opportunity to construct kinetic equations, taking into account
the correlations of particles in the initial state and also the description of
the process of propagation of initial correlations in scaling approximations.
In addition, this paper discusses the approach to describing the evolution
of a state by means of the state of a typical particle of a system of many
hard spheres, or, in other words, we consider the origin of the description
of the evolution of the state of hard spheres by the Enskog-type kinetic
equation [47]. One of the applications of the method is related to the chal-
lenge of the rigorous derivation of kinetic equations of the non-Markovian
type based on the dynamics of correlations, which allow us to describe the
memory effects in complex systems.
732
Thus, this review presents rigorous results in the theory of fundamental
evolution equations for many-particle systems with collisions, as well as
nonlinear kinetic equations describing their collective behavior in scaling
approximations.
1.1. A chronological overview of the theory of evolution equations
for many colliding particles. The theory of kinetic equations begins
with the work of L. Boltzmann [9], where an evolution equation for colli-
sion dynamics was formulated based on phenomenological models of kinetic
phenomena. Later, to generalize the Boltzmann equation for the case of
dense gases or fluids, D. Enskog [26] formulated a kinetic equation for a
system of many hard spheres, now known as the Enskog equation.
The idea that equations formulated on the basis of phenomenological
models of phenomena, such as hydrodynamics equations or kinetic equa-
tions, should be derived from fundamental evolutionary equations for sys-
tems of many particles, namely the Liouville equations, apparently goes
back to the works of D. Hilbert [76] and H. Poincaré [85]. At the Second
International Congress of Mathematicians, held in Paris at the beginning
of the 20th century, D. Hilbert formulated this idea in his list of open ques-
tions as follows: ”Boltzmann’s work on the principles of mechanics suggests
the problem of developing mathematically the limiting processes that lead
from the atomistic view to the laws of motion of continua”.
The approach to describe the evolution of the state of many-particle
systems in a way equivalent to the Liouville equation for the probability
distribution function based on the hierarchy of evolution equations for re-
duced distribution functions, known in our time as the BBGKY hierarchy,
was most consistently formulated in the work of M. M. Bogolyubov [6],
and independently by M. Born and H. S. Green [11], J. G. Kirkwood [77],
J. Yvon [100].
In his famous monograph ”Problems of the Dynamical Theory in Sta-
tistical Physics” [6], which was actually the manuscript of a 1945 report
at the Institute of Mathematics in Kyiv, M. M. Bogolyubov also formu-
lated a consistent approach to the problem of deriving kinetic equations
from the dynamics of many particles. Using the methods of perturba-
tion theory, an approach was developed to construct a generalization of
the Boltzmann equation, known as the Bogolyubov kinetic equation, as
well as justify the Vlasov and Landau kinetic equations for the first time.
Thanks to this work, the irreversibility mechanism of the evolution of sys-
tems of many particles, whose dynamics are described as reversible in time
by the fundamental equations of motion, became clear. A little later, in
the Proceedings of the Institute of Mathematics, M. M. Bogolyubov pub-
lished a paper on the derivation of the equations of hydrodynamics from
Evolution equations of colliding particles 733
the BBGKY hierarchy [5]. These works became widely known as a result
of G. E. Uhlenbeck’s lectures [16]. Bogolyubov’s ideas became the cradle
of modern kinetic theory, as M. H. Ernst noted in his review [27]. The Bo-
golyubov method of deriving the Boltzmann kinetic equation directly from
the BBGKY hierarchy is presented in modern terminology in the book [14].
Since these lines are written in the work dedicated to the 160th anniver-
sary of the birth of Dmytro Oleksandrovych Grave, the first academician
of the Ukraine Academy of Sciences in mathematics and the founder of the
Institute of Mathematics in 1920, it should be reminded that M. M. Bo-
golyubov was one of the students of D. O. Grave, thanks to whom he became
an outstanding scientist in the field of mathematical physics. It is known
that in 1922 at the age of thirteen, M. Bogolyubov became a participant in
Grave’s famous mathematician seminar. In 1925, at the request of professor
D. O. Grave, the Small Presidium of Ukrgolovnauka made a decision: ”In
view of his phenomenal abilities in mathematics, to consider M. Bogolyubov
as a post-graduate student of the research department of mathematics in
Kyiv from June, 1925”, and already in 1928 he defended his doctoral thesis.
By the way, this historical precedent convincingly illustrates the signifi-
cance of a scientific school for the development of mathematics.
Rigorous methods for the description of the equilibrium state by the
Gibbs distribution functions [71], i.e., by solutions of the steady BBGKY
hierarchy, originate from the works of M. M. Bogolyubov and D. Ya. Pet-
rina within the framework of a canonical ensemble [7] and D. Ruelle within
the framework of a grand canonical ensemble [90] and were investigated in
numerous works as a new direction of the progress of modern mathemati-
cal physics in the 70-80s. In our time, mention above work of M. M. Bo-
golyubov, D. Ya. Petrina and B. I. Khatset was included in the special
issue of the Ukrainian Journal of Physics dedicated to the 90th Academy of
Sciences of Ukraine, which was was republished the most significant works
of Ukrainian physicists over the entire period of the Academy’s existence,
in other words, works that contributed to the golden fund of world physical
science (Golden Pages of Ukrainian Physics [8]).
Note that the mathematical description of the Gibbs equilibrium states
for infinitely many particles forms the principal part of modern statistical
mechanics. The main rigorous results about the equilibrium Gibbs states
were presented in the book [64].
The mathematical theory of the BBGKY hierarchy originates from the
works of D. Ya. Petrina and V. I. Gerasimenko [56–58, 81, 82] in the early
80s. The dual BBGKY hierarchy for reduced functions of observables
was introduced by V. I. Gerasimenko in the middle of the 1980s, and the
734
theory of these evolution equations began to develop in the last two decades
(see [52] and references therein).
Mathematical methods for deriving nonlinear kinetic equations from the
BBGKY hierarchy began to develop intensively in the early 1980s [12, 79].
One of the achievements of this period was the formal derivation of the
Boltzmann equation from the dynamics of an infinite number of hard
spheres in the Boltzmann–Grad limit.
In the approach to the problem of deriving kinetic equations from particle
dynamics, which was formulated by H. Grad [73] and has now become
generally accepted, the philosophy of the description of kinetic evolution
looks like this: if the initial state is specified by a one-particle distribution
function, then the evolution of the state of many particles can be effectively
described by means of a one-particle distribution function governed by a
nonlinear kinetic equation in a certain scaling approximation.
The Boltzmann–Grad asymptotics of a solution of the BBGKY hierar-
chy, represented as an iteration series for infinitely many hard spheres, was
first constructed by C. Cercignani [12] and O. E. Lanford [79] and rigorously
justified in a series of papers [59–61,83] by D. Ya. Petrina and V. I. Gerasi-
menko (some details are given in sections 2.2 and 3.1). Incidentally, it
should be noted that the results of the papers [57, 59] were discussed with
Academician M. M. Bogolyubov at that time and were submitted by him
for publication.
Rigorous results of the theory of evolution equations for hard spheres and
the derivation of the Boltzmann equation from the BBGKY hierarchy in
the Boltzmann–Grad limit were summarized in monographs C. Cercignani,
V. I. Gerasimenko and D. Ya. Petrina [14], C. Cercignani, R. Illner and
M. Pulvirenti [15], H. Spohn [95] at the end of the 90th.
The last two decades of progress in solving the problem of rigorous deriva-
tion of kinetic equations from the collisional dynamics of particles are re-
presented in numerous recent works [4, 20, 25, 28, 29, 87–89, 92, 94]. The
challenges of this area of contemporary mathematical physics are also dis-
cussed in the latest review [52]. With respect to the modern progress in the
theory of evolution equations of quantum many-particle systems, we refer
to the overview [38].
In these notes, recent advances in the theory of evolutionary equations
for many colliding particles will be considered; more precisely, we focus on
the dynamics of many hard spheres with elastic collisions.
1.2. Evolution equations of finitely many hard spheres. The de-
scription of many-particle systems is based on the concepts of an observable
and a state. The mean value functional (expectation values) of observables
defines the duality between observables and a state, and as a result, there
Evolution equations of colliding particles 735
are two approaches to describing evolution. The evolution of the system
of finitely many colliding particles considered below is governed by such
fundamental evolution equations as the Liouville equation for observables
or its dual equation for a state.
Within the framework of a non-fixed, i.e., arbitrary but finite average
number of identical particles (non-equilibrium grand canonical ensemble),
the observables and the state of a hard sphere system are described by the
sequences of functions
A(t) = (A0, A1(t, x1), . . . , An(t, x1, . . . , xn), . . .)
at instant t P R and by the sequence
D(0) = (D0, D
0
1(x1), . . . , D
0
n(x1, . . . , xn), . . .)
of the probability distribution functions at the initial moment, respectively.
These functions are defined on the phase spaces of the corresponding num-
ber of particles, i.e., xi ” (qi, pi) P R3 ˆ R3 is phase coordinates that
characterize a center of the i hard sphere with a diameter of σ ą 0 in
the space R3 and its momentum and are symmetrical with respect to ar-
bitrary permutations of their arguments. For configurations of a system of
identical particles of a unit mass interacting as hard spheres the following
inequalities are satisfied: |qi ´ qj | ě σ, i ‰ j ě 1, i.e., the set
Wn ” ␣
(q1, . . . , qn) P R3n | |qi ´ qj | ă σu
for at least one pair (i, j) : i ‰ j P (1, . . . , n)
(
, n ą 1, is the set of forbidden
configurations.
A mean value functional of the observable of a hard sphere system is
represented by the series expansion [83]:
xAy(t) = (I,D(0))´1(A(t), D(0)), (1.1)
where the following abbreviated notation
(A(t), D(0)) ”
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
An(t, x1, . . . , xn)D
0
n(x1, . . . , xn)dx1 ¨ ¨ ¨ dxn
was used and the coefficient (I,D(0)) is a normalizing factor (grand canon-
ical partition function).
We remark that in the particular case of a system of N ă 8 hard spheres
the observables and a state are described by the one-component sequences:
A(N)(t) = (0, . . . , 0, AN (t), 0, . . .)
and
D(N)(0) = (0, . . . , 0, D0
N , 0, . . .),
736
respectively, and therefore, functional (1.1) has the following representation
xA(N)y(t) = (I,D(N)(0))´1(A(N)(t), D(N)(0)) ”
” (I,D(N)(0))´1 1
N !
ż
(R3ˆR3)N
AN (t, x1, . . . , xN )D0
N (x1, . . . , xN )dx1 ¨ ¨ ¨ dxN ,
where
(I,D(N)(0)) =
1
N !
ż
(R3ˆR3)N
D0
N (x1, . . . , xN )dx1 ¨ ¨ ¨ dxN
is the normalizing factor (canonical partition function), and it is usually
assumed that the normalization condition (I,D(N)(0)) = 1 holds.
Let Cγ be the space of sequences b = (b0, b1, . . . , bn, . . .) of bounded con-
tinuous functions bn = bn(x1, . . . , xn) that are symmetric with respect to
permutations of the arguments x1, . . . , xn, equal to zero on the set of for-
bidden configurations Wn and equipped with the norm:
}b}Cγ = max
ně0
γn
n!
}b}Cn = max
ně0
γn
n!
sup
x1,...,xn
|bn(x1, . . . , xn)|,
where 0 ă γ ă 1. We also introduce the space L1
α = ‘8
n=0α
nL1
n of sequences
f = (f0, f1, . . . , fn, . . .) of integrable functions fn = fn(x1, . . . , xn) that are
symmetric with respect to permutations of the arguments x1, . . . , xn, equal
to zero on the set Wn and equipped with the norm:
}f}L1
α
=
8ÿ
n=0
αn
ż
(R3ˆR3)n
|fn(x1, . . . , xn)|dx1 ¨ ¨ ¨ dxn,
where α ą 1 is a real number. If A(t) P Cγ and D(0) P L1
α mean value
functional (1.1) exists and determines the duality between observables and
states.
The evolution of the observables
A(t) = (A0, A1(t, x1), . . . , An(t, x1, . . . , xn), . . .)
is described by the Cauchy problem for the sequence of the weak formulation
of the Liouville equations for hard spheres with elastic collisions [14]. On
the space Cγ a non-perturbative solution A(t) = S(t)A(0) of the Liouville
equation of many hard spheres is determined by the following group of
Evolution equations of colliding particles 737
operators [83]:
(S(t)b)n(x1, . . . , xn) = Sn(t, 1, . . . , n)bn(x1, . . . , xn)
.
=
.
=
$
’&
’%
bn(X1(t, x1, . . . , xn), . . . , Xn(t, x1, . . . , xn)),
if(x1, . . . , xn) P (R3n ˆ (R3nzWn)),
0, if(q1, . . . , qn) P Wn,
(1.2)
where for arbitrary t P R the function Xi(t) is a phase trajectory of ith
particle constructed in book [14] and the set M0
n of the zero Lebesgue mea-
sure, which consists of the phase space points that are specified such initial
data that during the evolution generate multiple collisions, i.e., collisions of
more than two particles, more than one two-particle collision at the same
instant, and an infinite number of collisions within a finite time interval.
On the space Cγ one-parameter mapping (1.2) is an isometric ˚-weak
continuous group of operators, i.e., it is a C0̊ -group [19]. The infinitesimal
generator L = ‘8
n=0Ln of the group of operators (1.2) has the structure:
Ln =
nÿ
j=1
L(j) +
nÿ
j1ăj2=1
Lint(j1, j2),
where the operator L(j) defined on the set Cn,0 of continuously differen-
tiable functions with compact supports is the Liouville operator of free evo-
lution of the j hard sphere and for t ě 0 the operators L(j) and Lint(j1, j2)
are defined by the formulas [22,83]:
L(j) .= xpj , B
Bqj y,
Lint(j1, j2)bn
.
= σ2
ż
S2+
dηxη, (pj1 ´ pj2)y δ(qj1 ´ qj2 + ση)ˆ
ˆ(
bn(x1, . . . , qj1 , pj̊1 , . . . , qj2 , pj̊2 , . . . , xn) ´ bn(x1, . . . , xn)
)
,
(1.3)
respectively. In formulas (1.3) the symbol x¨, ¨y denotes a scalar product, δ
is the Dirac measure, S2+
.
= tη P R3 | |η| = 1, xη, (pj1 ´ pj2)y ą 0u and the
post-collision momenta: pj̊1 , pj̊2 are defined by the equalities
pj̊1
.
= pj1 ´ η xη, (pj1 ´ pj2)y ,
pj̊2
.
= pj2 + η xη, (pj1 ´ pj2)y .
(1.4)
For t ă 0 operator (1.3) is defined by the corresponding expression [14].
It should be noted that the structure of generator (1.3) is determined,
on the one hand, by the singular interaction potential of hard spheres and,
on the other, by the fact that the group of operators (1.2) is defined for
738
pairwise collisions outside the set M0
n of the zero Lebesgue measure defined
above in (1.2).
As mentioned above, the evolution of observables for many hard spheres,
i.e., the sequences of functions An(t) = Sn(t)A
0
n, n ě 1, is governed by the
Cauchy problem for the sequence of the weak formulation of the Liouville
equations with these generators [51]:
B
BtAn(t) =
( nÿ
j=1
L(j) +
nÿ
j1ăj2=1
Lint(j1, j2)
)
An(t),
An(t)|t=0 = A0
n, n ě 1.
(1.5)
For mean value functional (1.1) the following representation holds
(A(t), D(0)) = (A(0), D(t)),
where the sequence D(t) = (1, D1(t, x1), . . . , Dn(t, x1, . . . , xn), . . .) of dis-
tribution functions is defined as follows:
D(t) = S˚(t)D(0),
and the mapping S˚(t) is an adjoint operator to operator (1.2) in the sense
of mean value functional (1.1). We emphasize that this equality is a con-
sequence of a fundamental property of Hamiltonian systems, namely, the
validity of the Liouville theorem for phase trajectories [14], i.e., isometry of
the groups of operators (1.2).
On the space L1
α =
8‘
n=0
αnL1
n of sequences of integrable functions, the
group of operators S˚(t) =
8‘
n=0
Sn̊(t) is an adjoint to the group of opera-
tors (1.2) in the sense of functional (1.1) and is defined as follows [83]:
S˚(t) = S(´t). (1.6)
On the space L1
α, one-parameter mapping (1.6) is an isometric strong con-
tinuous group of operators, i.e., it is a C0-group [19]. The infinitesimal
generator L˚ = ‘8
n=0Ln̊ of this group of operators has the structure:
Ln̊
.
=
nÿ
j=1
L˚(j) +
nÿ
j1ăj2=1
Li̊nt(j1, j2),
Evolution equations of colliding particles 739
and for t ą 0 the operators L˚(j) and Li̊nt(j1, j2) are defined by the formu-
las:
L˚(j)fn
.
= ´xpj , B
Bqj yfn,
Li̊nt(j1, j2)fn
.
= σ2
ż
S2+
dηxη, (pj1 ´ pj2)yˆ
ˆ (
fn(x1, . . . , pj̊1 , qj1 , . . . , pj̊2 , qj2 , . . . , xn)δ(qj1 ´ qj2 + ση)´
´ fn(x1, . . . , xn)δ(qj1 ´ qj2 ´ ση)
)
,
(1.7)
respectively, the pre-collision momenta pj̊1 , pj̊2 are defined by relations (1.4)
and notations accepted in formula (1.3) are used. For t ă 0 these operators
are defined by the corresponding expressions [64].
We note that the evolution of the state, i.e., the sequence of probability
distribution functions Dn(t) = Sn̊(t)D
0
n, n ě 1, describes by the Cauchy
problem for the sequence of the weak formulation of the Liouville equations
for many hard spheres with these generators [83]:
B
BtDn(t) =
( nÿ
j=1
L˚(j) +
nÿ
j1ăj2=1
Li̊nt(j1, j2)
)
Dn(t),
Dn(t)|t=0 = D0
n, n ě 1.
(1.8)
Thus, the evolution of finitely many colliding particles is governed by
the fundamental evolution equations, such as the Liouville equation for
observables (1.5) or its dual equation for a state (1.8).
To formulate another representation of the mean value functional (1.1) in
terms of sequences of so-called reduced observables and reduced distribution
functions, on sequences of bounded continuous functions we introduce an
analog of the creation operator
(a+b)s(x1, . . . , xs)
.
=
sÿ
j=1
bs´1((x1, . . . , xs)z(xj)), (1.9)
and on sequences of integrable functions, we introduce an adjoint operator
to operator (1.9) in the sense of mean value functional (1.1) which is an
analogue of the annihilation operator
(af)n(x1, . . . , xn) =
ż
R3ˆR3
fn+1(x1, . . . , xn, xn+1)dxn+1. (1.10)
Then as a consequence of the validity of equalities:
(b, f) = (ea
+
e´a+b, f) = (e´a+b, eaf),
740
for mean value functional (1.1) the following representation holds:
xAy(t) = (I,D(0))´1(A(t), D(0)) = (B(t), F (0)), (1.11)
where a sequence of the reduced observables is defined by the formula
B(t) = e´a+S(t)A(0), (1.12)
and a sequence of so-called reduced distribution functions is defined as
follows (known as the non-equilibrium grand canonical ensemble [82])
F (0) = (I,D(0))´1eaD(0),
respectively.
Thus, according to the definition of the operator e´a+ , the sequence of
reduced observables (1.12) in component-wise form is represented by the
expansions:
Bs(t, x1, . . . , xs) =
=
sÿ
n=0
(´1)n
n!
sÿ
j1‰...‰jn=1
(S(t)A(0))s´n
(
(x1, . . . , xs)z(xj1 , . . . , xjn)),
s ě 1,
(1.13)
The mean value functional (1.11) also has the following representation:
(B(t), F (0)) = (B(0), F (t)). (1.14)
The sequence F (t) = (1, F1(t, x1), . . . , Fn(t, x1, . . . , xn), . . .) of reduced dis-
tribution functions is defined as follows (known as the non-equilibrium
grand canonical ensemble [82])
F (t) = (I,D(0))´1eaS˚(t)D(0), (1.15)
where the mapping S˚(t) is an adjoint operator (1.6) to operator (1.2).
According to the definition of the operator ea, the sequence of reduced
distribution functions (1.15) in component-wise form is represented by the
series:
Fs(t, x1, . . . , xs) = (I,D(0))´1ˆ
ˆ
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
(S˚(t)D(0))s+n(x1, . . . , xs+n)dxs+1 ¨ ¨ ¨ dxs+n, s ě 1,
where the coefficient (I,D(0)) is the normalizing factor as above.
We emphasize that a widely used approach to the description of the
evolution of many hard spheres is based on the evolution of a state de-
termined by the BBGKY hierarchy for reduced distribution functions [14].
An equivalent approach to describing evolution is based on reduced observ-
ables (1.12) governed by the dual hierarchy of evolution equations [52].
Evolution equations of colliding particles 741
2. HiERARCHiES OF EVOLUTiON EQUATiONS FOR COLLiDiNG PARTiCLES
As is well known, hierarchies of evolution equations for sequences of re-
duced functions of observables and, accordingly, of a state for a finitely
many hard spheres are equivalent to the Liouville equations. Their ad-
vantages consist in the possibility of rigorously describing the evolution of
infinitely many hard spheres whose collective behavior exhibits thermody-
namic (statistical) features, namely, the existence of an equilibrium state
in such a system as well as the kinetic or hydrodynamic behavior in corres-
ponding scaling approximations [14,21,64].
An alternative approach to the description of the evolution of the state
of a hard-sphere system is based on functions determined by the cluster
expansions of the probability distribution functions. The cumulants of
probability distribution functions are interpreted as correlation functions
and are governed by the Liouville hierarchy. The following outlines the
approach to the description of the evolution of a state by means of both
reduced distribution functions and reduced correlation functions, which is
based on the dynamics of correlations [53]. It should be emphasized that
on a microscopic scale, the macroscopic characteristics of fluctuations of
observables are directly determined by the reduced correlation functions.
2.1. Hierarchy of evolution equations for reduced observables. The
motivation for describing the evolution of many-particle systems in terms of
reduced observables is related to possible equivalent representations of the
mean value functional (mathematical expectation) of observables, namely
as (1.11) compared to the traditionally used form (1.14).
The evolution of sequence (1.12) of reduced observables of many hard
spheres is determined by the Cauchy problem of the following abstract
hierarchy of evolution equations [10,51]:
d
dt
B(t) = LB(t) +
[
L, a+
]
B(t), (2.1)
B(t)
ˇ̌
t=0
= B(0), (2.2)
where the operator L is generator (1.3) of the group of operators (1.2) for
hard spheres, the symbol
[¨, ¨] denotes the commutator of operators, which
in equation (2.1) has the following component-wise form:
([
L, a+
]
b
)
s
(x1, . . . , xs) =
=
sÿ
j1‰j2=1
Lint(j1, j2)bs´1(t, (x1, . . . , xs)zxj1), s ě 1.
In a component-wise form the hierarchy of evolution equations (2.1)) for
hard-sphere fluids, in fact, is a sequence of recurrence evolution equations
742
(in literature it is known as the dual BBGKY hierarchy [52]). We adduce
the simplest examples of recurrent evolution equations (2.1):
B
BtB1(t, x1) = L(1)B1(t, x1),
B
BtB2(t, x1, x2) =
( 2ÿ
i=1
L(j) + Lint(1, 2)
)
B2(t, x1, x2)+
+Lint(1, 2)
(
B1(t, x1) +B1(t, x2)
)
,
where the generators of these equations are defined by formula (1.3).
The non-perturbative solution of the Cauchy problem of the dual BBGKY
hierarchy (2.1),(2.2) for hard spheres is a sequence of reduced observables
represented by the following expansions [65,66]:
Bs(t, x1, . . . , xs) =
(
ea
+
A(t)B(0)
)
s
(x1, . . . , xs) =
=
sÿ
n=0
1
n!
sÿ
j1‰¨¨¨‰jn=1
A1+n
(
t, t(1, . . . , s)z(j1, . . . , jn)u, j1, . . . , jn
)ˆ
ˆB0
s´n(x1, . . . , xj1´1, xj1+1, . . . , xjn´1, xjn+1, . . . , xs), s ě 1,
(2.3)
where the mappings A1+n(t), n ě 0, are the generating operators which
are represented as cumulant expansions with respect of groups of opera-
tors (1.2). The simplest examples of reduced observables (2.3) are given by
the following expansions:
B1(t, x1) = A1(t, 1)B
0
1(x1),
B2(t, x1, x2) = A1(t, t1, 2u)B0
2(x1, x2) + A2(t, 1, 2)(B
0
1(x1) +B0
1(x2)).
To determine the generating operators of expansions of reduced observ-
ables (2.3), we will introduce the notion of dual cluster expansions of groups
of operators (1.2) in terms of operators interpreted as their cumulants. For
this end on sequences of one-parametric mappings
u(t) = (0, u1(t), . . . , un(t), . . .)
we define the following ‹-product [90]
(u(t) ‹ ru(t))s(1, . . . , s) =
ÿ
Y Ă(1,...,s)
u|Y |(t, Y )rus´|Y |(t, (1, . . . , s)zY ), (2.4)
where
ř
Y Ă(1,...,s) is the sum over all subsets Y of the set (1, . . . , s).
Using the definition of the ‹-product (2.4), the dual cluster expansions
of groups of operators (1.2) are represented by the mapping Exp‹ in the
form
S(t) = Exp‹ A(t) = I+
8ÿ
n=1
1
n!
A(t)‹n,
Evolution equations of colliding particles 743
where S(t) = (0, S1(t, 1), . . . , Sn(t, 1, . . . , n), . . .) and I = (1, 0, . . . , 0, . . .).
In component-wise form the dual cluster expansions are represented by the
following recursive relations:
Ss(t, (1, . . . , s)z(j1, . . . , jn), j1, . . . , jn) =
=
ÿ
P: (t(1,...,s)z(j1,...,jn)u, j1,...,jn)=Ť
i
Xi
ź
XiĂP
A|Xi|(t,Xi), n ě 0, (2.5)
where the set consisting of one element of indices (1, . . . , s)z(j1, . . . , jn) we
denoted by the symbol t(1, . . . , s)z(j1, . . . , jn)u and the symbol
ř
P means
the sum over all possible partitions P of the set
(t(1, . . . , s)z(j1, . . . , jn)u, j1, . . . , jn)
into |P| nonempty mutually disjoint subsets Xi Ă (1, . . . , s).
The solution of recursive relations (2.5) are represented by the inverse
mapping Ln˚ in the form of the cumulant expansion
A(t) = Ln‹(I+ S(t)) =
8ÿ
n=1
(´1)n´1
n
S(t)‹n.
Then the (1 + n)th-order dual cumulant of groups of operators (1.2) is
defined by the following expansion:
A1+n(t, t(1, . . . , s)z(j1, . . . , jn)u, j1, . . . , jn) .=
.
=
ÿ
P: (t(1,...,s)z(j1,...,jn)u,j1,...,jn)=Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
S|θ(Xi)|(t, θ(Xi)), (2.6)
where the above notation is used and the declusterization mapping θ is
defined by the formula:
θ(t(1, . . . , s)z(j1, . . . , jn)u) = (1, . . . , s)z(j1, . . . , jn).
The dual cumulants (2.6) of the first two orders have the form:
A1(t, t1, . . . , su) = Ss(t, 1, . . . , s),
A1+1(t, t(1, . . . , s)z(j)u, j) =
= Ss(t, 1, . . . , s) ´ Ss´1(t, (1, . . . , s)z(j))S1(t, j).
If bs P Cs, then for (1 + n)th-order cumulant (2.6) of groups of opera-
tors (1.2) the estimate is valid
››A1+n(t)bs
››
Cs ď
ÿ
P: (t(1,...,s)z(j1,...,jn)u,j1,...,jn)=Ť
i
Xi
(|P| ´ 1)!
››bs
››
Cs ď
ď
n+1ÿ
k=1
s(n+ 1, k)(k ´ 1)!
››bs
››
Cs ď n!en+2
››bs
››
Cs ,
(2.7)
744
where s(n+ 1, k) are the Stirling numbers of the second kind. Then accor-
ding to this estimate (2.7) for the generating operators of expansions (2.3)
provided that γ ă e´1 the inequality valid
››B(t)
››
Cγ
ď e2(1 ´ γe)´1
››B(0)
››
Cγ
. (2.8)
In fact, the following criterion holds.
Criterion. A solution of the Cauchy problem of the dual BBGKY hierar-
chy (2.1),(2.2) is represented by expansions (2.3) if and only if the generating
operators of expansions (2.3) are solutions of cluster expansions (2.5) of
the groups of operators (1.2) of the Liouville equations for hard spheres.
The necessity condition means that cluster expansions (2.5) hold for
groups of operators (1.2). These recurrence relations are derived from defi-
nition (1.13) of reduced observables, provided that they are represented as
expansions (2.3) for the solution of the Cauchy problem of the dual BBGKY
hierarchy (2.1),(2.2).
The sufficient condition means that the infinitesimal generator of one-
parameter mapping (2.3) coincides with the generator of the sequence of
recurrence evolution equations (2.1). Indeed, in the space Cγ the following
existence theorem is true [51].
Theorem. A non-perturbative solution of the Cauchy problem (2.1),(2.2)
is represented by expansions (2.3) in which the generating operators are
cumulants of the corresponding order (2.6) of groups of operators (1.2):
Bs(t, x1, . . . , xs) =
sÿ
n=0
1
n!
sÿ
j1‰¨¨¨‰jn=1
ÿ
P: (t(1,...,s)z(j1,...,jn)u,j1,...,jn)=Ť
i
Xi
(
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
S|θ(Xi)|(t, θ(Xi))ˆ
ˆB0
s´n(x1, ..., xj1´1, xj1+1, ..., xjn´1, xjn+1, ..., xs)
)
, s ě 1.
(2.9)
Under the condition γ ă e´1 for initial data B(0) P C0
γ of finite sequences
of infinitely differentiable functions with compact supports sequence (2.9) is
a unique global classical solution, and for arbitrary initial data B(0) P Cγ
is a unique global generalized solution.
We note that the one component sequences B(1)(0) = (0, b1(x1), 0, . . .)
of reduced observables correspond to the additive-type observable, and the
sequences
B(k)(0) = (0, . . . , bk(x1, . . . , xk), 0, . . .)
of reduced observables correspond to the k-ary-type observables [10].
Evolution equations of colliding particles 745
If initial data (2.2) is specified by the additive-type reduced observable,
then the structure of solution expansion (2.9) is simplified and attains the
form
B(1)
s (t, x1, . . . , xs) = As(t, 1, . . . , s)
sÿ
j=1
b1(xj), s ě 1, (2.10)
where the generating operator As(t) is the sth-order cumulant (2.6) of the
groups of operators (1.2).
An example of the additive-type observables is a number of particles,
i.e., the sequence N (1)(0) = (0, 1, 0, . . .), then
N (1)
s (t) = As(t, 1, . . . , s)s =
ÿ
P: (1,...,s)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
sÿ
j=1
1 =
=
sÿ
k=1
(´1)k´1s(s, k)(k ´ 1)!s = sδs,1 = N (1)
s (0),
where s(s, k) are the Stirling numbers of the second kind and δs,1 is a Kro-
necker symbol. Consequently, the observable of a number of hard spheres
is an integral of motion and, in particular, the average number of particles
is preserving in time.
In the case of initial k-ary-type, k ě 2, reduced observables solution
expansion (2.9) takes the form
B(k)
s (t) = 0, 1 ď s ă k,
B(k)
s (t, x1, . . . , xs) =
1
(s´ k)!
ˆ
ˆ
sÿ
j1‰¨¨¨‰js´k=1
A1+s´k
(
t, t(1, . . . , s)z(j1, . . . , js´k)u, j1, . . . , js´k
)ˆ
ˆ bk(x1, . . . , xj1´1, xj1+1, . . . , xjs´k´1, xjs´k+1, . . . , xs),
s ě k,
(2.11)
where the generating operator A1+s´k(t) is the (1 + s ´ k)th-order cumu-
lant (2.6) of the groups of operators (1.2).
We emphasize that cluster expansions (2.5) of the groups of operators (1.2)
underlie of the classification of possible solution representations of the
Cauchy problem (2.1),(2.2) of the dual BBGKY hierarchy. Indeed, us-
ing cluster expansions (2.5) of the groups of operators (1.2), other solution
representations can be constructed.
For example, let us express the cumulants A1+n(t), n ě 2, of groups of
operators (1.2) with respect to the 1st-order and 2nd-order cumulants. The
746
equalities are true:
A1+n(t, t(1, . . . , s)z(j1, . . . , jn)u, j1, . . . , jn) =
=
ÿ
H‰Y Ă(j1,...,jn)
A2(t, t(1, . . . , s)z(j1, . . . , jn)u, tY u)ˆ
ˆ
ÿ
P: (j1,...,jn)zY=
Ť
i
Xi
(´1)|P| |P|!
|P|ź
i=1
A1(t, tXiu), n ě 2,
where
ř
H‰Y Ă(j1,...,jn)
is a sum over all nonempty subsets Y Ă (j1, . . . , jn).
Then, taking into account the identity
ÿ
P: (j1,...,jn)zY=
Ť
i
Xi
(´1)|P||P|!
|P|ź
i=1
A1(t, tXiu)B0
s´n((x1, . . . , xs)z(xj1 , . . . , xjn)) =
=
ÿ
P: (j1,...,jn)zY=
Ť
i
Xi
(´1)|P| |P|!B0
s´n((x1, . . . , xs)z(xj1 , . . . , xjn)),
(2.12)
and the equalities
ÿ
P: (j1,...,jn)zY=
Ť
i
Xi
(´1)|P| |P|! = (´1)|(j1,...,jn)zY |, Y Ă (j1, . . . , jn), (2.13)
for solution expansions (2.3) of the dual BBGKY hierarchy we derive the
following representation:
Bs(t, x1, . . . , xs) = A1(t, t1, . . . , su)B0
s (x1, . . . , xs)+
+
sÿ
n=1
1
n!
sÿ
j1‰¨¨¨‰jn=1
ÿ
Y Ă(1,...,s)z(j1,...,jn),
Y ‰H
(´1)|(j1,...,jn)zY | A2(t, tj1, . . . , jnu, tY u)ˆ
ˆB0
s´n((x1, . . . , xs)z(xj1 , . . . , xjn)), s ě 1,
where notations accepted above are used.
Taking into account that initial reduced observables depend only from
the certain phase space arguments, we deduce the reduced representation
Evolution equations of colliding particles 747
of expansions (2.9):
B(t) =
8ÿ
n=0
1
n!
nÿ
k=0
(´1)n´k n!
k!(n´ k)!
(a+)n´kS(t)(a+)kB(0) =
= S(t)B(0) +
8ÿ
n=1
1
n!
[
. . .
[
S(t), a+
]
, . . . , a+looooomooooon
n-times
]
B(0) =
= e´a+S(t)ea
+
B(0).
(2.14)
Therefore, in component-wise form the generating operators of these expan-
sions represented as expansions (2.3) are the following reduced cumulants
of groups of operators (1.2):
U1+n(t, t1, . . . , s´ nu, s´ n+ 1, . . . , s) =
=
nÿ
k=0
(´1)k
n!
k!(n´ k)!
Ss´k(t, 1, . . . , s´ k).
(2.15)
Indeed, solutions of the recursive relations (2.5) with respect to first-
order cumulants can be represented as expansions in terms of cumulants
acting on variables on which the initial reduced observables depend, and in
terms of cumulants not acting on these variables
A1+n(t,t(1, . . . , s)z(j1, . . . , jn)u, j1, . . . , jn) =
=
ÿ
Y Ă(j1,...,jn)
A1(t, t(1, . . . , s)z((j1, . . . , jn) Y Y )u)ˆ
ˆ
ÿ
P: (j1,...,jn)zY=
Ť
i
Xi
(´1)|P| |P|!
|P|ź
i=1
A1(t, tXiu),
where
ř
Y Ă(j1,...,jn)
is the sum over all possible subsets Y Ă (j1, . . . , jn).
Then taking into account the identity (2.13) and the equalities (2.12) we
derive expansions (2.14) over reduced cumulants (2.15).
We note that traditionally the solution of the BBGKY hierarchy for
states of many hard spheres is represented by perturbation series [14,29,83].
The expansions (2.14) can also be represented as expansions (iterations) of
perturbation theory [10]:
B(t) =
8ÿ
n=0
tż
0
dt1 ¨ ¨ ¨
tn´1ż
0
dtn S(t´ t1)
[
L, a+
]ˆ
ˆ S(t1 ´ t2) ¨ ¨ ¨S(tn´1 ´ tn)
[
L, a+
]
S(tn)B(0).
748
Indeed, as a result of applying of analogs of the Duhamel equation to ge-
nerating operators (2.6) of expansions (2.3) we derive in component-wise
form, for examples,
U1(t, t1, . . . , su) = Ss(t, 1, . . . , s),
U2(t, t(1, . . . , s)z(j1)u, j1) =
=
tż
0
dt1 Ss(t´ t1, 1, . . . , s)
sÿ
j2=1,
j2‰j1
Lint(j1, j2)Ss´1(t1, (1, . . . , s)zj1).
Recall that the mean value functional (1.11) exists if B(0) P Cγ and
F (0) P L1
α. In the case of the observable of a number of hard spheres
N (1)(t) = (0, 1, 0, . . .), this means that
ˇ̌
(N (1)(t), F (0))
ˇ̌
=
ˇ̌
ˇ
ż
R3ˆR3
F 0
1 (x1)dx1
ˇ̌
ˇ ď ››F 0
1
››
L1
1
ă 8. (2.16)
Consequently, the states of a finite number of hard spheres are described by
sequences of functions from the space L1
α. To describe an infinite number
of hard spheres, it is necessary to consider reduced distribution functions
from appropriate function spaces, for example, from the space of sequences
of bounded functions with respect of the configuration variables [58,79,83].
2.2. The BBGKY hierarchy for reduced distribution functions.
As mentioned already, the evolution of systems of many particles is tradi-
tionally described as the evolution of the state of a system based on the
representation of the mean value functional for observables (1.14). In this
case, the sequence of reduced distribution functions is determined by the
hierarchy of evolution equations, known as the BBGKY hierarchy, whose
generator is the operator adjoint to the generator of the hierarchy of evo-
lution equations for reduced observables (2.1) in the sense of mean value
functional (1.11).
The evolution of sequence (1.15) of reduced distribution functions from
the space L1
α is governed by the Cauchy problem of the BBGKY hierarchy
for many hard spheres [6, 14,83]:
d
dt
F (t) = L˚F (t) +
[
a,L˚]F (t), (2.17)
F (t)
ˇ̌
t=0
= F (0), (2.18)
where the symbol
[¨, ¨] denotes the commutator of operator (1.10) and of
the Liouville operator (1.7), which is the generator of the isometric group
Evolution equations of colliding particles 749
of operators (1.6). Thus, in evolution equation (2.17), the second term has
the following component-wise form:
([
a,L˚]f
)
s
(x1, . . . , xs) =
=
sÿ
i=1
ż
R3ˆR3
dxs+1 Li̊nt(i, s+ 1)fs+1(t, x1, . . . , xs+1), s ě 1.
For t ą 0 in a one-dimensional space, i.e., for gas of hard rods, this term of
a generator has the form [82]:
sÿ
i=1
ż
RˆR
dxs+1Li̊nt(i, s+ 1)fs+1(t) =
=
sÿ
i=1
8ż
0
dP P
(
fs+1(t, x1, . . . , qi, pi ´ P, . . . , xs, qi ´ σ, pi)´
´ fs+1(t, x1, . . . , xs, qi ´ σ, pi + P )+
+ fs+1(t, x1, . . . , qi, pi + P, . . . , xs, qi + σ, pi)´
´ fs+1(t, x1, . . . , xs, qi + σ, pi ´ P )
)
,
(2.19)
and for t ă 0 this collision integral has the corresponding form [83].
It should be noted that for the system of a fixed, finite number of hard
spheres, the BBGKY hierarchy is an equation system for a finite sequence of
reduced distribution functions. Such an equation system is equivalent to the
Liouville equation for the distribution function, which describes all possible
states of finitely many hard spheres. For a system of an infinite number
of hard spheres, the BBGKY hierarchy is an infinite chain of evolution
equations, which can be derived as the thermodynamic limit of the BBGKY
hierarchy of a fixed finite number of hard spheres [14]. We note that since a
sequence of functions can be determined based on a generating functional,
the corresponding hierarchy of evolution equations can also be formulated
as the evolution equation for such a generating functional [44].
A non-perturbative solution of the Cauchy problem of the BBGKY hi-
erarchy (2.17),(2.18) is a sequence of reduced distribution functions repre-
sented by the following expansions [52,67]:
Fs(t, x1, . . . , xs) = (eaA˚(t)F (0))s(x1, . . . , xs) =
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
A1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n)ˆ
ˆF 0
s+n(x1, . . . , xs+n)dxs+1 ¨ ¨ ¨ dxs+n, s ě 1,
(2.20)
750
where the mappings A1̊+n(t), n ě 0, are the generating operators which are
represented by the cumulant expansions with respect to the group
S˚(t) =
8‘
n=0
Sn̊(t)
of operators (1.6).
Using the definition of the ‹-product (2.4), the cluster expansions of the
groups of operators (1.6) are represented by the mapping Exp‹ in the form
S˚(t) = Exp‹ A˚(t).
In component-wise form cluster expansions are represented by the following
recursive relations:
Ss̊+n(t, 1, . . . , s, s+ 1, . . . , s+ n) =
=
ÿ
P: (t1,...,su,s+1,...,s+n)=
Ť
i
Xi
ź
XiĂP
A˚
|Xi|(t,Xi), n ě 0, (2.21)
where the set consisting of one element of indices (1, . . . , s) we denoted by
the symbol t(1, . . . , s)u and the symbol
ř
P means the sum over all possible
partitions P of the set
(t1, . . . , su, s+ 1, . . . , s+ n)
into |P| nonempty mutually disjoint subsets Xi.
The solution of recursive relations (2.21) are represented by the inverse
mapping Ln‹ in the form of the cumulant expansion
A˚(t) = Ln‹(I+ S˚(t)).
Then the (1 + n)th-order cumulant of the group S˚(t) =
8‘
n=0
Sn̊(t) of oper-
ators (1.6) is defined by the following expansion:
A1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n)
.
=
.
=
ÿ
P: (t1,...,su,s+1,...,s+n)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
S˚
|θ(Xi)|(t, θ(Xi)), (2.22)
where the declusterization mapping θ is defined by the formula:
θ(t1, . . . , su) = (1, . . . , s)
Evolution equations of colliding particles 751
and the above notation is used. The simplest examples of cumulants (2.22)
of the groups of operators (1.6) have the form:
A1̊(t, t1, . . . , su) .= Ss̊ (t, 1, . . . , s),
A1̊+1(t, t1, . . . , su, s+ 1)
.
= Ss̊+1(t, 1, . . . , s+ 1) ´ Ss̊ (t, 1, . . . , s)S1̊ (t, s+ 1),
A1̊+2(t, t1, . . . , su, s+ 1, s+ 2)
.
= Ss̊+2(t, 1, . . . , s+ 2)´
´ Ss̊+1(t, 1, . . . , s+ 1)S1̊ (t, s+ 2) ´ Ss̊+1(t, 1, . . . , s, s+ 2)S1̊ (t, s+ 1)´
´ Ss̊ (t, 1, . . . , s)S2̊ (t, s+ 1, s+ 2) + 2!Ss̊ (t, 1, . . . , s)S1̊ (t, s+ 1)S1̊ (t, s+ 2).
If fs P L1
s, then taking into account that
››Sn̊(t)
››
L1
n
= 1, for the (1 + n)th-
order cumulant (2.22) the following estimate is valid:
››A1̊+n(t)fs+n
››
L1
s+n
ď
ÿ
P: (t1,...,su,s+1,...,s+n)=
Ť
i
Xi
(|P| ´ 1)!
››fs+n
››
L1
s+n
ď
ď
n+1ÿ
k=1
s(n+ 1, k)(k ´ 1)!
››fs+n
››
L1
s+n
ď n!en+2
››fs+n
››
L1
s+n
,
(2.23)
where s(n+ 1, k) are the Stirling numbers of the second kind.
Then, according to this estimate (2.23) for the generating operators of
expansions (2.20), provided that α ą e series (2.20) converges on the norm
of the space L1
α, and the inequality holds
}F (t)}L1
α
ď cα}F (0)}L1
α
,
where cα = e2(1 ´ e
α)
´1. The parameter α can be interpreted as the value
inverse to the average number of hard spheres.
In fact, the following criterion holds.
Criterion. A solution of the Cauchy problem of the BBGKY hierarchy
(2.17), (2.18) is represented by expansions (2.20) if and only if the generating
operators of expansions (2.20) are solutions of cluster expansions (2.21) of
the groups of operators (1.6).
The necessity condition means that cluster expansions (2.21) are take
place for groups of operators (1.6). These recurrence relations are derived
from definition (1.15) of reduced distribution functions, provided that they
are represented as expansions (2.20) for the solution of the Cauchy problem
of the BBGKY hierarchy (2.17),(2.18).
The sufficient condition means that the infinitesimal generator of one-
parameter mapping (2.20) coincides with the generator of the BBGKY
hierarchy (2.17). Indeed, in the space L1
α the following existence theorem
is true [65].
752
Theorem. If α ą e, a non-perturbative solution of the Cauchy prob-
lem of the BBGKY hierarchy (2.17),(2.18) is represented by series ex-
pansions (2.20) in which the generating operators are cumulants of the
corresponding order (2.22) of groups of operators (1.6):
Fs(t, x1, . . . , xs) =
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
ÿ
P: (t1,...,su,s+1,...,s+n)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!ˆ
ˆ
ź
XiĂP
S˚
|θ(Xi)|(t, θ(Xi))F
0
s+n(x1, . . . , xs+n)dxs+1 ¨ ¨ ¨ dxs+n, s ě 1.
(2.24)
For initial data F (0) P L1
0 of finite sequences of infinitely differentiable
functions with compact supports sequence (2.24) is a unique global clas-
sical solution and for arbitrary initial data F (0) P L1
α is a unique global
generalized solution.
We observe that cluster expansions (2.21) of the groups of operators (1.6)
underlie the classification of possible solution representations (2.20) of the
Cauchy problem of the BBGKY hierarchy (2.17),(2.18). In a particular
case, non-perturbative solution (2.24) of the BBGKY hierarchy for many
hard spheres can be represented in the form of the perturbation (iteration)
series as a result of applying analogs of the Duhamel equation to cumulants
(2.22) of groups of operators.
Indeed, let us put groups of operators in the expression of cumulant (2.22)
into a new order with respect to the groups of operators which act on the
variables (x1, . . . , xs)
A1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n) =
=
ÿ
Y Ă (s+1,...,s+n)
S˚
s+|Y |(t, (1, . . . , s) Y Y )ˆ
ˆ
ÿ
P :(s+1,...,s+n)zY=
Ť
i
Yi
(´1)|P||P|!
ź
YiĂP
S˚
|Yi|(t, Yi).
(2.25)
If Yi Ă (s + 1, . . . , s + n), then for the integrable functions F 0
s+n and the
unitary group of operators S˚(t) = ‘8
n=0Sn̊(t) the equality is valid
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n
ź
YiĂP
S˚
|Yi|(t;Yi)F
0
s+n(x1, . . . , xs+n) =
=
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+nF
0
s+n(x1, . . . , xs+n).
Evolution equations of colliding particles 753
Then, taking into account the validity for arbitrary Y Ă (s+ 1, . . . , s+ n)
the following equality:
ÿ
P :(s+1,...,s+n)zY=
Ť
i
Yi
(´1)|P||P|! = (´1)|(s+1,...,s+n)zY |,
according to expression (2.25) for series expansions (2.24) of the BBGKY
hierarchy, we obtain
Fs(t, x1, . . . , xs) =
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
U1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n)ˆ
ˆ F 0
s+n(x1, . . . , xs+n)dxs+1 ¨ ¨ ¨ dxs+n, s ě 1,
(2.26)
where U1̊+n(t) is the (1 + n)th-order reduced cumulant of the groups of
operators (1.6)
U1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n) =
=
ÿ
Y Ă(s+1,...,s+n)
(´1)|(s+1,...,s+n)zY | S˚
|(1,...,s)YY |(t, (1, . . . , s) Y Y ).
Using the symmetry property of initial reduced distribution functions, for
integrand functions in every term of series (2.26) the following equalities
are valid
ÿ
Y Ă(s+1,...,s+n)
(´1)|(s+1,...,s+n)zY )| S˚
|(1,...,s)YY |(t, (1, . . . , s) Y Y )F 0
s+n =
=
nÿ
k=0
(´1)k
s+nÿ
i1ă...ăin´k=s+1
Ss̊+n´k(t, 1, . . . , s, i1, . . . , in´k)F
0
s+n =
=
nÿ
k=0
(´1)k
n!
k!(n´ k)!
Ss̊+n´k(t, 1, . . . , s+ n´ k)F 0
s+n(x1, . . . , xs+n).
Thus, the (1 + n)th-order reduced cumulant represents by the following
expansion [80]:
U1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n) =
=
nÿ
k=0
(´1)k
n!
k!(n´ k)!
Ss̊+n´k(t, 1, . . . , s+ n´ k),
and consequently, we derive the representation for series expansions of a
solution of the BBGKY hierarchy [82] which is can be written down in
754
terms of an analogue of the annihilation operator (1.10):
F (t) =
8ÿ
n=0
1
n!
nÿ
k=0
(´1)k
n!
k!(n´ k)!
an´kS˚(t)akF (0) =
= S˚(t)F (0) +
8ÿ
n=1
1
n!
[
a, . . . ,
[
alooomooon
n-times
, S˚(t)
]
. . .
]
F (0) =
= eaS˚(t)e´aF (0).
(2.27)
Finally, in view of the validity of the equality
S˚(t´ τ)
[
a,L˚]S˚(τ)F (0) = d
dτ
S˚(t´ τ)aS˚(τ)F (0),
expansion (2.27) is represented in the form of perturbation (iteration) series
of the BBGKY hierarchy (2.17) for many hard spheres
F (t) =
8ÿ
n=0
tż
0
dt1 . . .
tn´1ż
0
dtnS
˚(t´ t1)
[
a,L˚]S˚(t1 ´ t2) . . .ˆ
ˆ S˚(tn´1 ´ tn)
[
a,L˚]S˚(tn)F (0),
or in component-wise form [6,58,79]:
Fs(t,x1, . . . , xs) =
8ÿ
n=0
tż
0
dt1 ¨ ¨ ¨
tn´1ż
0
dtn
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n Ss̊ (t´ t1)ˆ
ˆ
sÿ
j1=1
Li̊nt(j1, s+ 1)Ss̊+1(t1 ´ t2) ¨ ¨ ¨Ss̊+n´1(tn´1 ´ tn)ˆ
ˆ
s+n´1ÿ
jn=1
Li̊nt(jn, s+ n)Ss̊+n(tn)F
0
s+n(x1, . . . , xs+n), s ě 1.
(2.28)
Let us make some comments concerning the existence of solutions to
the Cauchy problem of the BBGKY hierarchy for initial data from various
function spaces.
In the spaces of sequences of integrable functions, the existence and
uniqueness of a global in time non-perturbative solution was proved in the
papers [67, 82] (see also book [14]). It should be noted that the first few
terms of the (2.24) series were established in papers [17, 18, 74, 75] as an
analog of cluster expansions of the reduced equilibrium distribution func-
tions.
The BBGKY hierarchy describes both the non-equilibrium and equilib-
rium states. Non-equilibrium states are described by the solution of the
initial value problem for this hierarchy, and, correspondingly, equilibrium
Evolution equations of colliding particles 755
states are solutions of the steady BBGKY hierarchy. The existence of
equilibrium solutions to the steady BBGKY hierarchy has been reviewed
in books [14,64].
As is known, to describe the evolution of a state of infinitely many par-
ticles, the suitable functional space is the space of sequences of functions
bounded with respect to the configuration variables and decreasing with
respect to the momentum ones; in particular, the equilibrium distribution
functions belong to this space. For the solution extension from the space
of sequences of integrable functions to this space, the method of the ther-
modynamic limit was developed [14,57,58].
For one-dimensional many-particle systems with short-range potential,
using the method of interaction region developed by Petrina [81] for solution
representation (2.26), the existence theorem for the BBGKY hierarchy was
proved for the first time in this functional space. By a similar method for
the initial reduced distribution functions from such a space, the existence
of a mean value functional for solution (2.9) of the dual BBGKY hierarchy
(2.1) was established in the paper [91].
As mentioned above, for a solution representation of the Cauchy problem
of the BBGKY hierarchy for hard spheres is widely used in the represen-
tation as a series of perturbation theory (2.28) (an iteration series over the
evolution of the state of selected groups of particles) [6,15,58,79,95]. In this
form, the solution is applied to construct its Boltzmann–Grad asymptotics,
which is governed by the Boltzmann kinetic equation (see section 3.1). The
justification of a solution represented as an iteration series for hard spheres
is based on giving a rigorous mathematical meaning to every term of the
iteration series and on the proof of its convergence. The main difficulty in
this problem is that the phase trajectories of particles for a system with a
singular interaction potential are defined almost everywhere in the phase
space, and initial distribution functions in the iteration series are concen-
trated on lower-dimensional manifolds. It is necessary to ensure that the
trajectories are defined on these manifolds. This problem was completely
solved in the papers [61,83].
In the case of infinitely many hard spheres a local in time solution [58]
of the Cauchy problem of the BBGKY hierarchy is represented by iteration
series for arbitrary initial data from the space of sequences of functions
bounded with respect to configuration variables and for initial data close
to the equilibrium state it is a global in time solution [83]. For such initial
data in a one-dimensional space for hard sphere system the existence of
global in time solution was proved in the paper [34].
756
In addition, we remark that the correlation decay property, known as
the Bogolyubov correlation weakening principle [6], for the solution of the
BBGKY hierarchy for hard spheres was proved in [68].
2.3. The Liouville hierarchy for correlation functions. An alterna-
tive approach to the description of a state of finitely many hard spheres
consists in the employment of functions determined by the cluster expan-
sions of the probability distribution functions. The solutions to such cluster
expansions are cumulants (semi-invariants) of probability distribution func-
tions and are interpreted, from a physical point of view, as correlations of
a state or correlation functions. The evolution of correlation functions is
governed by the so-called Liouville hierarchy [31]. Historically, there have
been several approaches to describing correlations in many-particle sys-
tems. Among them, we mention the well-known approach to the dynamics
of correlations by I. Prigogine [86] and R. Balescu [1] and its applications
in plasma theory.
Further, it will be established that the constructed dynamics of correla-
tion underlie the description of the dynamics of infinitely many hard spheres
governed by the BBGKY hierarchy for reduced distribution functions or the
hierarchy of nonlinear evolution equations for reduced correlation functions,
i.e., of the cumulants of reduced distribution functions.
We introduce a sequence of correlation functions g(t) = (1, g1(t, x1), . . . ,
gs(t, x1, . . . , xs), . . .) by means of cluster expansions of the probability dis-
tribution functions D(t) = (1, D1(t, x1), . . . , Dn(t, x1, . . . , xn), . . .), defined
on the set of allowed configurations R3nzWn as follows:
Dn(t, x1, . . . , xn) = gn(t, x1, . . . , xn)+
+
ÿ
P:(x1,...,xn)=
Ť
i
Xi,
|P|ą1
ź
XiĂP
g|Xi|(t,Xi), n ě 1, (2.29)
where
ř
P:(x1,...,xn)=
Ť
i
Xi, |P|ą1 is the sum over all possible partitions P of the
set of the arguments (x1, . . . , xn) into |P| ą 1 nonempty mutually disjoint
subsets Xi Ă (x1, . . . , xn).
On the set R3nzWn solutions of recursion relations (2.29) are given by
the following expansions:
gs(t, x1, . . ., xs) = Ds(t, x1, . . . , xs)+
+
ÿ
P:(x1,...,xs)=
Ť
i
Xi,
|P|ą1
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
D|Xi|(t,Xi), s ě 1. (2.30)
Evolution equations of colliding particles 757
The structure of expansions (2.30) is such that the correlation functions
can be treated as cumulants (semi-invariants) of the probability distribu-
tion functions [24]. Such an interpretation of these functions is due to
the fact that the probability distribution function of statistically indepen-
dent hard spheres on allowed configurations is described by the product of
single-particle correlation functions (probability distribution functions of
each particle):
gs(t, x1, . . . , xs) =
sź
i=1
g1(t, xi)XR3szWs
δs,1, s ě 1.
The evolution of the sequence of correlation functions (2.30) of many
hard spheres is determined by the Cauchy problem of the weak formulation
of the Liouville hierarchy of the following evolution equations [53]:
B
Btgs(t, x1, . . . , xs) = Ls̊ (1, . . . , s)gs(t, x1, . . . , xs)+
+
ÿ
P: (x1,...,xs)=X1
Ť
X2
ÿ
i1P pX1
ÿ
i2P pX2
Li̊nt(i1, i2)g|X1|(t,X1)g|X2|(t,X2),
(2.31)
gs(t, x1, . . . , xs)
ˇ̌
t=0
= g0s(x1, . . . , xs), s ě 1, (2.32)
where
ř
P: (x1,...,xs)=X1
Ť
X2
is the sum over all possible partitions P of the
set (x1, . . . , xs) into two nonempty mutually disjoint subsets X1 and X2,
the symbol pXi means the set of indexes of the set Xi of phase space co-
ordinates and the operator Ls̊ is defined on the subspace L1
0 Ă L1 by
formulas (1.7). It should be noted that the Liouville hierarchy (2.31) is the
evolution recurrence equations set.
For t ě 0 we give a few examples of recurrence equations set (2.31) for a
system of hard spheres:
B
Btg1(t, x1) = ´xp1, B
Bq1 yg1(t, x1),
B
Btg2(t, x1, x2) = ´
2ÿ
j=1
xpj , B
Bqj yg2(t, x1, x2)+
+ σ2
ż
S2+
dηxη, (p1 ´ p2)y
(
g2(t, q1, p1̊ , q2, p2̊)δ(q1 ´ q2 + ση)´
´g2(t, x1, x2)δ(q1 ´ q2 ´ ση)
)
+
+ σ2
ż
S2+
dηxη, (p1 ´ p2)y
(
g1(t, q1, p1̊)g1(t, q2, p2̊)δ(q1 ´ q2 + ση)´
´g1(t, x1)g1(t, x2)δ(q1 ´ q2 ´ ση)
)
,
758
where it was used notations accepted above in definition (1.7).
Thus, in terms of correlation functions (2.30), the evolution of the states
of a finite number of hard spheres is described by an equivalent method
compared to probability distribution functions, namely, within the frame-
work of the dynamics of correlations.
We note that because the Liouville hierarchy (2.31) is the recurrence
evolution equations set, we can construct a solution of the Cauchy prob-
lem (2.31),(2.32), integrating each equation of the hierarchy as the inho-
mogeneous Liouville equation. For example, as a result of the integration
of the first two equations of the Liouville hierarchy (2.31), we obtain the
following equalities:
g1(t, x1) = S1(´t, 1)g01(x1),
g2(t, 1, 2) = S2(´t, 1, 2)g02(x1, x2)+
=
tż
0
dt1S2(t1 ´ t, 1, 2)Li̊nt(1, 2)S1(´t1, 1)S1(´t1, 2)g01(x1)g01(x2).
Then for the corresponding term on the right-hand side of the second equal-
ity, an analog of the Duhamel equation holds
tż
0
dt1S2(t1 ´ t, 1, 2)Li̊nt(1, 2)S1(´t1, 1)S1(´t1, 2) =
= ´
tż
0
dt1
d
dt1
(
S2(t1 ´ t, 1, 2)S1(´t1, 1)S1(´t1, 2)
)
=
= S2(´t, 1, 2) ´ S1(´t, 1)S1(´t, 2) = A2̊(t, 1, 2),
where A2̊(t) is the second-order cumulant (2.22) of groups of operators
(1.6). As a result of similar transformations for s ą 2, the solution of the
Cauchy problem (2.31),(2.32), constructed using an iterative procedure, can
be represented as expansions in cumulants of groups of operators (1.6).
If the initial state is specified by the sequence
g(0) = (1, g01(x1), . . . , g
0
n(x1, . . . , xn), . . .),
of correlation functions g0n P L1
n, n ě 1, then the evolution of all possible
states, i.e., the sequence
g(t) = (1, g1(t, x1), . . . , gs(t, x1, . . . , xs), . . .)
Evolution equations of colliding particles 759
of the correlation functions gs(t), s ě 1, is represented by the following
expansions [53]:
gs(t, x1, . . . , xs) =
=
ÿ
P: (x1,...,xs)=
Ť
j
Xj
A˚
|P|(t, t pX1u, . . . , t pX|P|u)
ź
XjĂP
g0|Xj |(Xj), s ě 1, (2.33)
where the symbol
ř
P: (x1,...,xs)=
Ť
j
Xj
denotes the sum over all possible par-
titions P of the set (x1, . . . , xs) into |P| nonempty mutually disjoint subsets
Xj , the symbol pX means the set of indexes of the set X of phase space coor-
dinates and the set (t pX1u, . . . , t pX|P|u) consists of elements that are subsets
pXj Ă (1, . . . , s), i.e., |(t pX1u, . . . , t pX|P|u)| = |P|. The generating operator
A˚
|P|(t) of expansions (2.33) is the |P|th-order cumulant of the groups of
operators (1.6) which is defined by the expansion
A˚
|P|(t, t pX1u, . . . , t pX|P|u) .=
.
=
ÿ
P1 : (t pX1u,...,t pX|P|u)=
Ť
k
Zk
(´1)|P1 |´1(|P1 | ´ 1)!
ź
ZkĂP1
S˚
|θ(Zk)|(t, θ(Zk)), (2.34)
where the symbol θ is the declusterization mapping: θ(t pXiu) .
= ( pXi). The
simplest examples of correlation functions (2.33) are given as follows:
g1(t, x1) = A1̊(t, 1)g
0
1(x1),
g2(t, x1, x2) = A1̊(t, t1, 2u)g02(x1, x2) + A2̊(t, 1, 2)g
0
1(x1)g
0
1(x2).
The structure of expansions (2.33) is established as a result of the permu-
tation of the terms of cumulant expansions (2.30) for correlation functions
and cluster expansions (2.29) for initial probability distribution functions.
Thus, the cumulant origin of correlation functions induces the cumulant
structure of their dynamics (2.33).
In particular, in the absence of correlations between hard spheres at
the initial moment (initial state satisfying the chaos condition [14,95]) the
sequence of the initial correlation functions on allowed configurations has
the form g(c)(0) = (0, g01(x1), 0, . . . , 0, . . .). In terms of a sequence of the
probability distribution functions, the chaos condition means that initial
data is specified in the form
D(c)(0) =
(
1, D0
1(x1), D
0
1(x1)D
0
1(x2)XR6zW2
, . . . ,
nź
i=1
D0
1(xi)XR3nzWn
, . . .
)
,
where the function XR3nzWn
is the Heaviside step function of allowed con-
figurations of n hard spheres. In this case for (x1, . . . , xs) P R3s ˆ (R3szWs)
760
expansions (2.33) are represented as follows:
gs(t, x1, . . . , xs) = As̊ (t, 1, . . . , s)
sź
i=1
g01(xi)XR3szWs
, s ě 1, (2.35)
where the generating operator As̊ (t) of this expansion is the sth-order cu-
mulant of groups of operators (1.6) defined by the expansion
As̊ (t, 1, . . . , s) =
ÿ
P: (1,...,s)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
S˚
|Xi|(t,Xi), (2.36)
with notations accepted in formula (2.33). From the structure of series
(2.35) it is clear that in case of the absence of correlations at the initial
instant the correlations generated by the dynamics of a system of hard
spheres are completely determined by cumulants (2.36) of the groups of
operators (1.6).
We note that in the case of initial data g(c)(0) expansions (2.35) can be
rewritten in another representation that explains their physical meaning.
Indeed, for n = 1 we have
g1(t, x1) = A1̊(t, 1)g
0
1(x1) = g01(p1, q1 ´ p1t).
Then, according to formula (2.35) and the definition of the first-order cu-
mulant A1̊(t) = S1(´t), and its inverse group of operators S´1
1 (´t) = S1(t),
we express the correlation functions gs(t), s ě 2, in terms of the one-particle
correlation function g1(t). Therefore, for s ě 2 expansions (2.35) are repre-
sented in the following form:
gs(t, x1, . . . , xs) = pAs̊ (t, 1, . . . , s)
sź
i=1
g1(t, xi), s ě 2,
where pAs̊ (t, 1, . . . , s) is the s-order cumulant (2.36) of the scattering oper-
ators
pSn(t, 1, . . . , n) .= Sn(´t, 1, . . . , n)XR3nzWn
nź
i=1
S1(t, i), n ě 1.
On the subspace L1
n,0 a generator of the scattering operator pSn(t, 1, . . . , n)
is determined by the operator:
d
dt
pSn(t, 1, . . . , n) |t=0 =
nÿ
j1ăj2=1
Li̊nt(j1, j2),
where for t ě 0 the operator Li̊nt(j1, j2) is defined by formula (1.7).
Evolution equations of colliding particles 761
If g0n P L1
n, n ě 1, one-parameter mapping (2.33) generates strong conti-
nuous group of nonlinear operators
G(t; 1, . . . , s | g(0)) .= gs(t, x1, . . . , xs), (2.37)
and it is bounded, and the following estimate holds:
››G(t; 1, . . . , s | g)››
L1
s
ď s!cs,
where
c ” max
(
1, max
P: (1,...,s)=
Ť
i
Xi
}g|Xi|}L1
|Xi|
)
.
For gn P L1
n,0, n ě 1, the infinitesimal generator of this group of nonlinear
operators has the following structure
L(1, . . . , s | g) .= Ls̊ (1, . . . , s)gs(x1, . . . , xs)+
+
ÿ
P: (x1,...,xs)=X1
Ť
X2
ÿ
i1P pX1
ÿ
i2P pX2
Li̊nt(i1, i2)g|X1|(X1)g|X2|(X2), (2.38)
where we used the notation adopted above in expansions (2.33).
The following statement is true [53].
Theorem. If t P R, a unique solution of the Cauchy problem of the
Liouville hierarchy (2.31),(2.32) is represented by a sequence of expan-
sions (2.33). For g0n P L1
n,0 Ă L1
n, n ě 1, a sequence of functions (2.33) is
a classical solution and for arbitrary initial data g0n P L1
n, n ě 1, one has a
generalized solution.
The proof of the theorem is similar to the proof of the existence the-
orem for the BBGKY hierarchy in the space of sequences of integrable
functions [14, 67]. Indeed, if the initial data is g0n P L1
n,0, n ě 1, then the
infinitesimal generator of the group of nonlinear operators (2.37) coincides
with the operator (2.38) and hence the Cauchy problem (2.31),(2.32) has a
classical (strong) solution (2.33).
We remark that a steady solution of the Liouville hierarchy (2.31) is
a sequence of the Ursell functions on the allowed configurations of hard
spheres, i.e., it is the sequence g(eq) = (0, e´β
p21
2 , 0, . . . , ), where β is a
parameter inversely proportional to temperature [64].
Finally, we emphasize that the dynamics of correlations, that is, the
fundamental equations (2.31) describing the evolution of correlations of
states of hard spheres, can be used as a basis for describing the evolution
of the state of both a finite and an infinite number of hard spheres instead
of the Liouville equations (1.8).
762
In what follows, we outline an approach to describing the evolution of a
state using reduced distribution functions based on the dynamics of correla-
tions in a system of many hard spheres governed by the Liouville hierarchy
for correlation functions (2.31).
Remind that reduced distribution functions are defined by means of se-
quence (1.15) of the probability distribution functions:
Fs(t, x1, . . . , xs)
.
= (I,D(t))´1ˆ
ˆ
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+nDs+n(t, x1, . . . , xs+n), s ě 1, (2.39)
where the normalizing factor
(I,D(t))
.
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dx1 ¨ ¨ ¨ dxnDn(t, x1, . . . , xn)
is a grand canonical partition function. The possibility of redefining of the
reduced distribution functions naturally arises as a result of dividing the
series in expression (2.39) by the series of the normalization factor.
A definition of reduced distribution functions equivalent to definition
(2.39) is formulated on the basis of correlation functions (2.33) of a system
of hard spheres by means of the following series expansion [53]:
Fs(t, x1, . . . , xs)
.
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n ˆ
ˆg1+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n), s ě 1,
(2.40)
where on the set of allowed configurations R3(s+n)zWs+n the correlation
functions of clusters of hard spheres g1+n(t), n ě 0, are determined by the
expansions:
g1+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n) =
=
ÿ
P:(tx1,...,xsu,
xs+1,...,xs+n)=
Ť
i
Xi
A˚
|P|
(
t, tθ( pX1)u, . . . , tθ( pX|P|)u
) ź
XiĂP
g0|Xi|(Xi), n ě 0. (2.41)
We remind that in expansions (2.41) the generating operator A˚
|P|(t) is the
|P|th-order cumulant (2.34) of the groups of operators (1.6), and the symbolř
P:(tx1,...,xsu,xs+1,...,xs+n)=
Ť
i
Xi
means the sum over all possible partitions P
of the set (tx1, . . . , xsu, xs+1, . . . , xs+n) into nonempty mutually disjoint
subsets Xi.
Evolution equations of colliding particles 763
On allowed configurations the correlation functions of particle clusters in
series (2.40), i.e., the functions g1+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n), n ě 0,
are defined as solutions of generalized cluster expansions of a sequence of
solutions of the Liouville equations:
Ds+n(t, x1, . . . , xs+n) =
=
ÿ
P:(tx1,...,xsu,
xs+1,...,xs+n)=
Ť
i Xi
ź
XiĂP
g|Xi|(t,Xi), s ě 1, n ě 0, (2.42)
namely,
g1+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n) =
=
ÿ
P:(tx1,...,xsu,
xs+1,...,xs+n)=
Ť
i Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
D|θ(Xi)|(t, θ(Xi)), s ě 1, n ě 0,
where θ is the declusterization mapping defined in formula (2.34), the prob-
ability distribution function D|θ(Xi)|(t, θ(Xi)) is a solution of the Liouville
equation.
The correlation functions of particle clusters satisfy the Liouville hierar-
chy of evolution equations with the following generator
L(t1, . . . , su, s+ 1, . . . , s+ n | dtY ug(t))
.
=
.
= Ls̊+n(1, . . . , s+ n)g1+n(t,X)+
+
ÿ
P:X=X1
Ť
X2
ÿ
i1Pθ( pX1)
ÿ
i2Pθ( pX2)
Li̊nt(i1, i2)g|X1|(t,X1)g|X2|(t,X2), n ě 0,
(2.43)
where X ” (tY u, xs+1, . . . , xs+n) ” (tx1, . . . , xsu, xs+1, . . . , xs+n), the se-
quence of solutions of generalized cluster expansions (2.42) is denoted by
means of the mapping
(dtY ug)n(x1, . . . , xn)
.
= g1+n(tx1, . . . , xsu, xs+1, . . . , xs+n), n ě 0,
and we also used the notations adopted above in expansion (2.33).
We note that on the allowed configurations the correlation functions of
hard-sphere clusters can be expressed through correlation functions of hard
spheres (2.33) by the following relations:
g1+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n) =
=
ÿ
P:(tx1,...,xsu,
xs+1,...,xs+n)=
Ť
i Xi
(´1)|P|´1(|P| ´ 1)!ˆ
ˆ
ź
XiĂP
ÿ
P1: θ(Xi)=
Ť
ji
Zji
ź
Zji
ĂP1
g|Zji
|(t, Zji), n ě 0.
(2.44)
764
In particular case n = 0, i.e., the correlation function of a cluster of the s
hard spheres, these relations take the form
g1+0(t, tx1, . . . , xsu) =
ÿ
P: θ(tx1,...,xsu)=Ť
i
Xi
ź
XiĂP
g|Xi|(t,Xi).
As a consequence of these relations, for the initial state satisfying the chaos
condition, from (2.41) the following generalization of expansions (2.35)
holds:
gs+n(t, tx1, . . . , xsu, xs+1, . . . , xs+n) =
= A1+n(t, t1, . . . , su, s+ 1, . . . , s+ n)ˆ
ˆ
s+nź
i=1
g01(xi)XR3(s+n)zWs+n
, s ě 1, n ě 0.
(2.45)
As we noted above, the possibility of the description of the evolution
of a state based on the dynamics of correlations (2.40) occurs naturally in
consequence of dividing the series of expressions (2.39) by the series of the
normalizing factor. To provide evidence of this statement, we will introduce
the necessary notions and prove the validity of some auxiliary equalities.
On sequences of functions f, rf P L1 ‘8
n=0 L
1
n we define the following
˚-product [90]
(f ˚ rf)s(x1, . . . , xs) =
ÿ
ZĂ(x1,...,xs)
f|Z|(Z) rfs´|Z|((x1, . . . , xs)zZ), (2.46)
where
ř
ZĂ(x1,...,xs)
is the sum over all subsets Z of the set (x1, . . . , xs).
Using the definition of the ˚-product (2.46), we introduce the mapping
Exp˚ and the inverse mapping Ln˚ on sequences
h = (0, h1(x1), . . . , hn(x1, . . . , xn), . . .)
of functions hn P L1
n by the expansions:
(Exp˚ h)s(x1, . . . , xs) =
(
I+
8ÿ
n=1
h˚n
n!
)
s
(x1, . . . , xs) =
= δs,0 +
ÿ
P: (x1,...,xs)=
Ť
i
Xi
ź
XiĂP
h|Xi|(Xi),
(2.47)
Evolution equations of colliding particles 765
where we used the notations accepted in formula (2.29), I = (1, 0, . . . , 0, . . .)
and δs,0 is the Kronecker symbol, and respectively,
(Ln˚(I+ h))s(x1, . . . , xs) =
( 8ÿ
n=1
(´1)n´1 h
˚n
n
)
s
(x1, . . . , xs) =
=
ÿ
P: (x1,...,xs)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
h|Xi|(Xi).
(2.48)
Therefore in terms of sequences of functions recursion relations (2.29) are
rewritten in the form
D(t) = Exp˚ g(t),
where D(t) = I + (0, D1(t, x1), . . . , Dn(t, x1, . . . , xn), . . .). As a result, we
get
g(t) = Ln˚ D(t).
Thus, according to definition (2.46) of the ˚-product and mapping (2.48),
in the component-wise form solutions of recursion relations (2.29) are re-
presented by expansions (2.30).
For arbitrary f = (f0, f1, . . . , fn, . . .) P L1 and the set Y ” (x1, . . . , xs)
we define the linear mapping dY : f Ñ dY f , by the formula
(dY f)n(x1, . . . , xn)
.
= fs+n(x1, . . . , xs, xs+1, . . . , xs+n), n ě 0. (2.49)
For the set tY u consisting of the one element Y = (x1, . . . , xs), we have,
respectively
(dtY uf)n(x1, . . . , xn)
.
= f1+n(tx1, . . . , xsu, xs+1, . . . , xs+n), n ě 0. (2.50)
On sequences dY f and dY 1 rf we introduce the ˚-product
(dY f ˚ dY 1 rf)|X|(X)
.
=
ÿ
ZĂX
f|Z|+|Y |(Y, Z) rf|XzZ|+|Y 1|(Y 1, XzZ),
where X,Y, Y 1 are the sets, which characterize clusters of hard spheres,
and
ř
ZĂX is the sum over all subsets Z of the set X. In particular case
Y = H, Y 1 = H, this definition reduces to definition of ˚-product (2.46).
Let us establish some properties of introduced mappings (2.47) and (2.50).
If fn P L1
n, n ě 1 for the sequences f = (0, f1, . . . , fn, . . .), according to
definitions of mappings (2.47) and (2.50), the following equality holds
dtY uExp˚f = Exp˚f ˚ dtY uf, (2.51)
and for mapping (2.49), respectively
dY Exp˚f = Exp˚f ˚
ÿ
P:Y=
Ť
i Xi
dX1f ˚ . . . ˚ dX|P|f,
766
where
ř
P:Y=
Ť
i Xi
denotes the sum over all possible partitions P of the set
Y ” (x1, . . . , xs) into |P| nonempty mutually disjoint subsets Xi Ă Y .
Hence, in terms of mappings (2.49) and (2.50) generalized cluster expan-
sions (2.42) take the form
dYD(t) = dtY uExp˚ g(t). (2.52)
On sequences of functions f P L1 = ‘8
n=0L
1
n we also define the analogue
of the annihilation operator
(af)n(x1, . . . , xn) =
ż
R3ˆR3
dxn+1fn+1(x1, . . . , xn, xn+1). (2.53)
Then for sequences f, rf P L1, the following equality holds
(eaf ˚ rf)0 = (eaf)0(e
a rf)0, (2.54)
where such a notation was used
(eaf)0 =
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dx1 ¨ ¨ ¨ dxn fn(x1, . . . , xn). (2.55)
Now let us prove the equivalence of definition (2.39) of the reduced dis-
tribution functions and their definition (2.40) within the framework of the
dynamics of correlations.
In terms of mapping (2.49) and notation (2.55) the definition of reduced
distribution functions (2.39) is written as follows
Fs(t, x1, . . . , xs) = (eaD(t))´1
0 (eadYD(t))0.
Using generalized cluster expansions (2.52), and as a consequence of equal-
ities (2.51) and (2.54), we find
(eadYD(t))0 = (eadtY uExp˚ g(t))0 =
(eaExp˚g(t) ˚ dtY ug(t))0 = (eaExp˚g(t))0(eadtY ug(t))0.
Taking into account that, according to the particular case Y = H, of cluster
expansions (2.42), the equality holds
(eaExp˚g(t))0 = (eaD(t))0,
as a result, we establish the following representation for the reduced distri-
bution functions
Fs(t, x1, . . . , xs) = (eadtY ug(t))0.
Therefore, in componentwise-form we obtain relation (2.40).
Since the correlation functions g1+n(t), n ě 0, are governed by the corres-
ponding Liouville hierarchy for the cluster of hard spheres and hard spheres,
Evolution equations of colliding particles 767
the reduced distribution functions (2.40) are governed by the BBGKY hi-
erarchy for hard spheres
B
BtF (t) = eaL(t¨u, ¨ | e´aF (t)), (2.56)
where the operator L(t¨u, ¨ | f) is generator (2.43) of the Liouville hierarchy
for a cluster of hard spheres and hard spheres. For a generator of this
hierarchy of evolution equations takes place the following representation:
eaL(t¨u, ¨ | e´aF (t)) = eaL˚e´aF (t),
where the operator L˚ = ‘8
n=0Ln̊ is a direct sum of the Liouville operators
and the operator a is defined by formula (2.53). Due to the fact that
pairwise collisions occur during the evolution, a generator of this hierarchy
is reduced to the operator of such a structure [14]
eaL˚e´a = L˚ + [a,L˚],
where as above the bracket [¨, ¨] is the commutator of operators.
We note that for the first time the BBGKY hierarchy for many hard
spheres (2.56) was mathematically justified in paper [83] (see also [14]).
In consequence of definition (2.40) and the cumulant structure of rep-
resentation of a solution (2.33) for the Liouville hierarchy (2.31), if initial
state specified by the sequence of reduced distribution functions
F (0) =
(
1, F 0
1 (x1), . . . , F
0
n(x1, . . . , xn), . . .
)
,
then the evolution of all possible states, i.e., a sequence of the reduced distri-
bution functions Fs(t), s ě 1, is determined by the series expansions (2.24).
We remark that the representation (2.24) is directly established for the
initial states satisfying the chaos condition due to the validity in this case
of the representation (2.45) for the correlation functions of the hard-sphere
cluster and of the hard spheres.
Consequently, as follows from the above, the cumulant structure of ge-
nerating operators of expansions for correlation functions (2.33) or (2.41)
induces the cumulant structure (2.22) of generating operators of series ex-
pansions for reduced distribution functions (2.24) or in other words, the
evolution of the state of a system of an infinite number of hard spheres is
governed by the dynamics of correlations on a microscopic scale.
Thus, we have established relation (2.40) between the reduced distribu-
tion functions and correlation functions governed by the Liouville hierarchy.
2.4. The hierarchy of nonlinear evolution equations for reduced
correlation functions. As is known, on a microscopic scale, the macro-
scopic characteristics of fluctuations of observables are directly determined
by means of the reduced correlation functions. Assuming as a basis an
768
alternative approach to the description of the evolution of states of a hard-
sphere system within the framework of correlation functions (2.33), then
the reduced correlation functions are defined by means of a solution of the
Cauchy problem of the Liouville hierarchy (2.31),(2.32) as follows [53]:
Gs(t, x1, . . . , xs)
.
=
.
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n gs+n(t, x1, . . . , xs+n), s ě 1, (2.57)
where the generating function gs+n(t, x1, . . . , xs+n) is defined by expan-
sion (2.33), or in terms of mapping (2.49) and notation (2.55) this definition
takes the form
Gs(t, x1, . . . , xs) = (eadY g(t))0,
or in terms of sequences of functions this expression has the form
G(t) = eag(t).
We emphasize that nth term of expansions (2.57) of the reduced correlation
functions are determined by the (s+n)th-particle correlation function (2.33)
in contrast with the expansions of reduced distribution functions (2.40)
which are determined by the (1+n)th-particle correlation function of clus-
ters of hard spheres (2.41).
Such a representation for reduced correlation functions (2.57) can be
derived as a result of the fact that the reduced correlation functions are
cumulants of reduced distribution functions (2.40). Indeed, traditionally
reduced correlation functions are introduced by means of the cluster expan-
sions of the reduced distribution functions similar to the cluster expansions
of the probability distribution functions (2.29) and on the set of allowed
configurations R3nzWn they have the form:
Fs(t, x1, . . . , xs) =
ÿ
P:(x1,...,xs)=
Ť
i
Xi
ź
XiĂP
G|Xi|(t,Xi), s ě 1, (2.58)
where as above the symbol
ř
P:(x1,...,xs)=
Ť
i Xi
is the sum over all possible
partitions P of the set (x1, . . . , xs) into |P| nonempty mutually disjoint sub-
sets Xi Ă (x1, . . . , xs). As a consequence of this, the solution of recurrence
relations (2.58) are represented through reduced distribution functions as
follows:
Gs(t, x1, . . . , xs) =
=
ÿ
P:(x1,...,xs)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
F|Xi|(t,Xi), s ě 1. (2.59)
Evolution equations of colliding particles 769
Functions (2.59) are interpreted as the functions which describe the cor-
relations of hard-sphere states. The structure of expansions (2.59) is such
that the reduced correlation functions are cumulants (semi-invariants) of
the reduced distribution functions (2.24).
Thus, taking into account representation (2.40) of the reduced distribu-
tion functions, in consequence of the validity of relations (2.44) we derive
representation (2.57) of the reduced correlation functions through correla-
tion functions
Gs(t, x1, . . . , xs) =
=
ÿ
P:(x1,...,xs)=
Ť
i
Xi
(´1)|P|´1(|P| ´ 1)!
ź
XiĂP
(eadtXiug(t)) = (eadY g(t))0.
Since the correlation functions gs+n(t), n ě 0, are governed by the Li-
ouville hierarchy for hard spheres (2.31), the reduced correlation functions
defined as (2.57) are governed by the hierarchy of nonlinear equations for
hard spheres (the nonlinear BBGKY hierarchy) [53]:
B
BtGs(t, x1, . . . , xs) = Ls̊Gs(t, x1, . . . , xs)+
+
ÿ
P: (x1,...,xs)=X1
Ť
X2
ÿ
i1P pX1
ÿ
i2P pX2
Li̊nt(i1, i2)G|X1|(t,X1)G|X2|(t,X2))+
+
ż
R3ˆR3
dxs+1
( sÿ
i=1
Li̊nt(i, s+ 1)Gs+1(t, x1, . . . , xs+1)+
+
ÿ
P: (x1,...,xs+1)=X1
Ť
X2
ÿ
iP pX1
s+1P pX2
Li̊nt(i, s+ 1)G|X1|(t,X1)G|X2|(t,X2)
)
,
(2.60)
Gs(t, x1, . . . , xs)
ˇ̌
t=0
= G0
s(x1, . . . , xs), s ě 1, (2.61)
where the symbol
ř
P:(x1,...,xs+1)=X1
Ť
X2
means the sum over all possible
partitions of the set (x1, . . . , xs+1) into two mutually disjoint subsetsX1 and
X2, the sum over the index i which takes values from the subset pX1 provided
that the index s + 1 belongs to the subset pX2 is denoted by
ř
iP pX1,s+1P pX2
and notations accepted in the Liouville hierarchy (2.31) are used.
A generator of this hierarchy of nonlinear evolution equations has the
following structure:
B
BtG(t) = eaL(¨ | e´aG(t)),
where the operator L(¨ | f) = ‘8
n=0L(1, . . . , n | f) is a direct sum of genera-
tors (2.38) of the Liouville hierarchy (2.31). Here are some component-wise
770
examples of hierarchy (2.60):
B
BtG1(t, x1) = L1̊(1)G1(t, x1)+
+
ż
R3ˆR3
dx2Li̊nt(1, 2)
(
G2(t, x1, x2) +G1(t, x1)G1(t, x2)
)
,
B
BtG2(t, x1, x2) = L2̊(1, 2)G2(t, x1, x2) + Li̊nt(1, 2)G1(t, x1)G1(t, x2)+
+
ż
R3ˆR3
dx3
( 2ÿ
i=1
Li̊nt(i, 3)
(
G3(t, x1, x2, x3) +G2(t, x1, x2)G1(t, x3)
)
+
+ Li̊nt(2, 3)G2(t, x1, x3)G1(t, x2) + Li̊nt(1, 3)G2(t, x2, x3)G1(t, x1)
)
,
where it was used notations accepted above in definition (1.7).
If G(0) = (1, G0
1(x1), . . . , G
0
s(x1, . . . , xs), . . .) is a sequence of reduced cor-
relation functions at initial instant, then by means of mappings (2.37) the
evolution of all possible states, i.e., the sequence of the reduced correlation
functions Gs(t), s ě 1, is determined by the following series expansions:
Gs(t, x1, . . . , xs) =
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n ˆ
ˆ A1+n
(
t; t1, . . . , su, s+ 1, . . . , s+ n | G(0)), s ě 1,
(2.62)
where the generating operator A1+n(t; t1, . . . , su, s+ 1, . . . , s+ n | G(0)) of
this series is the (1 + n)th-order cumulant of groups of nonlinear opera-
tors (2.33):
A1+n(t; t1, . . . , su, s+ 1, . . . , s+ n | G(0)) .=
.
=
ÿ
P: (t1,...,su,s+1,...,s+n)=
Ť
k Xk
(´1)|P|´1(|P| ´ 1)!ˆ
ˆ G
(
t; θ(X1) | . . .G(t; θ(X|P|) | G(0)) . . .), n ě 0,
(2.63)
and where the composition of mappings (2.33) of the corresponding nonin-
teracting groups of particles was denoted by
G(t; θ(X1) | . . .G(t; θ(X|P|) | G(0)) . . .),
for example,
G
(
t; 1 | G(t; 2 | G(0))) = A1(t, 1)A1(t, 2)G
0
2(x1, x2),
G
(
t; 1, 2 | G(t; 3 | G(0))) = A1(t, t1, 2u)A1(t, 3)G
0
3(x1, x2, x3)+
Evolution equations of colliding particles 771
+ A2(t, 1, 2)A1(t, 3)
(
G0
1(x1)G
0
2(x2, x3) +G0
1(x2)G
0
2(x1, x3)
)
.
We will adduce examples of expansions (2.63). The first order cumulant
of the groups of nonlinear operators (2.33) is the group of these nonlinear
operators
A1(t; t1, . . . , su | G(0)) = G(t; 1, . . . , s | G(0)).
In case of s = 2 the second order cumulant of nonlinear operators (2.33)
has the structure
A1+1(t; t1, 2u, 3 | G(0)) =
= G(t; 1, 2, 3 | G(0)) ´ G
(
t; 1, 2 | G(t; 3 | G(0))) =
= A1̊+1(t, t1, 2u, 3)G0
3(1, 2, 3)+
+
(
A1̊+1(t, t1, 2u, 3) ´ A2(t, 2, 3)A1̊(t, 1)
)
G0
1(x1)G
0
2(x2, x3)+
+
(
A1+1(t, t1, 2u, 3) ´ A2̊(t, 1, 3)A1̊(t, 2)
)
G0
1(x2)G
0
2(x1, x3)+
+ A1̊+1(t, t1, 2u, 3)G0
1(x3)G
0
2(x1, x2) + A3̊(t, 1, 2, 3)G
0
1(x1)G
0
1(x2)G
0
1(x3),
where the operator
A3̊(t, 1, 2, 3) = A1̊+1(t, t1, 2u, 3) ´ A2̊(t, 2, 3)A1̊(t, 1) ´ A2̊(t, 1, 3)A1̊(t, 2)
is the third-order cumulant (2.36) of groups of operators (1.6) of a system
of hard spheres.
The following statement is true [53].
Theorem. Let G(0) P ‘8
n=0L
1
n, then for arbitrary t P R provided that
max
ně1
››G0
n
››
L1
n
ă (2e3)´1, the sequence of reduced correlation functions (2.62)
is a unique solution of the Cauchy problem of nonlinear hierarchy (2.60),
(2.61) for hard spheres.
In the particular case of the initial state specified by the sequence of
reduced correlation functions G(c) = (0, G0
1, 0, . . . , 0, . . .) on the allowed
configurations, that is, in the absence of correlations between hard spheres
at the initial moment of time [14], according to definition (2.63) of the
generating operators, reduced correlation functions (2.62) are represented
by the following series expansions:
Gs(t, x1, . . . , xs) =
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n ˆ
ˆ As̊+n(t; 1, . . . , s+ n)
s+nź
i=1
G0
1(xi)XR3(s+n)zWs+n
, s ě 1,
(2.64)
772
where the generating operator As̊+n(t) is the (s+n)th-order cumulant (2.36)
of the groups of operators (1.6).
We emphasize that in the absence of correlations of states of hard spheres
on allowed configurations at the initial moment of time, the generators of
expansions into a series of reduced correlation functions (2.64) and reduced
distribution functions (2.24) differ only in the order of cumulants of groups
of operators of hard spheres. Therefore, by means of such reduced distri-
bution functions or reduced correlation functions, the process of creating
correlations in a system of hard spheres is described.
We note that the reduced correlation functions give an equivalent appro-
ach to the description of the evolution of states of many hard spheres, along
with the reduced distribution functions. Indeed, the macroscopic characte-
ristics of fluctuations of observables are directly determined by the reduced
correlation functions on the microscopic scale [6] for example, the func-
tional of the dispersion of an additive-type observable, i.e., the sequence
A(1) = (0, a1(x1), . . . ,
nř
i1=1
a1(xi1), . . .), is represented by the formula
x(A(1) ´ xA(1)y)2y(t) =
ż
R3ˆR3
dx1 (a
2
1(x1) ´ xA(1)y2(t))G1(t, x1)+
+
ż
(R3ˆR3)2
dx1dx2 a1(x1)a1(x2)G2(t, x1, x2),
where
xA(1)y(t) =
ż
R3ˆR3
dx1 a1(x1)G1(t, x1)
is the mean value functional of an additive-type observable.
3. NONLiNEAR KiNETiC EQUATiONS FOR MANY HARD SPHERES
The conventional philosophy of the description of kinetic evolution is that
if the initial state is specified by a one-particle (reduced) distribution func-
tion, then at an arbitrary time the evolution of the state in an appropriate
scaling limit can be effectively described by means of a one-particle distri-
bution function that is governed by the nonlinear kinetic equation. Below,
we give an answer to the question about the description of the kinetic evo-
lution of colliding particles, not on the basis of a common interpretation
but within the framework of the evolution of the observables of many hard
spheres.
The problem of a rigorous description of the kinetic evolution by means
of hard sphere observables will be considered by giving the example of the
Evolution equations of colliding particles 773
Boltzmann–Grad asymptotics of a non-perturbative solution of the Cauchy
problem of the dual BBGKY hierarchy [51].
3.1. On the Boltzmann–Grad scaling approximation. The present
notion of the Boltzmann–Grad approximation was first introduced in Grad’s
paper [73]. From a physical point of view, this approximation means that
we deal with a low-density gas in a situation where the diameter of a hard
sphere, or, in other words, the radius of the short-range interaction poten-
tial, is sufficiently less in comparison with the average length of a free path
of hard spheres.
In a dimensionless form, the generator of the BBGKY hierarchy for hard
spheres contains a scaling parameter: the ratio of the diameter of hard
spheres to their mean free path [35]. The finite value of the mean free
path of hard spheres means that in this approximation the average number
of particles tends to infinity; in other words, according to the definition
of (2.16), the state must be described by functions from the appropriate
function spaces, for example, from the space to which the sequences of re-
duced equilibrium distribution functions belong [64, 90]. In this case, the
initial state is described by the reduced distribution functions from the
space L8
ξ of sequences of functions bounded with respect to the configura-
tion variables and decreasing with respect to the momentum ones, equipped
with the norm
}f}L8
ξ
= sup
ně0
ξ´n sup
x1,...,xn
|fn(x1, . . . , xn)| exp
(
β
nÿ
i=1
p2i
2
)
,
where ξ ą 0 and β ą 0 are parameters.
For such initial data, the Boltzmann–Grad asymptotics of a solution of
the Cauchy problem of the BBGKY hierarchy for hard spheres are described
by the so-called Boltzmann hierarchy [79]. As a consequence, for factorized
initial data, i.e., for the initial state without correlations, which describes
molecular chaos [14], the equation determining the evolution of an initial
state is a closed equation for a one-particle distribution function, that is to
say Boltzmann’s kinetic equation [13].
The detailed analysis of the problem of the construction of such asymp-
totics for a solution of the Cauchy problem of the BBGKY hierarchy shows
that the basic difficulty consists in proving the term-by-term convergence
of the iteration series that represents this solution to the corresponding
limit, that is, to the series representing the solution of the Cauchy problem
of the Boltzmann hierarchy. This difficulty is related to the fact that the
integrands in each term of the iteration series do not converge to the limit
uniformly across the whole domain of integration. We note that in early
works on the justification of the Boltzmann–Grad limit, attention was not
774
properly paid to this property, and a precise mathematical meaning was
not given to the individual terms of the iteration series representing a solu-
tion of the BBGKY hierarchy. In the papers [61,83] a complete discussion
of these problems was presented.
From a mathematical point of view, the existence of the Boltzmann–
Grad asymptotics of a perturbative solution of the BBGKY hierarchy for
hard spheres was discussed in Cercignani’s paper [12] and later in Lanford’s
work [79]. A rigorous mathematical proof of the Boltzmann–Grad limit
theorem has been given in a series of papers [57–59,61,83] by D. Ya. Petrina
and V. I. Gerasimenko. The Boltzmann–Grad limit theorem for equilibrium
states was proved in the paper [60].
Recently, there has been unflagging interest in the problem of deriv-
ing kinetic equations from the dynamics of many colliding particles as an
asymptotic behavior of the BBGKY hierarchy in the scaling limits. In
particular, progress in the rigorous solution of this problem on the basis
of perturbation theory was achieved in the Boltzmann–Grad limit in the
works [2–4,20,25,28,29,87–89]; also see links therein.
3.2. The Boltzmann–Grad limit of reduced observables. To deter-
mine the scaling parameter, we rewrite the dual BBGKY hierarchy in di-
mensionless form. Then generator (1.3) of the hierarchy takes the form:
L(j)bn .
= xpj , B
Bqj ybn,
Lint(j1, j2)bn
.
= ϵ2
ż
S2+
dηxη, (pj1 ´ pj2)yδ(qj1 ´ qj2 + ϵη)ˆ
ˆ (
bn(x1, . . . , qj1 , pj̊1 , . . . , qj2 , pj̊2 , . . . , xn) ´ bn(x1, . . . , xn)
)
,
(3.1)
where the coefficient ϵ ą 0 is a scaling parameter, which is the ratio of
the diameter σ ą 0 to the mean free path of hard spheres. For t ď 0, a
generator of the dimensionless dual BBGKY hierarchy is determined by the
corresponding expression [51].
Then the Boltzmann–Grad asymptotic behavior of dimensionless reduced
observables (2.3) is described by the following statement [51].
Theorem. Assume that for the initial data Bϵ,0
n P Cn, n ě 1, there is a
limit b0n P Cn in the sense of ˚-weak convergence of space Cn
w˚´ lim
ϵÑ0
(
ϵ´2nBϵ,0
n ´ b0n
)
= 0. (3.2)
Evolution equations of colliding particles 775
Then, for an arbitrary finite time interval, the Boltzmann–Grad limit of
dimensionless reduced observables (2.3) exists in the same sense
w˚´ lim
ϵÑ0
(
ϵ´2sBs(t) ´ bs(t)
)
= 0, (3.3)
and it is determined by the expansions:
bs(t, x1, . . . , xs) =
s´1ÿ
n=0
tż
0
dt1 ¨ ¨ ¨
tn´1ż
0
dtn
ź
jP(1,...,s)
S1(t´ t1, j)ˆ
ˆ
sÿ
i1‰j1=1
L0
int(i1, j1)
ź
jP(1,...,s)z(j1)
S1(t1 ´ t2, j) ¨ ¨ ¨
ź
jP(1,...,s)z(j1,...,jn´1)
S1(tn´1 ´ tn, j)ˆ
ˆ
sÿ
in‰jn=1,
in,jn‰(j1,...,jn´1)
L0
int(in, jn)
ź
jP(1,...,s)z(j1,...,jn)
S1(tn, j)b
0
s´n((x1, . . . , xs)z(xj1 , . . . , xjn)),
s ě 1,
(3.4)
where for the collision operator of point particles, the notation L0
int(j1, j2)
is used
L0
int(j1, j2)bn
.
=
ż
S2+
dηxη, (pj1 ´ pj2)yδ(qj1 ´ qj2)ˆ
ˆ (
bn(x1, . . . , qj1 , pj̊1 , . . . , qj2 , pj̊2 , . . . , xn) ´ bn(x1, . . . , xn)
)
.
(3.5)
Let us make several comments on this theorem.
Consider the existence of the Boltzmann–Grad limit for a special case
of reduced observables, namely additive-type reduced observables. Let
us say that for the initial additive-type dimensionless reduced observable
B(1)(0) = (0, bϵ1, 0, . . .) the following condition is satisfied:
w˚´ lim
ϵÑ0
(
ϵ´2bϵ1 ´ b01
)
= 0,
then, according to statement (3.3), for additive-type reduced observables (2.10)
we derive
w˚´ lim
ϵÑ0
(
ϵ´2sB(1)
s (t) ´ b(1)s (t)
)
= 0,
776
where the limit reduced observable b(1)s (t) is determined as a special case
of expansion (3.4):
b(1)s (t, x1, . . . , xs) =
tż
0
dt1 ¨ ¨ ¨
ts´2ż
0
dts´1
ź
jP(1,...,s)
S1(t´ t1, j)ˆ
ˆ
sÿ
i1‰j1=1
L0
int(i1, j1)
ź
jP(1,...,s)z(j1)
S1(t1 ´ t2, j) ¨ ¨ ¨
ź
jP(1,...,s)z(j1,...,js´2)
S1(ts´2 ´ ts´1, j)ˆ
ˆ
sÿ
is´1‰js´1=1,
is´1,js´1‰(j1,...,js´2)
L0
int(is´1, js´1)
ź
jP(1,...,s)z(j1,...,js´1)
S1(ts´1, j)b
0
1
(
(x1, . . . , xs)z(xj1 , . . . , xjs´1)
)
,
s ě 1.
(3.6)
We make several examples of expansions (3.6) of the limit additive-type
reduced observables:
b
(1)
1 (t, x1) = S1(t, 1) b
0
1(x1),
b
(1)
2 (t, x1, x2) =
ż t
0
dt1
2ź
i=1
S1(t´ t1, i)L0
int(1, 2)
2ÿ
j=1
S1(t1, j) b
0
1(xj).
Also suppose that the following condition is valid for the initial k-ary-
type reduced observable B(k)(0) = (0, . . . , bϵk, 0, . . .):
w˚´ lim
ϵÑ0
(
ϵ´2bϵk ´ b0k
)
= 0,
then, according to statement (3.3), for k-ary-type dimensionless reduced
observables (2.11), we derive
w˚´ lim
ϵÑ0
(
ϵ´2sB(k)
s (t) ´ b(k)s (t)
)
= 0,
where the limit reduced observable b(k)s (t) is determined as a special case
of expansion (3.4):
b(k)s (t, x1, . . . , xs) =
tż
0
dt1 ¨ ¨ ¨
ts´k´1ż
0
dts´k
ź
jP(1,...,s)
S1(t´ t1, j)
sÿ
i1‰j1=1
L0
int(i1, j1)ˆ
ˆ
ź
jP(1,...,s)z(j1)
S1(t1 ´ t2, j) ¨ ¨ ¨
ź
jP(1,...,s)z(j1,...,js´k´1)
S1(ts´k´1 ´ ts´k, j)
sÿ
is´k‰js´k=1,
is´k,js´k‰(j1,...,js´k´1)
L0
int(is´k, js´k)ˆ
ˆ
ź
jP(1,...,s)z(j1,...,js´k)
S1(ts´k, j)b
0
k((x1, . . . , xs)z(xj1 , . . . , xjs´k
)), 1 ď s ď k.
(3.7)
Evolution equations of colliding particles 777
If b0 P Cγ , then the sequence b(t) = (b0, b1(t), . . . , bs(t), . . .) of limit re-
duced observables (3.4) is a generalized global solution of the Cauchy prob-
lem of the dual Boltzmann hierarchy with hard sphere collisions [51]:
B
Btbs(t) =
sÿ
j=1
L(j) bs(t) +
sÿ
j1‰j2=1
L0
int(j1, j2) bs´1(t, (x1, . . . , xs)z(xj1)), (3.8)
bs(t, x1, . . . , xs) |t=0= b0s(x1, . . . , xs), s ě 1, (3.9)
where it was used notations accepted in (3.4).
This fact is proved similar to the case of an iteration series of the dual
BBGKY hierarchy [10].
It should be noted that equations set (3.8) has the structure of recur-
rence evolution equations. We make a few examples of the dual Boltzmann
hierarchy with hard sphere collisions (3.8):
B
Btb1(t, x1) = xp1, B
Bq1 y b1(t, x1),
B
Btb2(t, x1, x2) =
2ÿ
j=1
xpj , B
Bqj y b2(t, x1, x2)+
+
ż
S2+
dηxη, (p1 ´ p2)y
(
b1(q1, p1̊) ´ b1(x1) + b1(q2, p2̊) ´ b1(x2)
)
δ(q1 ´ q2).
Thus, in the Boltzmann–Grad scaling asymptotics, the kinetic evolution
of hard sphere observables is described in terms of limit reduced observ-
ables (3.4) governed by the dual Boltzmann hierarchy (3.8) with hard sphere
collisions.
3.3. The Boltzmann kinetic equation. We now establish the relation-
ship between the constructed Boltzmann–Grad asymptotics of the reduced
observables and the description of the kinetic evolution of states in terms of
the one-particle reduced distribution function described by the Boltzmann
kinetic equation.
In the case of the absence of correlations between particles at initial time,
i.e., initial states satisfying a chaos condition [14], in dimensionless form,
the sequence of initial reduced distribution functions for a system of hard
spheres has the form
F (c) ”
(
1, F ϵ,0
1 (x1), . . . ,
sź
i=1
F ϵ,0
1 (xi)XR3szWs
, . . .
)
, (3.10)
778
where XR3szWs
is the Heaviside step function of the allowed configurations.
This assumption about initial state is intrinsic for the kinetic theory, be-
cause in this case all possible states of gases are described by means of a
one-particle distribution function.
Let F 0,ϵ
1 P L8
ξ (R3 ˆ R3), i.e., the following inequality holds:
|F 0,ϵ
1 (xi)| ď ξ exp(´β p
2
i
2
),
where ξ ą 0, β ě 0 are parameters.
We assume that the Boltzmann–Grad limit of the initial one-particle
(reduced) distribution function F 0,ϵ
1 P L8
ξ (R3 ˆ R3) exists in the sense of a
weak convergence of the space L8
ξ (R3 ˆ R3), namely,
w´ lim
ϵÑ0
(ϵ2 F 0,ϵ
1 ´ f01 ) = 0, (3.11)
then the Boltzmann–Grad limit of the initial state (3.10) satisfies a chaos
property too, i.e.,
f (c) ” (
1, f01 (x1), . . . ,
sź
i=1
f01 (xi), . . .
)
.
We note that assumption (3.11) with respect to the Boltzmann–Grad
limit of initial states holds true for the equilibrium state [60].
If b(t) P Cγ and |f01 (xi)| ď ξ exp(´β p2i
2 ), then the Boltzmann–Grad limit
of mean value functional
(
B(t), F (c)
)
exists under the condition that [83]:
t ă t0 ” (
const(ξ, β)}f01 }L8
ξ (R3ˆR3)
)´1
,
and it is determined by the following series expansion:
(
b(t), f (c)
)
=
8ÿ
s=0
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs bs(t, x1, . . . , xs)
sź
i=1
f01 (xi).
For the limit of additive-type reduced observables (3.6) the following
equality holds [51]:
(
b(1)(t), f (c)
)
=
8ÿ
s=0
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs b(1)s (t, x1, . . . , xs)
sź
i=1
f01 (xi) =
=
ż
R3ˆR3
dx1 b
0
1(x1)f1(t, x1),
(3.12)
Evolution equations of colliding particles 779
where function b(1)s (t) is given by expansion (3.6) and the distribution func-
tion f1(t, x1) is represented by the series
f1(t, x1) =
8ÿ
n=0
tż
0
dt1 ¨ ¨ ¨
tn´1ż
0
dtn
ż
(R3ˆR3)n
dx2 ¨ ¨ ¨ dxn+1 S1̊ (t´ t1, 1)ˆ
ˆ L0,˚
int (1, 2)
2ź
j1=1
S1̊ (t1 ´ t2, j1) ¨ ¨ ¨
nź
in=1
S1̊ (tn´1 ´ tn, in)ˆ
ˆ
nÿ
kn=1
L0,˚
int (kn, n+ 1)
n+1ź
jn=1
S1̊ (tn, jn)
n+1ź
i=1
f01 (xi),
(3.13)
and the following operator was introduced:
ż
R3ˆR3
dxn+1L0,˚
int (i, n+ 1)fn+1(x1, . . . , xn+1) ”
”
ż
R3ˆS2+
dpn+1dη xη, (pi ´ pn+1)yˆ
ˆ (
fn+1(x1, . . . , qi, pi̊ , . . . , xs, qi, pn̊+1)´
´fn+1(x1, . . . , xs, qi, pn+1)
)
.
(3.14)
A one-particle distribution function represented as a series (3.13) is a
solution of the Cauchy problem of the Boltzmann kinetic equation:
B
Btf1(t, x1) = ´xp1, B
Bq1 yf1(t, x1)+
+
ż
R3ˆS2+
dp2 dη xη, (p1 ´ p2)yˆ
ˆ (
f1(t, q1, p1̊)f1(t, q1, p2̊) ´ f1(t, x1)f1(t, q1, p2)
)
,
(3.15)
f1(t, x1)|t=0 = f01 (x1). (3.16)
Thus, we establish that the dual Boltzmann hierarchy (3.8) for additive-
type reduced observables and initial state (3.11) describe the evolution of
hard sphere systems just as the Boltzmann kinetic equation (3.15).
We remark that in a one-dimensional space, the collision integral of the
Boltzmann equation with elastic hard sphere collisions identically equals
zero. In a one-dimensional space, the Boltzmann–Grad limit is not trivial
in the case of hard sphere dynamics with inelastic collisions. In the pa-
per [43] for one-dimensional granular gas, the process of the creation and
propagation of correlations in the Boltzmann–Grad scaling limit was also
described (see also section 5.1).
780
Correspondingly, if the initial state of hard spheres is given by a sequence
of reduced distribution functions (3.10), then in the Boltzmann–Grad limit,
the property of the propagation of initial chaos holds [51]. It is a result of
the validity of the following equality for the limit k-ary reduced observables,
i.e., for the sequences b(k)(0) = (0, . . . , b0k(x1, . . . , xk), 0, . . .),
(
b(k)(t), f (c)
)
=
8ÿ
s=0
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs b(k)s (t, x1, . . . , xs)
sź
i=1
f01 (xi) =
=
1
k!
ż
(R3ˆR3)k
dx1 ¨ ¨ ¨ dxk b0k(x1, . . . , xk)
kź
i=1
f1(t, xi), k ě 2,
(3.17)
where the limit one-particle reduced distribution function f1(t) is defined
by expansion (3.13) and therefore it is governed by the Cauchy problem of
the Boltzmann kinetic equation (3.15),(3.16).
Thus, in the Boltzmann–Grad scaling limit, an equivalent approach to
the description of the kinetic evolution of hard spheres in terms of the
Cauchy problem of the Boltzmann kinetic equation (3.15),(3.16) is given
by the Cauchy problem of the dual Boltzmann hierarchy with hard sphere
collisions (3.8),(3.9) for the additive-type reduced observables. In the case
of non-additive-type reduced observables, a solution of the dual Boltzmann
hierarchy with hard sphere collisions (3.8) is equivalent to the property of
the propagation of initial chaos in the sense of equality (3.17).
3.4. The Boltzmann kinetic equation with initial correlations. We
now consider the case of the more general initial state of a hard sphere
system specified by the one-particle reduced distribution function F 0,ϵ
1 P
L8
ξ (R3 ˆ R3) in the presence of correlations, i.e., the initial state that is
specified by the following sequence of reduced distribution functions:
F (cc) =
(
1, F 0,ϵ
1 (x1), g
ϵ
2
2ź
i=1
F 0,ϵ
1 (xi), . . . , g
ϵ
n
nź
i=1
F 0,ϵ
1 (xi), . . .
)
, (3.18)
where the functions gϵn ” gϵn(x1, . . . , xn) P Cn(R3n ˆ (R3nzWn)), n ě 2,
specify the initial correlations. Since many-particle systems in condensed
states are characterized by correlations, sequence (3.18) describes the initial
state of the kinetic evolution of hard sphere fluids.
We assume that the Boltzmann–Grad limit of the initial one-particle
reduced distribution function F 0,ϵ
1 P L8
ξ (R3 ˆ R3) exists in the sense as
above, i.e., in the sense of a weak convergence the equality holds:
lim
ϵÑ0
(ϵ2 F 0,ϵ
1 ´ f01 ) = 0,
Evolution equations of colliding particles 781
and in the case of correlation functions, suppose that:
lim
ϵÑ0
(gϵn ´ gn) = 0, n ě 2.
Then in the Boltzmann–Grad limit, initial state (3.18) is specified by the
following sequence of the limit reduced distribution functions:
f (cc) =
(
1, f01 (x1), g2
2ź
i=1
f01 (xi), . . . , gn
nź
i=1
f01 (xi), . . .
)
. (3.19)
We now consider relationships between the constructed Boltzmann–Grad
asymptotic behavior of reduced observables and the nonlinear Boltzmann-
type kinetic equation in the case of initial state specified by sequence (3.19).
For the limit additive-type reduced observables (3.6) and initial states
(3.19) the following equality is true:
(
b(1)(t), f (cc)
)
=
=
8ÿ
s=0
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs b(1)s (t, x1, . . . , xs)gs(x1, . . . , xs)
sź
i=1
f01 (xi) =
=
ż
R3ˆR3
dx1 b
0
1(x1)f1(t, x1),
where the functions b(1)s (t) are represented by expansions (3.6) and the
limit reduced distribution function f1(t) is represented by the following
series expansion:
f1(t, x1) =
8ÿ
n=0
tż
0
dt1 ¨ ¨ ¨
tn´1ż
0
dtn
ż
(R3ˆR3)n
dx2 ¨ ¨ ¨ dxn+1S1̊ (t´ t1, 1)ˆ
ˆ L0,˚
int (1, 2)S1̊ (t1 ´ t2, j1) ¨ ¨ ¨
nź
in=1
S1̊ (tn ´ tn, in)ˆ
ˆ
nÿ
kn=1
L0,˚
int (kn, n+ 1)
n+1ź
jn=1
S1̊ (tn, jn)g1+n(x1, . . . , xn+1)
n+1ź
i=1
f01 (xi).
(3.20)
Series (3.20) is uniformly convergent for a finite time interval under the
condition as above.
782
The function f1(t) represented by series (3.20) is a weak solution of the
Cauchy problem of the Boltzmann kinetic equation with initial correla-
tions [30,55]:
B
Btf1(t, x1) = ´xp1, B
Bq1 yf1(t, x1) +
ż
R3ˆS2+
dp2 dη xη, (p1 ´ p2)yˆ
ˆ (
g2(q1 ´ p1̊t, p1̊ , q2 ´ p2̊t, p2̊)f1(t, q1, p1̊)f1(t, q1, p2̊)´
´ g2(q1 ´ p1t, p1, q2 ´ p2t, p2)f1(t, x1)f1(t, q1, p2)
)
,
(3.21)
f1(t, x1)
ˇ̌
t=0
= f01 (x1). (3.22)
This fact is proved similarly to the case of a perturbative solution of the
BBGKY hierarchy for hard spheres represented by the iteration series [14,
58].
Thus, in the case of initial states specified by a one-particle reduced dis-
tribution function (3.19) we establish that the dual Boltzmann hierarchy
with hard sphere collisions (3.8) for additive-type reduced observables de-
scribes the evolution of a hard sphere system just as the Boltzmann kinetic
equation with initial correlations (3.21).
The property of the propagation of initial correlations is a consequence
of the validity of the following equality for the mean value functional of the
limit k-ary reduced observables in the case of k ě 2
(
b(k)(t), f (cc)
)
=
=
8ÿ
s=0
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs b(k)s (t, x1, . . . , xs)gs(x1, . . . , xs)
sź
j=1
f01 (xj) =
=
1
k!
ż
(R3ˆR3)k
dx1 ¨ ¨ ¨ dxk b0k(x1, . . . , xk)ˆ
ˆ
kź
i1=1
S1̊ (t, i1)gk(x1, . . . , xk)
kź
i2=1
(S1̊ )
´1(t, i2)
kź
j=1
f1(t, xj),
(3.23)
where the one-particle reduced distribution function f1(t, xj) is solution
(3.20) of the Cauchy problem of the Boltzmann kinetic equation with initial
correlations (3.21), (3.22), and the inverse group to the group of operators
S1̊ (t) we denote by
(S1̊ )
´1(t) = S1̊ (´t) = S1(t).
This fact is proved similarly to the proof of a property on the propagation
of initial chaos (3.17).
Evolution equations of colliding particles 783
We note that, according to equality (3.23), in the Boltzmann–Grad limit,
the reduced correlation functions are defined as cluster expansions of re-
duced distribution functions, namely,
fs(t, x1, . . . , xs) =
ÿ
P:(x1,...,xs)=
Ť
i
Xi
ź
XiĂP
g|Xi|(t,Xi), s ě 1,
and they have the explicit form:
g1(t, x1) = f1(t, x1),
gs(t, x1, . . . , xs) =
= g̃s(q1 ´ p1t, p1, . . . , qs ´ pst, ps)
sź
j=1
f1(t, xj), s ě 2,
(3.24)
where for initial correlation functions (3.19) it is used the following nota-
tions:
g̃s(x1, . . . , xs) =
ÿ
P:(x1,...,xs)=
Ť
i
Xi
ź
XiĂP
g|Xi|(Xi),
the symbol
ř
P means the sum over possible partitions P of the set of
arguments (x1, . . . , xs) on |P| nonempty subsets Xi, and the one-particle
reduced distribution function f1(t) is a solution of the Cauchy problem of
the Boltzmann kinetic equation with initial correlations (3.21),(3.22).
Thus, in the case of the limit k-ary reduced observables, a solution of the
dual Boltzmann hierarchy with hard sphere collisions (3.8) is equivalent to a
property of the propagation of initial correlations for the k-particle reduced
distribution function in the sense of equality (3.23) or in other words, the
Boltzmann–Grad scaling dynamics does not create new correlations.
4. ORiGiN OF KiNETiC EQUATiONS
One of the challenges of kinetic theory, as mentioned above, is under-
standing the nature of the possibility of describing the evolution of the state
of a system of many hard spheres by means of the state of a typical hard
sphere. More precisely, we further focus on the problem of the origin of the
description of the evolution of the state of hard spheres by the Enskog-type
kinetic equation.
In the circumstances where the initial state is specified by a one-particle
reduced distribution function, for the mean value functional of observables
at an arbitrary instant, the representation is also valid in terms of a one-
particle reduced distribution function, the evolution of which is governed
by a non-Markovian nonlinear evolution equation. In other words, for such
initial data, the Cauchy problem of the BBGKY hierarchy for hard spheres
784
is equivalent to the nonlinear Enskog-type kinetic equation and a sequence
of reduced functionals determined by the solution of this evolution equation.
4.1. The generalised Enskog kinetic equation. In the case of initial
state (3.10) the dual picture of the evolution to the picture described by
employing observables governed by the dual BBGKY hierarchy (2.1) for
hard spheres is the picture of the evolution of a state described by means of
the non-Markovian Enskog kinetic equation and by a sequence of explicitly
defined functionals of the solution of such a kinetic equation that describe
the evolution of all possible correlations in a system of hard spheres [46,47].
In view of the fact that the initial state is completely specified by a one-
particle reduced distribution function on allowed configurations (3.10), for
mean value functional (1.11) the following representation holds [46]:
(
B(t), F (c)
)
=
(
B(0), F (t | F1(t))
)
, (4.1)
where F (c) is the sequence of initial reduced distribution functions (3.10),
and the sequence F (t | F1(t))=
(
1, F1(t), F2(t | F1(t)), . . . , Fs(t | F1(t)), . . .
)
is a sequence of the reduced functionals of the state Fs(t, x1, . . . , xs | F1(t)),
s ě 2, represented by the series expansions over the products of the one-
particle distribution function F1(t), namely
Fs(t, x1, . . . , xs | F1(t))
.
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+nˆ
ˆ V1+n(t, t1, . . . , su, s+ 1, . . . , s+ n)
s+nź
i=1
F1(t, xi), s ě 2.
(4.2)
where the generating operators of series (4.2) are the (1 + n)th-order oper-
ators V1+n(t), n ě 0, determined by the following expansions [47]:
V1+n
(
t, t1, . . . , su, s+ 1, . . . , s+ n
)
=
= n!
nÿ
k=0
(´1)k
nÿ
n1=1
¨ ¨ ¨
n´n1´¨¨¨´nk´1ÿ
nk=1
1
(n´ n1 ´ ¨ ¨ ¨ ´ nk)!
ˆ
ˆ pA1+n´n1´¨¨¨´nk
(t, t1, . . . , su, s+1, . . . , s+n´n1´¨ ¨ ¨´nk)ˆ
ˆ
kź
j=1
ÿ
Dj :Zj=
Ť
lj
Xlj
|Dj |ďs+n´n1´¨¨¨´nj
1
|Dj |!ˆ
ˆ
s+n´n1´...´njÿ
i1‰...‰i|Dj |=1
ź
Xlj
ĂDj
1
|Xlj |!
pA1+|Xlj
|(t, ilj , Xlj ),
(4.3)
Evolution equations of colliding particles 785
In expansion (4.3) the symbol
ř
Dj :Zj=
Ť
lj
Xlj
means the sum over all pos-
sible dissections of the linearly ordered set
Zj ” (s+ n´ n1 ´ . . .´ nj + 1, . . . , s+ n´ n1 ´ . . .´ nj´1)
on no more than s + n ´ n1 ´ . . . ´ nj linearly ordered subsets, and the
(1 + n)th-order scattering cumulant we denoted by the operator:
pA1+n(t, t1, . . . , su, s+ 1, . . . , s+ n)
.
=
.
= A1̊+n(t, t1, . . . , su, s+ 1, . . . , s+ n)XR3(s+n)zWs+n
s+nź
i=1
A1̊(t, i)
´1,
(4.4)
where the operator A1̊+n(t) is the (1 + n)th-order cumulant of the groups
of operators (1.6) of hard spheres.
We provide some examples of expressions for the generating operators of
series (4.2) for reduced functionals of the state:
V1(t, t1, . . . , su) = pA1(t, t1, . . . , su) .=
.
= Ss̊ (t, 1, . . . , s)XR3(s)zWs
sź
i=1
S1̊ (t, i)
´1,
V2(t, t1, . . . , su, s+ 1) = pA2(t, t1, . . . , su, s+ 1)´
´ pA1(t, t1, . . . , su)
sÿ
i1=1
pA2(t, i1, s+ 1).
The method of constructing reduced state functionals (4.2) is based on
the application of the variation of cluster expansions, the so-called kinetic
cluster expansions, to generating operators (2.22) of series representing so-
lutions of hierarchies of evolution equations [47].
The one-particle distribution function F1(t) in dimensionless form, i.e.,
the first element of the sequence F (t | F1(t)), is determined by series (2.20),
namely
F1(t, x1) =
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dx2 ¨ ¨ ¨ dxn+1ˆ
ˆ A1̊+n(t, 1, . . . , n+ 1)
n+1ź
i=1
F ϵ,0
1 (xi)XR3(1+n)zW1+n
,
(4.5)
where the generating operator A1̊+n(t) is the (1 + n)th-order cumulant of
the groups of operators (1.6).
Let us note that in particular case of initial data (2.2) specified by the
additive-type reduced observables, according to solution expansion (2.10),
786
equality (4.1) takes the form
(
B(1)(t), F (0)
)
=
ż
R3ˆR3
dx1 b
ϵ
1(x1)F1(t, x1), (4.6)
where the one-particle distribution function F1(t) is determined by series
(4.5). In the case of initial data (2.2) specified by the s-ary reduced observ-
able s ě 2, equality (4.1) has the form
(
B(s)(t), F (0)
)
=
1
s!
ż
(R3ˆR3)s
dx1 ¨ ¨ ¨ dxs bϵs(x1, . . . , xs)Fs(t, x1, . . . , xs | F1(t)),
where the reduced functionals of the state Fs(t, x1, . . . , xs | F1(t)) are de-
termined by series (4.2).
Thus, for the initial state specified by a one-particle distribution function,
the evolution of all possible states of a system of many hard spheres can
be described by means of the state of a typical particle without any scaling
approximations. We emphasize that reduced functionals of the state(4.2)
describe all possible correlations created during the evolution of many hard
spheres in terms of the state of a typical hard sphere.
For t ě 0 the one-particle distribution function (4.5) is a solution of the
following Cauchy problem of the non-Markovian generalized Enskog kinetic
equation [47,53]:
B
BtF1(t, q1, p1) = ´xp1, B
Bq1 yF1(t, q1, p1) + ϵ2
ż
R3ˆS2+
dp2dηxη, (p1 ´ p2)yˆ
ˆ (
F2(t, q1, p1̊ , q1´ϵη, p2̊ | F1(t))´F2(t, q1, p1, q1+ϵη, p2 | F1(t))
)
,
(4.7)
F1(t)
ˇ̌
t=0
= F ϵ,0
1 , (4.8)
where the collision integral is determined by the reduced functional of the
state (4.2) in the case of s = 2 and the expressions p1̊ and p2̊ are the pre-
collision momenta of hard spheres (1.4). The series on the right-hand side
of this equation converges under the condition: }F1(t)}L1(RˆR) ă e´8 .
Hence in the case of the additive-type reduced observables the generalized
Enskog kinetic equation (4.7) is dual to the dual BBGKY hierarchy of hard
spheres (2.1) with respect to bilinear form (1.11).
We observe that the structure of the collision integral of the generalized
Enskog equation (4.7) is such that the first term of its expansion is the
collision integral of the Boltzman–Enskog kinetic equation, and the next
terms describe all possible correlations that are created by the dynamics of
hard spheres and by the propagation of initial correlations connected with
Evolution equations of colliding particles 787
the forbidden configurations, indeed
B
BtF1(t, x1) = ´xp1, B
Bq1 yF1(t, x1) + IGEE ,
where the collision integral is determined by the following series expansion:
IGEE
.
= ϵ2
8ÿ
n=0
1
n!
ż
R3ˆS2+
dp2dη
ż
(R3ˆR3)n
dx3 ¨ ¨ ¨ dxn+2xη, (p1 ´ p2)yˆ
ˆ (
V1+n(t, t1˚, 2˚́ u, 3, . . . , n+2)F1(t, q1, p1̊)F1(t, q1´ϵη, p2̊)
n+2ź
i=3
F1(t, xi)´
´ V1+n(t, t1, 2+u, 3, . . . , n+ 2)F1(t, x1)F1(t, q1 + ϵη, p2)
n+2ź
i=3
F1(t, xi)
)
,
and the notations adopted for the conventional notation of the Enskog col-
lision integral were used: indices (17, 27
˘) denote that the evolution operator
V1+n(t) acts on the corresponding phase points (q1, p
7
1) and (q1 ˘ ϵη, p7
2),
and the (n + 1)th-order evolution operator V1+n(t), n ě 0, is determined
by expansion (4.3) in the case of s = 2.
We note that in the work [47] for the initial-value problem (4.7),(4.8)
the existence theorem was proved in the space of integrable functions. The
accordance of the generalized Enskog equation (4.7) and of the Markovian
Enskog-type kinetic equation was also established there. By the point, we
remark that in the paper [97] the explicit soliton-like solutions of kinetic
equation (4.7) were found.
Thus, if the initial state is specified by a one-particle distribution func-
tion on allowed configurations, then the evolution of many hard spheres
governed by the dual BBGKY hierarchy (2.1) for reduced observables can
be completely described by the generalized Enskog kinetic equation (4.7)
and by a sequence of reduced functionals of the state (4.2).
We remark also that in the case of the initial state that involves corre-
lations (3.18) considered approach permits to take into consideration the
initial correlations in the kinetic equations [30,52].
Further, we sketch out the Boltzmann–Grad scaling behavior of the non-
Markovian Enskog kinetic equation (4.7) and reduced state functional (4.2).
Taking into account the validity of assumption (3.11) for the initial one-
particle distribution function (3.10), in that case for a finite time interval,
the Boltzmann–Grad limit of dimensionless solution (4.5) of the Cauchy
problem of the non-Markovian Enskog kinetic equation (4.7),(4.8) exists in
788
the same sense, namely
lim
ϵÑ0
(
ϵ2F1(t, x1) ´ f1(t, x1)
)
= 0,
where the limit one-particle distribution function f1(t) is a weak solution
of the Cauchy problem of the Boltzmann kinetic equation (3.15),(3.16).
Taking into consideration the fact of the existence of the Boltzmann–
Grad scaling limit of a solution of the non-Markovian Enskog kinetic equa-
tion (4.7), for reduced functionals of the state (4.2) the following statement
holds [47]:
lim
ϵÑ0
(
ϵ2sFs
(
t, x1, . . . , xs | F1(t)
) ´
sź
j=1
f1(t, xj)
)
= 0,
where the limit one-particle distribution function f1(t) is governed by the
Boltzmann kinetic equation with hard sphere collisions (3.15). Because
all possible correlations of many hard spheres with elastic collisions are
described by reduced functionals of the state (4.2), as noted above, this
property means the propagation of the initial chaos in the Boltzmann–Grad
limit.
The proof of these statements is based on the properties of cumulants of
asymptotically perturbed groups of operators (1.6) and the explicit struc-
ture (4.3) of the generating operators of series expansions (4.2) for reduced
functional of state and of series (4.5).
4.2. Dynamics of correlations governed by kinetic equations. Let
the initial state be specified by a one-particle reduced correlation function,
namely, the initial state specified by a sequence of reduced correlation func-
tions satisfying the chaos property stated above, i.e., by the sequence
G(c) = (G0, G
0
1, 0, . . . , 0, . . .).
We note that such an assumption about initial states is intrinsic to the
contemporary kinetic theory of many-particle systems [14,15].
Since the initial data G(c) is completely specified only by a one-particle
correlation function, the Cauchy problem (2.60),(2.61) of the nonlinear hi-
erarchy for hard spheres is not a completely well-defined Cauchy problem
because the initial data is not independent for every unknown function
determined of the hierarchy of mentioned evolution equations. As a con-
sequence, it becomes possible to reformulate such a Cauchy problem as a
new Cauchy problem for a one-particle correlation function with indepen-
dent initial data and explicitly defined functionals of the solution of this
Cauchy problem for the kinetic equation.
Evolution equations of colliding particles 789
We formulate such a restated Cauchy problem and the sequence of the
suitable functionals. In the case under consideration, the reduced correla-
tion functionals Gs(t | G1(t)), s ě 2, are represented with respect to the
one-particle correlation function (2.64), i.e.,
G1(t, x1) =
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dx2 ¨ ¨ ¨ dx1+nA1̊+n(t, 1, . . . , n+ 1)ˆ
ˆ
n+1ź
i=1
G0
1(xi)XR3(n+1)zWn+1
,
(4.9)
as the following series expansions:
Gs
(
t, x1, . . . , xs | G1(t)
)
=
=
8ÿ
n=0
1
n!
ż
(R3ˆR3)n
dxs+1 ¨ ¨ ¨ dxs+n ˆ
ˆ Vs+n
(
t, 1, . . . , s+ n
) s+nź
i=1
G1(t, xi), s ě 2.
(4.10)
The generating operator Vs+n(t), n ě 0, of the (s+n)th-order of this series
is determined by the following expansion:
Vs+n
(
t, 1, . . . , s, s+ 1, . . . , s+ n
)
=
= n!
nÿ
k=0
(´1)k
nÿ
n1=1
¨ ¨ ¨
n´n1´...´nk´1ÿ
nk=1
1
(n´ n1 ´ . . .´ nk)!
ˆ
ˆ Âs+n´n1´...´nk
(t, 1, . . . , s+ n´ n1 ´ . . .´ nk)ˆ
ˆ
kź
j=1
ÿ
Dj :Zj=
Ť
lj
Xlj
,
|Dj |ďs+n´n1´¨¨¨´nj
1
|Dj |!ˆ
ˆ
s+n´n1´...´njÿ
i1‰...‰i|Dj |=1
ź
Xlj
ĂDj
1
|Xlj |! Â1+|Xlj
|(t, ilj , Xlj ),
(4.11)
where
ř
Dj :Zj=
Ť
lj
Xlj
is the sum over all possible dissections of the linearly
ordered set
Zj ” (s+ n´ n1 ´ . . .´ nj + 1, . . . , s+ n´ n1 ´ . . .´ nj´1)
on no more than s+n´n1 ´ . . .´nj linearly ordered subsets, the (s+n)th-
order scattering cumulant is defined by the formula
Âs+n(t, 1, . . . , s+ n)
.
= As̊+n(t, 1, . . . , s+ n)XR3(s+n)zWs+n
s+nź
i=1
(A1̊)
´1(t, i),
790
and notations accepted above were used.
We adduce simplest examples of generating operators (4.11):
Vs(t, 1, . . . , s) = As(t, 1, . . . , s)XR3szWs
sź
i=1
(A1̊)
´1(t, i),
Vs+1(t, 1, . . . , s, s+1) = As+1(t, 1, . . . , s+1)XR3(s+1)zWs+1
s+1ź
i=1
(A1̊)
´1(t, i)´
´ As(t, 1, . . . , s)XR3szWs
sź
i=1
(A1̊)
´1(t, i)ˆ
ˆ
sř
j=1
A2(t, j, s+ 1)XR6zW2
(A1̊)
´1(t, j)(A1̊)
´1(t, s+ 1).
If }G1(t)}L1(R3ˆR3) ă e´(3s+2), for arbitrary t P R series (4.10) converges
in the norm of the space L1
s [47].
We note that in the case of initial state specified by a one-particle correla-
tion function the reduced correlation functionals (4.10) describe all possible
correlations generated by the dynamics of many hard spheres in terms of a
one-particle correlation function.
Thus, according to the representation (2.64) of reduced correlation func-
tions, the cumulant structure of their generating operators induces a gene-
ralized cumulant structure of the generating operators for series (4.10) of
reduced correlation functionals.
In this case, the method of constructing reduced correlation function-
als (4.10) is based on the application of the variation of cluster expansions,
the so-called kinetic cluster expansions [47], to generating operators (2.36)
of series representing reduced correlation functions (2.64).
Indeed, taking into account relations of kinetic cluster expansions for
scattering cumulants (4.4):
pAs+n(t, 1, . . . , s+ n) =
nÿ
n1=0
n!
(n´ n1)!
Vs+n´n1
(
t, 1, . . . , s+ n´ n1
)ˆ
ˆ
ÿ
D:Z=
Ť
l Xl,
|D|ďs+n´n1
1
|D|!
s+n´n1ÿ
i1‰...‰i|D|=1
ź
XlĂD
1
|Xl|!
pA1+|Xl|(t, il, Xl),
where
ř
D:Z=
Ť
l Xl, |D|ďs+n´n1
is the sum over all possible dissections D of
the linearly ordered set
Z ” (s+ n´ n1 + 1, . . . , s+ n)
Evolution equations of colliding particles 791
on no more than s+n´n1 linearly ordered subsets, we derive the expansions
of the reduced correlation functionals
Gs(t, x1, . . . , xs | G1(t)), s ě 2,
on the basis of solution expansions (2.64) of the hierarchy of nonlinear
evolution equations (2.60).
Note that the structure of kinetic cluster expansions of scattering cu-
mulants of the groups of operators is similar to the structure of virial ex-
pansions of equilibrium distribution functions, i.e., as a power series in the
density.
If initial data G0
1 P L1
1, then for arbitrary t P R one-particle correla-
tion function (4.9) is a weak solution of the Cauchy problem of the non-
Markovian Enskog kinetic equation [53]:
B
BtG1(t, x1) = L˚(1)G1(t, x1) +
ż
R3ˆR3
dx2 Li̊nt(1, 2)G1(t, x1)G1(t, x2)+
+
ż
R3ˆR3
dx2 Li̊nt(1, 2)G2
(
t, x1, x2 | G1(t)
)
,
(4.12)
G1(t, x1)
ˇ̌
t=0
= G0
1(x1), (4.13)
where the first part of the collision integral in equation (4.12) has the
Boltzmann–Enskog structure, and the second part of the collision integral
is determined in terms of the two-particle correlation functional represented
by series expansion (4.10) which describes all possible correlations that are
created by hard-sphere dynamics and by the propagation of initial correla-
tions related to the forbidden configurations.
In the paper [47], similar statements were proved for the state evolution
of a hard-sphere system described in terms of reduced distribution functions
governed by the BBGKY hierarchy. We emphasize that the nth term of
expansions (4.10) of the reduced correlation functionals are determined by
the (s+n)th-order generating operator (4.3) in contradistinction to the ex-
pansions of reduced distribution functionals of the state constructed in [47]
which are determined by the (1 + n)th-order generating operator (4.3).
Thus, for the initial state specified by a one-particle correlation function,
the evolution of all possible states of the system of hard spheres can be de-
scribed without any approximations within the framework of a one-particle
correlation function governed by the non-Markovian Enskog-type kinetic
equation (4.12), and by a sequence of explicitly defined functionals (4.10)
of its solution.
792
5. CONCLUSiON
In conclusion, the challenges of the evolution of many hard spheres with
inelastic collisions will be reviewed, as will some applications of the methods
outlined above to complex systems of various natures.
5.1. On the dynamics of inelastic collisions. According to contempo-
rary concept [96, 99] on a microscopic scale, the characteristic properties
of granular media are determined by dissipative collisional dynamics and
can be described as the evolution of a system of many hard spheres with
inelastic collisions.
Since the characteristic features of the collective behavior of inelastically
colliding particles in one-dimensional space reflect the main properties of
granular gases, an approach to the rigorous derivation of the Boltzmann-
type equation for one-dimensional granular gases will be presented below.
We note that, in contrast to the system of hard rods with inelastic collisions,
in one-dimensional space the evolution of hard rods with elastic collisions
is trivial in the Boltzmann–Grad scaling limit; it is known as so-called free
molecular motion or the Knudsen flow [14].
In the case of a one-dimensional granular gas for t ě 0 in dimensionless
form the Cauchy problem of the non-Markovian generalized Enskog kinetic
equation (4.7),(4.8) takes the form [37,42,43]:
B
BtF1(t, q1, p1) = ´p1 B
Bq1F1(t, q1, p1)+
+
8ż
0
dP P
( 1
(1 ´ 2ε)2
F2(t, q1, p1̨(p1, P ), q1 ´ ϵ, p2̨(p1, P ) | F1(t))´
´F2(t, q1, p1, q1 ´ ϵ, p1 + P | F1(t))
)
+
+
8ż
0
dP P
( 1
(1 ´ 2ε)2
F2(t, q1, p̃1̨(p1, P ), q1 + ϵ, p̃2̨(p1, P ) | F1(t))´
´F2(t, q1, p1, q1 + ϵ, p1 ´ P | F1(t))
)
,
(5.1)
F1(t)|t=0 = F ϵ,0
1 , (5.2)
where ε = 1´e
2 P [0, 12) and e P (0, 1] is a restitution coefficient, ϵ ą 0 is a
scaling parameter (the ratio of a hard sphere diameter (the length) σ ą 0
to the mean free path), the collision integral is determined by reduced
functional (4.2) of the state F1(t) in the case of s = 2 and the expressions:
p1̨(p1, P ) = p1 ´ P +
ε
2ε´ 1
P,
p2̨(p1, P ) = p1 ´ ε
2ε´ 1
P
Evolution equations of colliding particles 793
and the expressions
p̃1̨ (p1, P ) = p1 + P ´ ε
2ε´ 1
P,
p̃2̨ (p1, P ) = p1 +
ε
2ε´ 1
P,
are transformed pre-collision momenta of inelastically colliding particles in
a one-dimensional space.
The solution of the Cauchy problem (5.1),(5.2) is represented by the
following series:
F ϵ
1(t, x1) =
8ÿ
n=0
1
n!
ż
RnˆRn
dx2 ¨ ¨ ¨ dxn+1A1̊+n(t)
n+1ź
i=1
F ϵ,0
1 (xi)XR(1+n)zW1+n
, (5.3)
where the generating operator A1̊+n(t) is the (1+n)th-order cumulant (2.22)
of the semigroups of operators (1.6) of inelastically colliding hard rods in
a one-dimensional space. Let the initial one-particle distribution function
satisfy the following condition: |F ϵ,0
1 (x1)| ď Ce´β
2 p
2
1 , where β ą 0 is a
parameter and C ă 8 is some constant. Then every term of series (5.3)
exists; for a finite time interval it is the uniformly convergent series with
respect to x1 from an arbitrary compact, and function (5.3) is a weak
solution of the Cauchy problem (5.1),(5.2) of the non-Markovian Enskog-
type equation with inelastic collisions.
We assume that, in the sense of weak convergence, there exists a limit
w´ lim
ϵÑ0
(
F ϵ,0
1 (x1) ´ f01 (x1)
)
= 0.
Then, for a finite time interval, the Boltzmann–Grad limit of solution (5.3)
of the Cauchy problem of the non-Markovian Enskog-type equation for a
one-dimensional granular gas (5.1) exists in the sense of a weak convergence
w´ lim
ϵÑ0
(
F ϵ
1(t, x1) ´ f1(t, x1)
)
= 0, (5.4)
where the limit of one-particle distribution function (5.3) is determined by
the following series uniformly convergent on an arbitrary compact set
f1(t, x1) =
8ÿ
n=0
1
n!
ż
RnˆRn
dx2 ¨ ¨ ¨ dxn+1A
0
1+n(t)
n+1ź
i=1
f01 (xi), (5.5)
and the generating operator A0
1+n(t) ” A0
1+n(t, 1, . . . , n+1) is the (n+1)th-
order cumulant of semigroups (1.6) of point particles with inelastic colli-
sions. For t ě 0 an infinitesimal generator of this semigroup of operators is
794
determined by the operator:
(L˚,0
n fn)(x1, . . . , xn) = ´
nÿ
j=1
pj
B
Bqj fn(x1, . . . , xn)+
+
nÿ
j1ăj2=1
|pj2 ´ pj1 |δ(qj1 ´ qj2)ˆ
ˆ
( 1
(1 ´ 2ε)2
fn(x1, . . . , xj̨1 , . . . , xj̨2 , . . . , xn) ´ fn(x1, . . . , xn)
)
,
where xj̨ ” (qj , pj̨ ) and the pre-collision momenta pj̨1 , pj̨2 of inelastically
colliding particles are determined by the following expressions:
pj̨1 = pj2 +
ε
2ε´ 1
(pj1 ´ pj2),
pj̨2 = pj1 ´ ε
2ε´ 1
(pj1 ´ pj2).
For t ě 0 the limit one-particle distribution function represented by
series (5.5) is a weak solution of the Cauchy problem of the Boltzmann-
type kinetic equation of point particles with inelastic collisions [43]
B
Btf1(t, q, p) = ´p B
Bq f1(t, q, p) +
+8ż
´8
dp1 |p´ p1|ˆ
ˆ
( 1
(1´2ε)2
f1(t, q, p
˛)f1(t, q, p1̨)´f1(t, q, p) f1(t, q, p1)
)
+
8ÿ
n=1
I(n)
0 .
(5.6)
In kinetic equation (5.6) the remainder
ř8
n=1 I
(n)
0 of the collision integral
is determined by the following expressions:
I(n)
0 ” 1
n!
8ż
0
dP P
ż
RnˆRn
dq3dp3 ¨ ¨ ¨ dqn+2dpn+2V1+n(t)ˆ
ˆ
( 1
(1 ´ 2ε)2
f1(t, q, p1̨(p, P ))f1(t, q, p2̨(p, P )) ´ f1(t, q, p)f1(t, q, p+ P )
)
ˆ
ˆ
n+2ś
i=3
f1(t, qi, pi)+
+
8ż
0
dP P
ż
RnˆRn
dq3dp3 ¨ ¨ ¨ dqn+2dpn+2V1+n(t)ˆ
ˆ
( 1
(1 ´ 2ε)2
f1(t, q, p̃1̨(p, P ))f1(t, q, p̃2̨(p, P )) ´ f1(t, q, p)f1(t, q, p´ P )
)
ˆ
ˆ
n+2ś
i=3
F1(t, qi, pi),
Evolution equations of colliding particles 795
where the generating operators
V1+n(t) ” V1+n(t, t1, 2u, 3, . . . , n+ 2), n ě 0,
of the series for a collision integral are represented by expansions (4.3) with
respect to the cumulants of semigroups of scattering operators of point hard
rods with inelastic collisions in a one-dimensional space
pS0
n(t, 1, . . . , n)
.
= S˚,0
n (t, 1, . . . , s)
nź
i=1
(S˚,0
1 )´1(t, i).
In fact, series expansions for the collision integral of the non-Markovian
Enskog equation for a granular gas (5.6) or solution (5.3) are represented
as the power series over the density, so that the terms I(n)
0 , n ě 1, of the
collision integral in kinetic equation (5.6) are corrections with respect to
the density to the Boltzmann collision integral of one-dimensional granular
gases formulated in the paper [96].
Since the scattering operator of point hard rods is an identity operator in
the approximation of elastic collisions, namely, in the limit ε Ñ 0, the col-
lision integral of the Boltzmann kinetic equation (5.6) in a one-dimensional
space is identical to zero. In the quasi-elastic approximation [96] the limit
one-particle distribution function (5.5)
lim
εÑ0
εf1(t, q, p) = f0(t, q, p),
satisfies the nonlinear friction kinetic equation for one-dimensional granular
gases [96]:
B
Btf
0(t, q, p) = ´p B
Bq f
0(t, q, p)+
+
B
Bp
8ż
´8
dp1 |p1 ´ p| (p1 ´ p) f0(t, q, p1)f
0(t, q, p).
Taking into consideration the result (5.4) on the Boltzmann–Grad as-
ymptotic behavior of the non-Markovian Enskog equation (5.1), for reduced
functionals of the state (4.2) in a one-dimensional space, the following state-
ment is true [43]:
w´ lim
ϵÑ0
(
Fs
(
t, x1, ..., xs | F ϵ
1(t)
) ´ fs
(
t, x1, ..., xs | f1(t)
))
= 0, s ě 2, (5.7)
where in equality (5.7) the limit reduced functionals of the limit one-particle
distribution function (5.5) are determined by the series expansions with a
structure similar to series (4.2) and the generating operators represented by
expansions (4.3) over the cumulants of semigroups of scattering operators
of point hard rods with inelastic collisions in a one-dimensional space.
796
As mentioned above, in the case of a system of hard rods with elastic
collisions, the limit reduced functionals of the state are the product of the
limit one-particle distribution functions, describing the free motion of point
particles.
Thus, the Boltzmann–Grad asymptotic behavior of solution (5.3) of the
non-Markovian Enskog equation (5.1) is governed by the Boltzmann kinetic
equation (5.6) for a one-dimensional granular gas.
We emphasize that the Boltzmann-type equation (5.6) describes the
memory effects in a one-dimensional granular gas. In addition, the limit of
reduced functionals of the state fs
(
t, x1, . . . , xs | f1(t)
)
, s ě 2, which are
defined above, describe the process of the propagation of initial chaos in a
one-dimensional granular gas, or, in other words, the process of creatiing
correlations in a system of hard rods with inelastic collisions.
It should be noted that the Boltzmann–Grad asymptotic behavior of the
non-Markovian Enskog equation with inelastic collisions in a multidimen-
sional space is analogous to the Boltzmann–Grad asymptotic behavior of
a hard sphere system with the elastic collisions [43], i.e., it is governed
by the Boltzmann equation for a granular gas [98, 99], and the asymptotic
behavior of the reduced functionals of the state (4.2) is described by the
product of one-particle distribution functions of its solution, i.e., describes
the propagation of initial chaos.
5.2. Some bibliographic notes on collisional dynamics. Above, it
was studied systems of identical colliding particles, which are described
by means of functions of observables and distribution functions, which are
symmetrical with respect to arbitrary permutations of their arguments. In
papers [32,33,67,70,82], the theory of the hierarchies of evolution equations
for systems of many colliding particles described by non-symmetric func-
tions was developed. An example of such a system is a one-dimensional
system of particles interacting with their nearest neighbors, so-called non-
symmetric systems of particles [32].
As is known, many-entity systems of active soft condensed matter are
dynamic systems exhibiting a collective behavior that differs from the sta-
tistical behavior of ordinary gases. To describe the nature of entities (or self-
propelled particles), in the paper [78], collision dynamics based on Markov
jump processes, which should reflect the internal properties of living crea-
tures, were proposed. In works [36, 45] an approach was developed to de-
scribe the collective behavior of complex systems of mathematical biology
within the framework of the evolution of observables of many colliding sto-
chastic processes, and the dual Vlasov hierarchy was constructed in the
mean field approximation. This representation of the kinetic evolution
seems, in fact, to be the direct mathematically fully consistent formulation
Evolution equations of colliding particles 797
modeling the kinetic evolution of biological systems, since the notion of
the state is more subtle and is an implicit characteristic of populations of
living creatures. In the paper [36] the processes of creation of correlations
generated by the dynamics of active soft matter and propagation of ini-
tial correlations have also been described by means of the non-Markovian
generalized kinetic equation with initial correlations, and, in particular, in
the mean-field scaling approximation, the Vlasov-type kinetic equation for
many colliding stochastic processes was constructed.
The study of systems of colliding particles in interaction with the envi-
ronment, the so-called open systems, involves a number of unsolved fun-
damental problems. One of them is related to the challenge of the ori-
gin of stochastic behavior in dynamical systems of many particles. In pa-
pers [48–50], based on the approaches to the derivation of kinetic equations
outlined above, a generalization of the Fokker–Planck equation for open
systems of colliding particles was justified.
In previous decades, a lot of work has been performed on discrete-velocity
models of the Boltzmann equation, which are of significant conceptual in-
terest for the kinetic theory of gases and, at the same time, represent a
fascinating mathematical subject [84]. In connection with this topic of re-
search, we note the works [39–41, 54], in which the discrete-velocity model
was studied, related to the problem of deriving a model of the Enskog
discrete-velocity kinetic equation.
An overview of some modern applications of kinetic equations to the de-
scription of non-equilibrium processes in complex systems of various natures
is presented in the monograph [23].
5.3. Outlook. The purpose of this review was to analyze the development
and current advances of the theory of evolution equations for systems of
many colliding particles, in particular, kinetic equations and their relations
to the fundamental equations that describe the laws of nature.
The problem of constructing a solution to the Cauchy problem for hier-
archies of evolution equations of observables (2.1) and the state (2.17) of a
system of hard spheres with elastic collisions for initial data belonging to
some functional spaces is considered. As was established, solutions of hier-
archies of evolution equations are determined by groups of operators, which
are represented by expansions over the groups of particles whose evolution
is described by cumulants of the corresponding order of the groups of oper-
ators of the Liouville equations. Due to the fact that the cumulants of the
groups of operators are determined by cluster expansions of the groups of
operators of the Liouville equations, in the corresponding function spaces
there are different representations for solutions to the hierarchies of evolu-
tion equations. These cluster expansions of the groups of operators underlie
798
the classification of possible solution representations to the Cauchy problem
for the hierarchies of evolution equations of many colliding particles.
To describe the evolution of the state of a many-particle system, there is
an alternative approach that is based on the dynamics of correlations. In
this approach, a state of finitely many hard spheres is described with the
employment of functions determined by the cluster expansions of the prob-
ability distribution functions that are governed by the so-called Liouville
hierarchy (2.31). It was above established that the constructed dynamics of
correlation underlie the description of the dynamics of infinitely many hard
spheres governed by the BBGKY hierarchy for reduced distribution func-
tions (2.17) or the hierarchy of nonlinear evolution equations for reduced
correlation functions (2.60), i.e., of the cumulants of reduced distribution
functions. We emphasize the importance of the mathematical description of
the processes of the creation and propagation of correlations, in particular,
for numerous applications [2, 3, 86].
To describe the evolution of many hard spheres within the framework of
the evolution of states for an initial state close to ”kinetic,” i.e., a state de-
scribed in terms of the state of a typical particle, there is another possibility:
by means of the so-called non-Markovian Enskog kinetic equation (4.7). In
other words, the origin of the collective behavior of a hard-sphere system
on a microscopic scale was examined above. As already mentioned, one of
the advantages of such an approach to the derivation of kinetic equations
from underlying collisional dynamics is the opportunity to construct the ki-
netic equations with initial correlations, which makes it possible to describe
the creation of correlations and propagation of initial correlations. Another
advantage of this approach is related to the rigorous derivation of the Boltz-
mann equation (3.15) with higher-order corrections to the canonical term
of the collision integral.
Thus, the concept of cumulants of the groups of operators of Liouville
equations underlies non-perturbative expansions of solutions to hierarchies
of fundamental evolution equations that describe the evolution of observ-
ables and a state of many colliding particles, as well as underlies the ki-
netic description of their collective behavior. We note that for quantum
many-particle systems the concept of cumulants of groups of operators is
considered in review [38].
In the paper, possible approaches to the rigorous derivation [35] and
justification [62, 63] of the kinetic equations for many colliding particles
were considered. One of them is an approach to the description of the
kinetic evolution within the framework of the evolution of the observables
of many colliding particles [51]. The advances of the method based on the
dual Boltzmann hierarchy (3.8) are the opportunity to construct kinetic
Evolution equations of colliding particles 799
equations (3.21), taking into account the correlations of particles of the
initial state, and the description of the process of propagation of initial
correlations in scaling approximations (3.24).
The paper [69] considered the challenge of deriving hydrodynamic equa-
tions from the dual BBGKY hierarchy for reduced observed microscopic
phase densities. We notice that the rigorous derivation of hydrodynamic
equations from the dynamics of many colliding particles is still an open
problem. Regarding the classical problem of rigorous derivation of the
hydrodynamic equations from the Boltzmann kinetic equation in scaling
limits, we refer to the books [72,93].
Acknowledgements. Glory to Ukraïne!
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V. I. Gerasimenko
INSTiTUTE OF MATHEMATiCS OF NAS OF UKRAiNE, KYiV
Email: gerasym@imath.kiev.ua
ORCID: 0000-0003-2577-2237
I. V. Gapyak
TARAS SHEVCHENKO NATiONAL UNiVERSiTY OF KYiV
Email: gapjak@ukr.net
ORCID: 0000-0003-2102-1583
|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English Ukrainian |
| last_indexed | 2026-08-04T01:08:55Z |
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| spelling | oai:trim.imath.kiev.ua:article-5282023-08-17T09:06:54Z Advances in theory of evolution equations of many colliding particles Досягнення в теорії рівнянь еволюції багатьох частинок із зіткненнями Gerasimenko, Viktor Gapyak, Igor Герасименко, Віктор Гап'як, Ігор The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations. В огляді подано строгі результати теорії фундаментальних еволюційних рівнянь систем багатьох частинок із зіткненнями, а також розглянуто їх зв’язок із нелінійними кінетичними рівняннями, які описують колективну поведінку частинок у скейлінгових наближеннях. Інститут математики НАН України 2023-08-17 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/528 10.3842/trim.v20n1.528 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 20 No. 1 (2023): Modern problems of mathematics and its applications, III; 729-804 Сборник Трудов Института математики НАН Украины; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 729-804 Збірник Праць Інституту математики НАН України; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 729-804 3083-7529 1815-2910 en uk https://trim.imath.kiev.ua/index.php/trim/article/view/528/510 Авторське право (c) 2023 Віктор Герасименко, Ігор Гап'як http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Gerasimenko, Viktor Gapyak, Igor Герасименко, Віктор Гап'як, Ігор Advances in theory of evolution equations of many colliding particles |
| title | Advances in theory of evolution equations of many colliding particles |
| title_alt | Досягнення в теорії рівнянь еволюції багатьох частинок із зіткненнями |
| title_full | Advances in theory of evolution equations of many colliding particles |
| title_fullStr | Advances in theory of evolution equations of many colliding particles |
| title_full_unstemmed | Advances in theory of evolution equations of many colliding particles |
| title_short | Advances in theory of evolution equations of many colliding particles |
| title_sort | advances in theory of evolution equations of many colliding particles |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/528 |
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