Regularization Method for Solution of BBGKY Hierarchy

T. V. Ryabukha
Institute of Mathematics
01601 Kyiv-4, Ukraine
Abstract. We suggest a regularization method for the solution in the cumulant representation for the initial value problem
of the BBGKY hierarchy for a one-dimensional systems of hard spheres interacting via a short-range potential. An existence
theorem of a local in time weak solution is proved for the initial data from the space of sequences of bounded functions.
Keywords: BBGKY Hierarchy; Cumulant; Regularized Solution.
PACS: 05.20.-y
INTRODUCTION
It is well known the evolution of states of many-particle systems is described by the initial value problem for the
BBGKY hierarchy [1, 2]. While solving the initial value problem for the BBGKY hierarchy of the classical systems of
particles with the initial data from the space of sequences of bounded functions, one is faced with certain difficulties
related to the divergence of integrals with respect to configuration variables in each term of an expansion of the solution
[3] (see also [1, 2]). The same problem arises also in the case of the cumulant representation of the solution constructed
in [4, 5]. In this paper, we suggest a regularization method for the solution in the cumulant representation for the
BBGKY hierarchy. Due to this method, the structure of the solution expansions guarantees the mutual compensation
of the divergent integrals in every term of the series. We establish convergence conditions for the series of the solution
and prove an existence theorem of a local in time weak solution of the BBGKY hierarchy for the initial data from the
space of sequences of functions which are bounded with respect to the configuration variables and by the Maxwellian
distribution with respect to the momentum variables.
CUMULANT REPRESENTATION FOR SOLUTION OF BBGKY HIERARCHY
Let us consider a one-dimensional system of identical particles (intervals with length s and unit mass m = 1)
interacting as hard spheres via a short range pair potential F: Every particle i is characterized by phase coordinates
(qi; pi) ´ xi 2 R£R; i ¸ 1: For the configurations qi 2 R1 of such a system (qi is the position of the center of
the ith particle), the following inequalities must be satisfied: jqi ¡qjj ¸ s ; i 6= j ¸ 1. The set Wn ´ ffq1; : : : ;qng j
9(i; j); i 6= j 2 f1; : : : ;ng : jqi¡qjj <s g defines the set of forbidden configurations in the phase space of a system of
n particles. The phase trajectories of such hard sphere system are determined almost everywhere in the phase space
fx1; : : : ;xng 2 Rn £(Rn nWn), namely, outside a certain set M0
n of the Lebesgue measure zero [1]. The initial data
fx1; : : : ;xng 2 Rn£(Rn nWn) belong to the setM0
n if (a) there is more than one pair collision at the same moment of
time t 2 (¡¥;+¥) or (b) there is infinite number of collisions occur within a finite time interval.
We assume that the interaction between the hard spheres is given by a potential F with a finite range R such that the
following conditions are satisfied.

REGULARIZATION METHOD
For F(0) 2 L¥
x ;b every term n ´ jX nYj of expansion (4) contains divergent integrals with respect to the configuration
variables. Let us show that the above-stated cumulant nature of the solution expansions (4) for the initial value
problem of the BBGKY hierarchy guarantees the compensation of the divergent integrals, i. e., the cumulants are
determined terms of expansion (4) as the sum of summands with divergent integrals that compensate one another. In
order to prove this fact, let us rearrange the terms of expansion (4) so that they be represented by the simplest mutually
compensating groups of summands. Such procedure will be called a regularization of the solution (4). In this case,
the regularization will be based on expressing cumulants (6) of higher order in terms of the first and second order
cumulants. For fixed initial data the second-order.

ACKNOWLEDGMENTS
The author is pleased to thank her supervisor Prof. Victor Gerasimenko for the problem setting and many useful
discussions.
This work was partially supported by the National Academy of Sciences of Ukraine through grant No. 0105U005666
for young scientists and partially supported by INTAS.
REFERENCES
1. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics.
Continuous Systems, Second ed., Taylor & Francis Inc., London and N. Y., 2002, 352 p.
2. C. Cercignani, V. I. Gerasimenko, and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Acad. Publ.,
1997, 256 p.
3. D. Ya. Petrina, TMP, 38, No 2, pp. 230-250 (1979).
4. V. I. Gerasimenko, and T. V. Ryabukha, Ukr. Math. J., 54, No 10, pp. 1583–1601 (2002).
5. V. I. Gerasimenko, T. V. Ryabukha, and M. O. Stashenko, J. Phys. A: Math. Gen., 37, No 42, pp. 9861-9872 (2004).

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