Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions

The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operators. The infinitesimal generator of the semigroup coincides on a certain everywhere dense set with the operator on the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data, they are generalized ones. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 371–380, March, 2006.     

Equilibrium time evolutions and BBGKY hierarchy equations of large classical systems

Summary We show the existence of a time evolution {P _t ; t of a locally perturbed equilibrium state P of infinitely many particles in {suv}, v1, evolving under the action of the infinite Newtonian dynamics associated to the same smooth, finite range pair potential as the equilibrium state itself. Moreover, it is shown that {P _t ; t solves the weak BBGKY hierarchy equations. The treatment of this problem will be done in the general setting of so called ( ) point processes developed in [11, 10] and [4] and will require the method of moments.     

Solutions of Bogolyubov equations for one-dimensional system of hard spheres

For a one-dimensional system of particles that interact as hard elastic spheres the existence of global solutions to the Cauchy problem for the Bogolyubov equations is proved for initial data in spaces of sequences of summable and bounded functions.Institute of Mathematics, Uzb. SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 120–128, April, 1992.     

Differential-difference equations for the radial distribution function of hard rods and hard spheres under Percus-Yevick approximation

It is shown that under the assumption of the Percus-Yevick approximation, linear homogeneous retarded differential-difference equations for the radial distribution functions of hard rods and hard spheres can be derived from Baxter's relations. The solutions obtained by the standard method in differential-difference equation are identical to that obtained by Wertheim, Thiel, and Baxter.     

BBGKY hierarchy and its evolution operator for one class of integrable systems

We define the abstract BBGKY hierarchy and its formal evolution operator. The existence of the latter is established in a special Banach space for a system of charged particles with Chern-Simons interaction regularized for small and large distances. An analog of the high-temperature expansion of equilibrium statistical mechanics is applied.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 853–858, June, 1995.     

Chain of BBGKY equations for bimolecular chemical reactions

A dynamically verified statistical theory of moderately dense gases developed by Bogoliubov and others is generalized to the case of bimolecular chemical reactions in a gas. The corresponding chain of BBGKY equations is derived. From this chain, the kinetic equations for one-molecule distribution functions are obtained in the approximation of bimolecular and trimolecular interactions. Deceased. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 163–178, May, 1997.     

A reduction method for semilinear elliptic equations and solutions concentrating on spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A⊆R^2m$A\subseteq {\mathbb{R}}^{2m}$, m?2$m?2$, invariant by the action of a certain symmetry group can be reduced to a nonhomogeneous similar problem in an annulus D⊂R^m+1$D\subset {\mathbb{R}}^{m+1}$, invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A   which concentrate on one or two (m−1)$(m-1)$ dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.     

Nonlocal boundary-value problem for parabolic equations

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1621–1626, December, 1994.The work was supported by the Foundation for Fundamental Studies of Ukrainian State Committee on Science and Technology.     

Nonlocal reaction--diffusion equations and nucleation

A nonlocal reaction-diffusion equation is presented and analysedusing matched asymptotic expansions and multiple timescales.The problem models a binary mixture undergoing phase separation.The particular form of the equation is motivated by argumentsfrom the calculus of variations, with the nonlocality arisingfrom an enforcement of conservation of mass. It is shown thatthe evolution of the solution can be characterized by trackingthe motion of fronts separating phases. The propagation of theinterfaces is found to be a coarsening process which dependsin a nonlocal fashion on mean curvature. Several special featuresof the equations of motion for the fronts are studied, and therelation of this evolution to Cahn-Hilliard theory and nucleationis discussed     

A nonsmooth program for jamming hard spheres

We study packings of $n$ hard spheres of equal radius in the $d$ -dimensional unit cube. We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there are only finitely many radii of such jammed configurations. We propose an algorithm for the maximization of this maximal radius function and present examples for five to eight disks in the unit square and four to six spheres in the unit cube. The method allows straightforward generalization to packings of spheres in other compact containers.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.