Skrypnik W. I.
Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 519-533
We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line.
Two-dimensional Coulomb dynamics of two and three equal negative charges in the field of two equal fixed positive charges
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1528-1539
Periodic and bounded for positive time solutions of the planar Coulomb equation of motion for two and three identical negative charges in the field of two equal fixed positive charges are found. The systems possess equilibrium configurations to which the found bounded solutions converge in the infinite time limit. The periodic solutions are obtained with the help of the Lyapunov center theorem.
Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges
Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1273-1285
Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed and equal positive charges are found with the help of the Lyapunov center theorem.
Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line
Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 679–693
Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied.
Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 689–697
We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions.
Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 546-554
The existence of holomorphic (in time) solutions of the nonrelativistic Darwin equations of motion of point charges is proved with the help of the Cauchy theorem.
Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 270-280
The Maxwell - Lorenz system of an electromagnetic field interacting with charged particles (point charges) is considered in the Darwin approximation which is characterized by the Lagrangian and Hamiltonian of the particles both uncoupled with the field. The solution of the equation of motion of the particles with the approximated Darwin Hamiltonian is found on a finite time interval with the use of the Cauchy theorem. Components of this solution are represented as holomorphic functions of time.
Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1687–1704
We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.
Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials
Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 689-700
For a Gibbs system of one-dimensional quantum oscillators on a d-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.
Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 538-547
Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ.
Solutions of the Kirkwood–Salsburg equation for a lattice classical system of one-dimensional oscillators with positive finite-range many-body interaction potentials
Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1427–1433
For a system of classical one-dimensional oscillators on the d-dimensional hypercubic lattice interacting via pair superstable and many-body positive
finite-range potentials, the (lattice) Kirkwood–Salsburg equation is proposed for the first time and is solved.
Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1138–1143
For a system of classical particles interacting via stable pairwise integrable and positive many-body (nonpairwise) finite-range potentials, we prove the existence of a solution of the symmetrized Kirkwood-Salsburg equation.
Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1407–1424
The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools.
Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 976–996
For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.
Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 388–405
The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools.
Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 810–817
Long-range order is proved to exist for lattice linear oscillator systems with ferromagnetic potential energy containing a term with strong nearest-neighbor (n-n) quadratic pair potential. A contour bound and a generalized Peierls argument are used in the proof.
On Polymer Expansion for Gibbsian States of Nonequilibrium Systems of Interacting Brownian Oscillators
Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1356-1377
The convergence of polymer cluster expansions for correlation functions of general Gibbs oscillator-type systems and related nonequilibrium systems of Brownian oscillators is established. The initial states for the latter are Gibbsian. It is proved that the sequence of the constructed correlation functions of the nonequilibrium system is a generalized solution of a diffusion BBGKY-type hierarchy.
Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1664-1685
By using a high-temperature cluster expansion, we construct the evolution operator of the BBGKY-type gradient diffusion hierarchy for plane rotators that interact via a summable pair potential in a Banach space containing the Gibbs (stationary) correlation functions. We prove the convergence of this expansion for a sufficiently small time interval. As a result, we prove that weak solutions of the hierarchy exist in the same Banach space. If the initial correlation functions are locally perturbed Gibbs correlation functions, then these solutions are defined on an arbitrary time interval.
Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1532-1544
For Gibbs lattice systems characterized by a measurable space at sites of a d-dimensional hypercubic lattice and potential energy with pair complex potential, we formulate conditions that guarantee the convergence of polymer (cluster) expansions. We establish that the Gibbs correlation functions and reduced density matrices of classical and quantum systems of linear oscillators with ternary interaction can be expressed in terms of correlation functions of these systems.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1404–1421
Finite volume grand canonical correlation functions of nonequilibrium systems of d-dimensional Brownian particles, interacting through a regular (long-range) pair potential with integrable first partial derivatives, are expressed in terms of the expectation values of a Gaussian random field. The initial correlation functions coincide with the Gibbs correlation functions corresponding to a more general pair long-range potential. Nonequilibrium Euclidean action is introduced, satisfying a thermodynamic stability property.
Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 691–698
Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy).