# Volume 57, № 6, 2005

### On the Drift-Diffusion Model for a Two-Band Quantum Fluid at Zero Temperature

Ali G., Frosali G., Manzini Ch.

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 723–730

By using a scale transformation, we obtain hydrodynamic equations in the quasiclassical approximation from the two-band Schrodinger equation.

### Kinetic Equations and Integrable Hamiltonian Systems

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 731–741

A survey of interrelations between kinetic equations and integrable systems is presented. We discuss common origin of special classes of solutions of the Boltzmann kinetic equation for Maxwellian particles and special solutions for integrable evolution equations. The thermodynamic limit and soliton kinetic equation for the integrable Korteweg-de Vries equation are considered. The existence of decaying and degenerate dispersion laws and the appearance of additional integrals of motion for the interacting waves is discussed.

### Different Approaches for Multiband Transport in Semiconductors

Borgioli G., Frosali G., Modugno M., Morandi O.

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 742–748

We compare the well-known Kane model with a new multiband envelope function model, which presents many advantages with respect to the first one.

### Quantum-Classical Wigner-Liouville Equation

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 749–756

We consider a quantum system that is partitioned into a subsystem and a bath. Starting from the Wigner transform of the von Neumann equation for the quantum-mechanical density matrix of the entire system, the quantum-classical Wigner-Liouville equation is obtained in the limit where the masses *M* of the bath particles are large as compared with the masses *m* of the subsystem particles. The structure of this equation is discussed and it is shown how the abstract operator form of the quantum-classical Liouville equation is obtained by taking the inverse Wigner transform on the subsystem. Solutions in terms of classical trajectory segments and quantum transition or momentum jumps are described.

### Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 757–769

We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particles becomes stochastic. Under the assumption that the empirical velocity distribution of the small particles is governed by a probability density ?, the mean-field force can be represented as the negative gradient of a scaled version of ?. The stochastic motion is described by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition, the so-called correlation length. In this set-up, the separate motion of each particle is a classical Brownian motion (Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is not Gaussian. The motion of two particles is analyzed in detail and a diffusion equation is deduced for the difference in the positions of the two particles. The diffusion coefficient in the latter equation is spatially dependent, which allows us to determine regions of attraction and repulsion of the two particles by computing the probability fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get attracted to one another if the distance between them is smaller than a critical small parameter. In our case, this parameter is shown to be proportional to the aforementioned correlation length.

### Stochastic Semigroups and Coagulation Equations

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 770–777

A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation process in terms of stochastic semigroups.

### Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 778–789

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.

### A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 790–817

We present simple proofs of several basic facts of the global regime (the existence and the form of the non-random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices, whose probability law involves the Gaussian distribution.
The main difference with previous proofs is the systematic use of the Poincare - Nash inequality,
allowing us to obtain the *O*(*n* - 2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter.

### Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions

Caraffini G. L., Petrina D. Ya.

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 818–839

Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions.

### On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 840–851

In this paper, we study the lower and upper bounds for solutions of the limit problem for the plane vacuum diode in the magnetic field in the statement by N. Ben Abdallah, P. Degond, and F. Mehats. This problem was finally set by physicists in the late 1980s and was extensively studied by numerous mathematicians in the 1990s.

### Equations of Electrodynamics in a Hydrodynamic Medium with Regard for Nonequilibrium Fluctuations

Sokolovs'kyi O. I., Stupka A. A.

Ukr. Mat. Zh. - 2005. - 57, № 6. - pp. 852–864

We investigate the kinetics of the electromagnetic field in a hydrodynamic medium containing charged particles. A closed system of equations of the electromagnetic field and hydrodynamic equations taking into account dissipative processes is constructed. To describe the electromagnetic field, we use its average values and the corresponding binary correlation functions as new independent variables. The reverse influence of the field on the medium is studied. The investigation is based on quasirelativistic quantum electrodynamics in the Hamilton gauge and on the Bogolyubov method of reduced description of nonequilibrium processes.