Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x ^2m+2, m=1,2,..., and of the Riesz spatial fractional derivative of order , 12, are studied analytically and numerically. We show that for 1<2, the stationary distribution functions have power-law asymptotic approximations decreasing as x ^–(+2m+1) for large values of the argument. We also show that these distributions are bimodal.     

Lévy Flights in a Steep Potential Well

Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67:010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U(x)∝|x| ^c with c>2. Apart from the bifurcation into bimodality, we find the interesting result that for c>4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient trimodal distribution of the Lévy flight. These properties of Lévy flights in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process.     

On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems

This paper is devoted to the asymptotic analysis of functions depending on a small parameter characterizing the microinhomogeneous structure of the domain on which the functions are defined. We derive the Friedrichs inequality for these functions and prove the convergence of solutions to corresponding problems posed in a domain perforated aperiodically along the boundary. Moreover, we use numerical simulation to illustrate the results. The work of the first author was partially supported by RFBR. The work of the first and the second authors was partially supported by the program “Leading Scientific Schools” (NSh-1698.2008.1).     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

Negative viscosity for Rossby wave and drift wave turbulence

The possible occurrence of a “negative viscosity effect” is studied for Rossby wave and drift wave turbulence. It is assumed that (i) the space and time scales of the wave field are much smaller than the scales of the mean field, and (ii) the small-scale field is sufficiently weak, stationary, and maintained by an external source. Such a formulation is fruitful for studying the effects (characterized by the effective viscosity) of smaller-scale motions upon larger-scale ones. The criteria of large-scale instability due to the negative effective viscosity are derived for the coherent wave motions as well as for small-scale isotropic wave turbulence. Zh. éksp. Teor. Fiz. 113, 646–663 (February 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Percolation of the boundary layer of a newtonian fluid through a perforated obstacle

We study the behavior of an inhomogeneous conducting fluid percolating through a porous obstacle. For a family of boundary value problems with micro-inhomogeneities on the boundary and nontrivial internal microstructure we construct the homogenized problem and prove the convergence of solutions to the initial problem to the solution of the homogenized problem. Bibliography: 7 titles. Illustrations: 2 figures.     

Neutron lifetime measurement on setups with gravitational trap

Currently, the best accuracy of neutron lifetime measurements has been attained in the experiment with a gravitational trap for ultracold neutrons (UCNs), performed at the Petersburg Nuclear Physics Institute (PNPI); the measured lifetime was 878.5 ± 0.8 s. A new setup with a big gravitational trap has been designed to continue the methods and approaches used in the previous experiment. It is planned to reduce the measurement error to 0.2 s, i.e., improve the existing accuracy by a factor of 4. The spectrometer was designed at PNPI and installed on the PF2/MAM beam at the Institute Laue–Langevin. Test experiments have been performed.