Langevin equation for the extended Rayleigh model with an asymmetric bath

In this paper a one-dimensional model of two infinite gases separated by a movable heavy piston is considered. The non-linear Langevin equation for the motion of the piston is derived from first principles for the case when the thermodynamic parameters and/or the molecular masses of gas particles on left and right sides of the piston are different. Microscopic expressions involving time correlation functions of the force between bath particles and the piston are obtained for all parameters appearing in the non-linear Langevin equation. It is demonstrated that the equation has stationary solutions corresponding to directional fluctuation-induced drift in the absence of systematic forces. In the case of ideal gases interacting with the piston via a quadratic repulsive potential, the model is exactly solvable and explicit expressions for the kinetic coefficients in the non-linear Langevin equation are derived. The transient solution of the non-linear Langevin equation is analyzed perturbatively and it is demonstrated that previously obtained results for systems with the hard-wall interaction are recovered.Comment: 10 pages. To appear in Phys. Rev.

Evolution of collision numbers for a chaotic gas dynamics

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time-integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger number of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.Comment: 4 pages, published versio

Evolution of microstructure and microhardness of dispersion-hardened V-Cr-Zr-W alloy during deformation by torsion under pressure

Results of the study of the microstructural evolution and microhardness changes of dispersion-hardened V–Cr–Zr–W alloy under severe deformation during torsion on Bridgman anvils are presented. Typical structural states and mechanisms of their formation are revealed for basic evolution stages as well as appropriate microhardness values are determined. It was shown that at true logarithmic strain values (e) in the range 0.7 ≤ e 2, the anisotropic submicrocrystalline structure is observed and the formation of two-level nanostructural states was found within grains. In the strain range (e) from 3 to 6.6, submicrocrystal sizes hardly change, but changes of two-level nanostructural state parameters are observed: the nanofragment size decreases and values of elastic curvature of the crystal lattice increases

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

Microstructure and mechanical properties of heat-resistant 12 % Cr ferritic-martensitic steel EK-181 after thermomechanical treatment

The effect of high-temperature thermomechanical treatment (TMT) with the deformation in the austenitic region on the features of microstructure, phase transformations and mechanical properties of low-activation 12% Cr ferritic-martensitic steel EK-181 is investigated. It is established, that directly after thermomechanical treatment (without tempering) the sizes and density of V(CN) particles are comparable with those after a traditional heat treatment (air quenching and tempering at 720°C, 3 h), where these particles are formed only during tempering. It causes the increasing of the yield strength of the steel up to ≈1450 MPa at room temperature and up to ≈430 MPa at the test temperature T = 650°C. The potential of microstructure modification by this treatment aimed at improving heat resistance of steel is discussed

Influence of temperature on microstructure parameters and microhardness of dispersion-hardened V–Cr–Zr–W alloy after deformation by torsion under pressure

Study of microstructure transformation and microhardness changes of dispersion-hardened V–Cr–Zr–W alloy after severe plastic deformation by torsion on Bridgman anvils and subsequent heat treatments was conducted. Basic stages of relaxation processes were revealed: at 800°C recovery processes take place and primary recrystallization begins; at 900°C primary recrystallization intensifies; in range of 950–1050°C collective recrystallization processes activate; at 1200°C secondary recrystallization starts. Microhardness measurement and comparison of its values with structural states features were conducted. Strengthening mechanisms and their contribution at various stages of defect substructure relaxation are discussed. It is shown that increase of thermal stability of V–Cr–Zr–W alloy microstructure is a consequence of the formation of high density of thermally stable Zr (O–N–C)-based nanoparticles

Microstructure and mechanical properties of V–Cr–Zr alloy with carbide and oxide strengthening

A comparative study of the effectiveness of carbide and oxide types of strengthening of V–Cr–Zr alloy was carried out by means of a comprehensive certification of structural-phase state parameters and measuring the mechanical properties characteristics. It has been shown that the use of chemical-heat treatment contributes to a significant increase in the thermal stability of the microstructure and mechanical properties of V–Cr–Zr alloy in comparison with carbide strengthening under the conditions of thermomechanical treatment. A controlled increase in the volume fraction of fine particles based on ZrO2, along with an increase in the concentration of oxygen in the solid solution, leads to a decrease in the rate of oxides coagulation and an increase in the thermal stability of high disperse heterophase structure. These effects contribute to the retention of high defect structural states with nonzero values of crystal lattice curvature even after high-temperature (0.67 Tmelt) anneals. The high efficiency of dispersion and substructural strengthening is a consequence of blocking dislocation slip by fine particles stabilized by oxygen in a solid solution

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page