Statistical properties of the periodic Lorentz gas. Multidimensional case

In 1981 Bunimovich and Sinai established the statistical properties of the planar periodic Lorentz gas with finite horizon. Our aim is to extend their theory to the multidimensional Lorentz gas. In that case the Markov partitions of the Bunimovich-Sinai type, the main tool of their theory, are not available. We use a crude approximation to such partitions, which we call Markov sieves. Their construction in many dimensions is essentially different from that in two dimensions; it requires more routine calculations and intricate arguments. We try to avoid technical details and outline the construction of the Markov sieves in mostly qualitative, heuristic terms, hoping to carry out our plan in full detail elsewhere. Modulo that construction, our proofs are conclusive. In the end, we obtain a stretched-exponential bound for the decay of correlations, the central limit theorem, and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite horizon.     

Remarks on the notion of mean free path for a periodic array of spherical obstacles

In this note, we explain in detail how the notion of mean free path is related to the mathematical results obtained in our previous paper.     

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.     

In Search of Alternatives to Higgs Fields

Field equations of the S² sigma model (“the A3 model”) with spontaneously broken Z(2) symmetry are presented for (D+1)-dimensional space–time. The A3 model is an extension of the sine-Gordon equation (SGE) and supports kink-like U(1) charged solitons which are a generalization of neutral solitons of the SGE. The natural question arises — is the A3 model completely integrable in (1+1)-dimensional space–time? The Lorentz-invariant scalar A3 field can be viewed as a promising alternative to the Higgs field.     

三维Minkowski空间中的曲面及其可积性

利用可积系统的思想,借助三维Minkowski空间L3的矩阵模型,研究了L3中具有调和逆平均曲率的类空曲面和洛伦兹调和逆平均曲率类时曲面的可积性及其形变.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

The dirac-schwinger covariance condition in classical field theory

A straightforward derivation of the Dirac-Schwinger covariance condition is given within the framework of classical field theory. The crucial role of the energy continuity equation in the derivation is pointed out. The origin of higher order derivatives of delta function is traced to the presence of higher order derivatives of canonical coordinates and momenta in the energy density functional.     

Derivation of a new class of distributions for the microcanonical ensemble

A new class of distributions for the microcanonical ensemble, which are shown to be stable laws, are derived by applying the central limit theorem to the canonical ensemble. This opens up a whole new host of phenomena that can be treated from a unified thermodynamic point of view. Pressure broadening of line shapes is used as an illustration.1. Work supported, in part, by contributions from the Consiglio Nazionale di Ricerche and the Ministero dell'Università e della Ricerca Scientifica e Tecnologica.2. The interaction parameterC is proportional to the mean square dipole moment. As an order of magnitude of the dipole moment, we can take it as the product of the electric charge and a typical atomic radius for a moderately excited state, which is several times the Bohr radius. Multiplication by the fine structure constant converts the Bohr radius into the Compton wavelength, thereby reducing the magnitude of the interaction by 1/137.     

Generalized Navier-Stokes equations with initial data in local Q-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Navier-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of . This result applies to the usual incompressible Navier-Stokes equations.