Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

Rückbezüge des Mathematikunterrichts und der Methodik des Mathematikunterrichts in der DDR auf historische Vorausentwicklungen

The author discusses at some examples exemplarily, how the pedagogy of mathematics education in the GDR from curricular and educational/psychological knowledge from the time before 1945 has developed scientifically.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

Reaction-diffusion problems in electrolysis

Supported in part by Schweizerischer Nationalfonds (20-25263.88) and European Community (Science Plan, Project Evolutionary Systems)     

Exact Scaling Functions for One-Dimensional Stationary KPZ Growth

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann–Hilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore⁽¹⁾ obtained from the mode coupling approximation.     

Measurement of the sputter yield after mild ion erosion of a pristine Cu(001) surface

Using the STM technique we have determined the sputter yield on a pristine Cu(001) surface after mild (fluence less than 0.044 ions per surface atom) bombardment of the pristine surface with 800 eV Ar⁺ions at normal incidence. The experiments have been performed at substrate temperatures ranging from 200 to 350 K. Making use of the positional correlation of adatoms and surface vacancies, at 200 K and 325 K, we concluded that about 1/3 of the surface adatoms originate from interstitials arriving at the surface and they give a direct indication of the buried bulk vacancies. A careful analysis of the different areas for surface vacancies and adatom then allowed a quantitative evaluation of the sputter yield at 1.2 Cu atoms per 800 eV Ar⁺ ion.     

Identification of the Isomeric Dideutero p-Benzoquinones by 13C-H and 13C-D Coupling Constants

2,5- and 2,6-dideuterobenzoquinone-1,4 were prepared by a direct catalytic (NaBD₄/PdCl₂) exchange of bromine in the corresponding dibromo derivatives and the subsequent oxidation with Ag₂O. The structure of the products was unambiguously established by proton-carbon and deuteron-carbon splitting pattern of their NMR spectra.     

The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics

For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f-function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.The resulting phonon Boltzmann equation has been hardly studied on a rigorous level. As one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal Wigner functions. For three phonon processes such a result requires extra conditions on the dispersion law. We also outline the reasoning leading to Fourier’s law for heat conduction.     

Theoretical results concerning the influence of non‐random‐field defects

We shall focus on extended defect systems and review their critical behavior. Primarily, with two aims, one of which is to understand phase transitions and how to derive effective dimension of extended defects with various structures, and the other is to propose a new research-method for defect systems, we let extended defects grow on a triangular lattice with frustration in a similar fashion to diffusion-limited aggregation, and discuss the situation. The existence of phase transitions, phase diagram, effective defect dimension, etc. will be shown. Furthermore, we shall summarize theoretical studies of extended defect systems on phase diagrams, critical behavior, tricritical behavior, and crossover behavior as static properties, and on nonconserved systems and conserved systems as dynamic properties.