Boltzmann-Type Equation in Anelastic Case: A Convergence Result

In this Letter, a convergence result for the BBGKY hierarchy to a Boltzmann-like equation, in the case of an Anelastic collision, is shown. Boltzmann-like equations are often used to model dissipative dynamical systems such as granular media. This convergence result aims to make a contribution towards a mathematical foundation to these applications.     

Fourfold Symmetric Solutions to the Ginzburg Landau Equation for d-Wave Superconductors

We find and investigate the structure of solutions to the Ginzburg Landau equation for a high temperature superconductor with tetragonal symmetry. This is done near an isolated, rotationally symmetric d-wave vortex state with its core at the origin defined on all of \mathbbR²{\mathbb{R}^2}. We prove that the solution’s s-wave component nucleates near the vortex core for temperatures just below the d-wave critical temperature. We further show that this causes the rotational symmetry to break and that the solution develops a fourfold symmetry with respect to a rotation by an angle of \fracp2{\frac{\pi}{2}}.     

Spray Forming: Controlling the Atomization Result with Regard to Particle Properties

Spray forming is a new production process for manufacturing semi‐finished metal products. The procedure combines the liquid metal atomization with the compaction event of the generated metal droplets on a substrate. During spray forming, the heat emission of the molten metal particles is one of the most important transfer operations defining the thermal conditions in the spray and deposit. Definite thermal conditions throughout the manufacturing process lead to fine equi‐axed grain structures and therewith to characteristic and desired material properties. Thus, in particular, the disintegration of the molten metal occurring during spray forming is an essential step.     

Numerical Solution of the Quantum Landau Equation for Dense Plasmas

The numerical solution of the quantum Landau kinetic equation for a dense electron gas is considered. Being one of the most simple kinetic equations, it still retains essential physical features of a correlated many-particle system, such as selfconsistent static screening and Pauli blocking, and, at the same time, it is a good test case for the efficiency of numerical methods. Two schemes for the evaluation of the scattering rates and the collision integral are discussed. To illustrate our results, we present time-dependent calculations i) for the relaxation of a nonequilibrium electron distribution function and ii) for the stopping of fast electrons.     

From the von Neumann Equation to the Quantum Boltzmann Equation II: Identifying the Born Series

In a previous paper [Ca1], the author studied a low density limit in the periodic von Neumann equation with potential, modified by a damping term. The model studied in [Ca1], considered in dimensions d3, is deterministic. It describes the quantum dynamics of an electron in a periodic box (actually on a torus) containing one obstacle, when the electron additionally interacts with, say, an external bath of photons. The periodicity condition may be replaced by a Dirichlet boundary condition as well. In the appropriate low density asymptotics, followed by the limit where the damping vanishes, the author proved in [Ca1] that the above system is described in the limit by a linear, space homogeneous, Boltzmann equation, with a cross-section given as an explicit power series expansion in the potential. The present paper continues the above study in that it identifies the cross-section previously obtained in [Ca1] as the usual Born series of quantum scattering theory, which is the physically expected result. Hence we establish that a von Neumann equation converges, in the appropriate low density scaling, towards a linear Boltzmann equation with cross-section given by the full Born series expansion: we do not restrict ourselves to a weak coupling limit, where only the first term of the Born series would be obtained (Fermi's Golden Rule).     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

A Non-Existence Result for Supersonic Travelling Waves in the Gross-Pitaevskii Equation

We prove the non-existence of non-constant travelling waves of finite energy and of speed in the Gross-Pitaevskii equation in dimension N2.     

From semiclassical transport to quantum Hall effect under low-field Landau quantization

The crossover from the semiclassical transport to the quantum Hall effect is studied by examining a two-dimensional electron system in an AlGaAs/GaAs heterostructure. By probing the magneto-oscillations, it is shown that the semiclassical Shubnikov-de Haas (SdH) formulation can be valid even when the minima of the longitudinal resistivity approach zero. The extension of the applicable range of the SdH theory could be due to the damping effects resulting from disorder and temperature. Moreover, we observed plateau-plateau transition-like behavior with such an extension. From our study, it is important to include positive magnetoresistance to refine the SdH theory.     

From Rosochatius System to KdV Equation

The Rosochatius system on the sphere, an integrable mechanical system discovered in the nineteenth century, is investigated in a suitably chosen framework with the sphere as an invariant set, to avoid the complicated constraint presentations. Higher order Rosochatius flows are defined and straightened out in the Jacobi variety of the associated hyperelliptic curve. A relation is found between these flows and the KdV equation, whose finite genus solution is calculated in the context of the Rosoehatius hierarchy.     

From the Von-Neumann Equation to the Quantum Boltzmann Equation in a Deterministic Framework

In this paper, we investigate the rigorous convergence of the Density Matrix Equation (or Quantum Liouville Equation) towards the Quantum Boltzmann Equation (or Pauli Master Equation). We start from the Density Matrix Equation posed on a cubic box of size L with periodic boundary conditions, describing the quantum motion of a particle in the box subject to an external potential V. The physics motivates the introduction of a damping term acting on the off-diagonal part of the density matrix, with a characteristic damping time ^–1. Then, the convergence can be proved by letting successively L tend to infinity and to zero. The proof relies heavily on a lemma which allows to control some oscillatory integrals posed in large dimensional spaces. The present paper improves a previous announcement [CD].     

From Rosochatius System to KdV Equation

The Rosochatius system on the sphere, an integrable mechanical system discovered in the nineteenth century, is investigated in a suitably chosen framework with the sphere as an invariant set, to avoid the complicated constraint presentations. Higher order Rosochatius flows are defined and straightened out in the Jacobi variety of the associated hyperelliptic curve. A relation is found between these flows and the KdV equation, whose finitegenus solution is calculated in the context of the Rosochatius hierarchy.     

Diffusive Mixing of Stable States in the Ginzburg–Landau Equation

The Ginzburg–Landau equation on the real line has spatially periodic steady states of the form , with and . For , , we construct solutions which converge for all t>0 to the limiting pattern as . These solutions are stable with respect to sufficiently small perturbations, and behave asymptotically in time like , where is uniquely determined by the boundary conditions . This extends a previous result of [BrK92] by removing the assumption that should be close to zero. The existence of the limiting profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term. Received: 22 January 1998 / Accepted: 19 April 1998     

Particle Spectrum Implied by the Dirac Equation

There is a process that starts from the Lagrangian of a set of field equations and leads to a spectrum of particle states. The process is applied in this article to a Lagrangian for the Dirac equation. It leads to a differential equation with solutions that describe particles with definite mass, angular momentum J, charge, and isotopic spin I, having I = J. There is no suggestion of strangeness. The theory is in rough agreement with the masses of many of the 0⁺ (0^–+) and 0⁺ (0⁺⁺) mesons and with the masses of the nonstrange 1/2 (1/2⁺) and 3/2 (3/2⁺) baryons.     

From Discreteness to Continuity： Dislocation Equation for Two-Dimensional Triangular Lattice

A systematic method from the discreteness to the continuity is presented for the dislocation equation of the triangular lattice. A modification of the Peierls equation has been derived strictly. The modified equation includes the higher order corrections of the discrete effect which are important for the core structure of dislocation. It is observed that the modified equation possesses a universal form which is model-independent except the factors. The factors, which depend on the detail of the model, are related to the derivatives of the kernel at its zero point in the wave-vector space. The results open a way to deal with the complicated models because what one needs to do is to investigate the behaviour near the zero point of the kernel in the wave-vector space instead of calculating the kernel completely.     

Landau Damping for the Linearized Vlasov Poisson Equation in a Weakly Collisional Regime

In this paper, we consider the linearized Vlasov–Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter \({\varepsilon }\) in front of the collision operator which will tend to 0. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker–Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter \({\varepsilon }\) as it goes to 0.     

Entropy Production: From Open Volume-Preserving to Dissipative Systems

We generalize Gaspard's method for computing the -entropy production rate in Hamiltonian systems to dissipative systems with attractors considered earlier by Tél, Vollmer, and Breymann. This approach leads to a natural definition of a coarse-grained Gibbs entropy which is extensive, and which can be expressed in terms of the SRB measures and volumes of the coarse-graining sets which cover the attractor. One can also study the entropy and entropy production as functions of the degree of resolution of the coarse-graining process, and examine the limit as the coarse-graining size approaches zero. We show that this definition of the Gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems. We apply the method to the case of a two-dimensional map, based upon a model considered by Vollmer, Tél, and Breymann, that is a deterministic version of a biased-random walk. We treat both volume-preserving and dissipative versions of the basic map, and make a comparison between the two cases. We discuss the -entropy production rate as a function of the size of the coarse-graining cells for these biased-random walks and, for an open system with flux boundary conditions, show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases. This work describes in some detail the relation between the results of Gaspard, those of of Tél, Vollmer, and Breymann, and those of Ruelle, on entropy production in various systems described by Anosov or Anosov-like maps.