Evaporation of droplets in the two-dimensional Ginzburg–Landau equation

We consider the problem of coarsening in two dimensions for the real (scalar) Ginzburg–Landau equation. This equation has exactly two stable stationary solutions, the constant functions +1 and −1. We assume most of the initial condition is in the “−1” phase with islands of “+1” phase. We use invariant manifold techniques to prove that the boundary of a circular island moves according to Allen–Cahn curvature motion law. We give a criterion for non-interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces.     

Landau damping in a bounded magnetized plasma column

The damping decrement of Landau damping and the effect of thermal velocity on the frequency spectrum of a propagating wave in a bounded plasma column are investigated.The magnetized plasma column partially filling a cylindrical metallic tube is considered to be collisionless and non-degenerate.The Landau damping is due to the thermal motion of charge carriers and appears whenever the phase velocity of the plasma waves exceeds the thermal velocity of carriers.The analysis is based on a self-consistent kinetic theory and the solutions of the wave equation in a cylindrical plasma waveguide are presented using Vlasov and Maxwell equations.The hybrid mode dispersion equation for the cylindrical plasma waveguide is obtained through the application of appropriate boundary conditions to the plasma-vacuum interface.The dependence of Landau damping on plasma parameters and the effects of the metallic tube boundary on the dispersion characteristics of plasma and waveguide modes are investigated in detail through numerical calculations.     

A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers

We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green–Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.     

Fixed-phase diffusion Monte Carlo study of the quantum-Hall effect on the Haldane sphere

A generalized diffusion Monte Carlo method for solving the many-body Schrödinger equation on curved manifolds is introduced and used to perform a ‘fixed-phase’ simulation of the integer and fractional quantum-Hall effect on the Haldane sphere. The effect of Landau level mixing on the and energy gaps and the relative stability of spin-polarized and spin-reversed quasielectron excitations are studied using the new method.     

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 variational principle for the Landau equation

The Landau equation describing the collective modes of an infinite many-fermion system is cast into a form which displays explicitly its structure as a homogeneous Fredholm equation of the second kind with non-symmetric kernel. A variational principle appropriate to this equation is constructed by noting the analogy to the integral form of the Schrödinger equation. The Landau equation is also re-expressed in terms of two coupled equations with symmetric kernels.     

On the modification of a model of minority charge-carrier diffusion in semiconductor materials based on the use of recursive trigonometric functions and the estimation of the stability of solutions for the modified model

A modified model of the diffusion of minority charge carriers generated by a broad electron beam in semiconductor materials is proposed. New methods for the approximation of step functions are used, and an estimate of the stability of solutions to the diffusion equation is obtained for this model.     

The Landau Equation in a Periodic Box

 The Landau equation, which was proposed by Landau in 1936, is a fundamental equation to describe collisions among charged particles interacting with their Coulombic force. In this article, global in time classical solutions near Maxwellians are constructed for the Landau equation in a periodic box. Our result also covers a class of generalized Landau equations, which describes grazing collisions in a dilute gas. Received: 4 February 2002 / Accepted: 10 July 2002 Published online: 29 October 2002     

Localization and the integer-quantized Hall effect: Diffusion constant for high Landau levels and white noise potential

The diffusion constant and the diagonal conductivity for non-interacting electrons in a two-dimensional, disordered system are studied. A homogeneous magnetic field perpendicular to the electron system is assumed. For weak short-range random potentials and high fields the Landau quantum numbern can be used as expansion parameter. In the limit of high Landau levels the system shows metallic behaviour. Corrections for finiten decrease the conductivity and indicate localized states in the whole energy band. A breakdown of the expansion and stronger localization are observed only for the lowest Landau levels if the typical experimental length scale of the quantized Hall effect is used.     

Second-phase nucleation on an edge dislocation

A model for nucleation of second phase at or around a dislocation in a crystalline solid is considered. The model employs the Ginzburg–Landau theory of phase transitions comprising the sextic term in the order parameter (η⁶) in the Landau free energy. The ground state solution of the linearised time-independent Ginzburg–Landau equation is derived, through which the spatial variation of the order parameter is delineated. Moreover, a generic phase diagram indicating tricritical behaviour near and away from the dislocation is depicted. The relation between classical nucleation theory and the Ginzburg–Landau approach is discussed, for which the critical formation energy of the nucleus is related to the maximum of the Landau potential energy. A numerical example illustrating the application of the model to the case of nucleation of hydrides in zirconium alloys is provided.     

Splitting Scheme Integration of the Two-and-One-Half-Dimensional Vlasov Equation

A numerical splitting scheme technique is applied to the integration of the Vlasov equation in two-space and three velocity dimensions. Calculations are performed for a cold, stationary ion, mobile electron plasma in the presence of a constant external magnetic field. Results for Landau damping and the two-stream instability are presented in both one and two velocity dimensions. The Landau damping results show a plateau structure in the electron velocity distribution function in both the parallel and perpendicular directions formed as a result of the damping of a high initial electric field propagating at 45° to the parallel velocity axis. The two-stream velocity distribution function shows pitch angle scattering and velocity diffusion for an electron beam initially propagating at 45° to the magnetic field direction through a low-temperature equal density background. The two-stream distribution function evolves to a stable quasi-isotropic plateau structure similar to that found in auroral sounding rocket data.     

Probing microscopic chaotic dynamics by observing macroscopic transport processes

We derive the Kramers equation, namely, the Fokker-Planck equation for an oscillator, from a completely deterministic picture. The oscillator is coupled to a “booster”, i.e., a deterministic system in a fully chaotic state, wherein diffusion is derived from the sensitive dependence of chaos on initial conditions and friction is a consequence of the linear response of the booster to the action exerted on it by the oscillator. To deal with the Hamiltonian nature of the system of interest and of its coupling to the booster, we extend the earlier theoretical derivation of macroscopic transport coefficients from deterministic dynamics. We show that the frequency of the oscillator can be tuned to the microscopic frequencies of the booster without affecting the canonical nature of the “macroscopic” statistics. The theoretical predictions are supported by numerical simulations.     

Calculation of electric transport close to a Lifshitz transition in a high magnetic field

The effects of the Landau quantization and interactions on a Lifshitz transition are studied. The Landau quantization leads to a quasi-one-dimensional behavior for the direction parallel to the field. The repulsive Coulomb interactions give rise to a gas of strongly correlated carriers. Consequently, in the ground state, an electron pocket is emptied in a discontinuous fashion as a function of the chemical potential or magnetic field. This discontinuity is gradually smeared by temperature, in agreement with experiments for CeIn₃. We further calculate the conductivity and the Hall conductivity in the presence of nonmagnetic impurities, the Landau quantization and interactions.     

Saturation spectroscopy of electronic states in a magnetic field in InAs/AlxGa1−xSb single quantum wells

We have carried out saturation spectroscopy of cyclotron resonance in a semiconducting InAs/Al_0.5Ga_0.5Sb single quantum well using the UCSB free electron laser and have extracted an effective Landau level lifetime using an n-level rate equation model. The effective lifetime shows strong oscillations (>an order of magnitude) with frequency. Minima are shifted to higher frequencies than those given by the simple parabolic magnetophonon resonance condition due to large nonparabolicity in the InAs conduction band. We have also used this technique to investigate the origins of two lines: the X-line and cyclotron resonance in a “semimetallic” InAs/Al_0.1Ga_0.9Sb single quantum-well structure. Results show that the two lines are of different origin.     

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

We consider the solutions lying on the global attractor of the two-dimensional Navier–Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg–Landau equation, the Kuramoto–Sivashinsky equation, and reaction–diffusion equations with analytic nonlinearities.     

Dissipative solitons with energy and matter flows: Fundamental building blocks for the world of living organisms

We consider a combined model of dissipative solitons that are generated due to the balance between gain and loss of energy as well as to the balance between input and output of matter. The system is governed by the generic complex Ginzburg–Landau equation, which is coupled to a common reaction–diffusion (RD) system. Such a composite dynamical system may describe nerve pulses with a significant part of electromagnetic energy involved. We present examples of such composite dissipative solitons and analyse their internal balances between energy and matter generation and dissipation.     

中子星中简并电子气体的临界磁化

中子星内部的电子处于高度简并或完全简并的状态,电子磁矩(包括内禀磁矩和朗道反磁矩)的取向不是随机的,而是呈现出极强的磁化行为.考虑了磁化后的磁诱导方程要改写,改写后的方程添加了新的磁场生成项,更重要的改变是等效磁扩散系数变小了(顺磁情况),在临界情况(等效扩散系数等于零),磁场在磁生成项的作用下增加直到抑制机理出现,朗道反磁矩就是在这个时候变得越来越重要.磁场增加的最终结果使中子星局域磁场成为振荡的,对外看来有可能成为磁星. 关键词： 中子星 简并 磁化     

On the vacuum structure in the Coulomb and Landau gauges

Vacuum structure in the SU(N) Coulomb and Landau gauges is studied by using the methods of harmonic maps. A systematic way for solving the Gribov vacuum copy equation is presented and many examples are discussed in both the Coulomb and Landau gauges as applications of the method. Finally, the physical interpretation of Gribov ambiguities is shortly reviewed from a topological point of view.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.