A testbench for the nested dipole hypothesis of Kosterlitz and Thouless

We consider the two-dimensional one-component plasma without a background and confined to a half-plane near a metal wall. The particles are also subjected to an external potential acting perpendicular to the wall with an inverse-power-law Boltzmann factor. The model has a known solvable isotherm which exhibits a Kosterlitz-Thouless-type transition from a conductive to an insulator phase as the power law is varied. This allows predictions of theoretical methods of analyzing the Kosterlitz-Thouless transition to be compared with the exact solution. In particular, we calculate the asymptotic density profile by resumming its low-fugacity expansion near the zero-density critical coupling in the insulator phase, and solving a mean-field equation deduced from the first BGY equation. Agreement with the exact solution is obtained. As the former calculation makes essential use of the nested dipole hypothesis of Kosterlitz and Thouless, the validity of this hypothesis is explicitly verified.     

Transient response of a spherical shell in an acoustic medium—Comparison of exact and approximate solutions

The transient response of a spherical shell in an acoustic medium is studied. The exact solution is obtained by expressing the classical spherical wave equation in terms of a residual potential. Approximate solutions are obtained from the use of eight functions; namely, doubly asymptotic approximations, δ-sequence functions, and spherical wave approximations. The advantages and disadvantages of these approximate solutions are pointed out. The third member of the class of the modal doubly asymptotic approximations for a spherical surface and an improved spherical wave approximation are introduced.     

Field theoretical approach to inhomogeneous ionic systems: thermodynamic consistency with the contact theorem,Gibbs adsorption and surface tension

Using a statistical field approach we investigate the structure of an electrolyte solution in contact with a neutral impenetrable wall. The Hamiltonian contains the Coulomb interaction and the ideal entropy. At the level of the quadratic approximation, the Hamiltonian yields the Debye-Hückel theory in the bulk. Analytic expressions of the charge-charge and potential-potential inhomogeneous correlation functions are obtained. Exact asymptotic results for point ion charge correlation functions are obtained and the profile for the fluctuation of the electric potential is calculated. We also consider the term beyond the quadratic expansion of the ideal entropy in the Hamiltonian. With this term a higher order coupling between charge density and number density produces a non-trivial profile for the total ion density. This density profile is consistent with the contact theorem and the related surface tension calculated from the Gibbs adsorption isotherm.     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

Non-linear dynamics of spinodal decomposition

We develop a new technique describing the non linear growth of interfaces. We apply this analytical approach to the one dimensional Cahn-Hilliard equation. The dynamics is captured through a solvability condition performed over a particular family of quasi-static solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the non-linear growth is also well characterized. Numerical simulations are compared in a satisfactory way with the analytical results through three different fitting methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold. Received 16 October 2001 / Received in final form 15 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: josseran@lmm.jussieu.fr RID="b" ID="b"UMR CNRS 7607     

Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions

We formulate the exact non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point of the field-theoretic action, and the effects of counter-ion fluctuations are included by a loop-wise expansion around this saddle point. The Poisson equation is obeyed at each order in this loop expansion. We explicitly give the expansion of the Gibbs potential up to two loops. We then apply our field-theoretic formalism to the case of a single impenetrable wall with counter ions only (in the absence of salt ions). We obtain the fluctuation corrections to the electrostatic potential and the counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. The effective interactions and correlation functions between charged particles close to the charged wall are obtained on the one-loop level. Received 8 February 1999 and Received in final form 15 May 1999     

A general solution to the Schrödinger–Poisson equation for a charged hard wall: Application to potential profile of an AlN/GaN barrier structure

A general, system-independent, formulation of the parabolic Schrödinger–Poisson equation is presented for a charged hard wall in the limit of complete screening by the ground state. It is solved numerically using iteration and asymptotic boundary conditions. The solution gives a simple relation between the band bending and sheet charge density at an interface. Approximative analytical expressions for the potential profile and wave function are developed based on properties of the exact solution. Specific tests of the validity of the assumptions leading to the general solution are made. The assumption of complete screening by the ground state is found be a limitation; however, the general solution provides a fair approximate account of the potential profile when the bulk is doped. The general solution is further used in a simple model for the potential profile of an AlN/GaN barrier structure. The result compares well with the solution of the full Schrödinger–Poisson equation.     

Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

We consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle’s momentum density, a generalized Ohm’s-like equation; and for the particle’s energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann’s entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.     

A geometric wave function for a few interacting bosons in a harmonic trap

We establish a new geometric wave function that combined with a variational principle efficiently describes a system of bosons interacting in a one-dimensional trap. By means of a combination of the exact wave function solution for contact interactions and the asymptotic behavior of the harmonic potential solution we obtain the ground state energy, probability density and profiles of a few boson system in a harmonic trap. We are able to access all regimes, ranging from the strongly attractive to the strongly repulsive one with an original and simple formulation.     

一类广义Canard系统的近似解析解

利用变分迭代理论研究了一类广义Canard系统. 首先引入一组泛函,然后构造了原方程解的迭代关系式,最后得到了问题鸭解的近似解析式. 关键词： 鸭解 解析解 变分迭代     

Variational Principle in the Theory of Partial Distribution Functions

We have constructed thermodynamically consistent theory of correlations in a classical equilibrium system. The theory is based on a variational principle for thermodynamic potential, as functional of correlation functions. Both the thermodynamic potential and correlation functions are determined simultaneously from a uniform variation problem for this functional. We have offered also methods of consecutive approximations system construction for the solution of the obtained variational problem.     

Deuterium isotopic effect in 13C-N.M.R. spectra

We obtain the grand potential and the associated distribution functions for a system of hard rods on a line of finite length. These functions are shown to be related to those of a semi-infinite system bounded by one hard wall. It follows that the contact values of the density and distribution functions at the walls of the finite system are related to those at the wall of the semi-infinite system, and that the force on the walls is the density at contact. We obtain also the density and distribution functions for a system with repeating boundary conditions, which are also expressible in terms of those of the finite and semi-infinite systems.     

Nonlinear asymptotic impedance model for a Helmholtz resonator liner

The usual nonlinear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the nonlinear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including nonlinear effects. The amplitude regime considered is such that when we stay away from the resonance condition, the nonlinear terms are relatively small and the solution obtained is of the linear equation (formed after neglecting the nonlinear terms). Close to the resonance frequency, the nonlinear terms can no longer be neglected and algebraic equations are obtained that describe the corresponding nonlinear impedance. Sample results are presented including a few comparisons with measurements available in the literature. The validity of the model is understood in the near resonance and non-resonance regimes.     

Eigenmodes for electromagnetic waves propagating in a toroidalcavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived     

Renormalized Surface Charge Density for a Strongly Charged Plate in Asymmetric Electrolytes: Exact Asymptotic Expansion in Poisson Boltzmann Theory

The Poisson-Boltzmann equation for a strongly charged plate inside a generic charge-asymmetric electrolyte is solved using the method of asymptotic matching. Both near field and far field asymptotic behaviors of the potential are systematically analyzed. Using these expansions, the renormalized surface charge density is obtained as an asymptotic series in terms of the bare surface charge density.     

A variational treatment of the mixed boundary condition for the equilibrium diffusion equation

We formulate the asymptotic boundary layer analysis which leads to a mixed (Robbin) boundary condition for the equilibrium diffusion equation of radiative transfer. The information required from the nonlinear boundary layer equations to obtain the boundary condition is obtained by deriving and applying an appropriate variational principle. Based upon an examination of certain limiting cases and the good accuracy previously reported in the literature for similar variational treatments, the accuracy of this variational procedure is expected to be quite good over a wide range of conditions.     

Explicit formulae for cutoff values on fibres and couplers

Very simple expressions are derived which give accurate approximations for the cutoff values of all modes of optical fibres and couplers from a knowledge of the refractive-index profile alone. The derivation uses the exact solution of the Laplace equation for the field in the cladding, and approximates the field in the core using Laguerre-Gauss functions. Simple superposition expresses the coupler fields in terms of the fields of the individual fibres.