On the use of variational methods for solving Boltzmann equations involving non-Hermitian operators

Variational principles yielding upper and lower bounds on transport coefficients can readily be applied to the Boltzmann equation, provided it has the form of a linear, inhomogeneous integrodifferential equation with a Hermitian operator acting on the deviation from equilibrium of the distribution function. In transport problems involving a magnetic field or an alternating electric field, this operator is non-Hermitian. By suitably transforming the transport equation, we show how Variational principles may still give upper and lower bounds. The bounds are used for considering the frequency-dependent conductivity associated with a general scattering operator, and the longitudinal magnetoresistivity in the relaxation time approximation for the scattering operator. Explicit results are presented for (1) the frequency-dependent conductivity of a charged Fermi liquid and (2) the longitudinal magnetoresistivity for a weakly anisotropic Fermi surface.     

An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.     

A numerical study of the classical one-dimensional anisotropic ferromagnet

We present a numerical solution to the equation of motion of the classical onedimensional Heisenberg ferromagnet with two anisotropies in the presence of an applied magnetic field in the small velocities limit. Our results will permit us to obtain upper bounds to the Double sine-Gordon approximation.     

An integral equation approach to the fundamental frequency of vibrating beams

An approximation to the lowest natural frequency of vibrating beams is obtained analytically by applying eigenvalue, eigenfunction theory to the defining integral equation. The method produces successively closer values for both upper and lower bounds to the fundamental frequency. It is found that the second lower bound provides in itself a good approximation to published values and a graph is derived which provides a bound for the error in this approximation without further computation. The application of integral equations to the formulation of mechanical engineering problems is increasing and one aim of the paper is to draw attention to the possibility of obtaining analytical solutions.     

Association of evaluation methods of the effective permittivity of heterogeneous media on the basis of a generalized singular approximation

Various methods for evaluation of the effective permittivity of heterogeneous media, namely, the effective medium approximation (Bruggeman’s approximation), the Maxwell-Garnett approximation, Wiener’s bounds, and the Hashin-Shtrikman variational bounds (for effective static characteristics) are combined on the basis of a generalized singular approximation.     

Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids

This paper focuses on flux-continuous pressure equation approximation for strongly anisotropic media. Previous work on families of flux-continuous schemes for solving the general geometry–permeability tensor pressure equation has focused on single-parameter families. These schemes have been shown to remove the O(1) errors introduced by standard two-point flux reservoir simulation schemes when applied to full-tensor flow approximation. Improved convergence of the schemes has also been established for specific quadrature points. However these schemes have conditional M-matrices depending on the strength of the off-diagonal tensor coefficients. When applied to cases involving full-tensors arising from strongly anisotropic media, the point-wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution.New double-family flux-continuous locally conservative schemes are presented for the general geometry–permeability tensor pressure equation. The new double-family formulation is shown to expand on the current single-parameter range of existing conditional M-matrix schemes. The conditional M-matrix bounds on a double-family formulation are identified for both quadrilateral and triangle cell grids. A quasi-positive QM-matrix analysis is presented that classifies the behaviour of the new schemes with respect to double-family quadrature in regions beyond the M-matrix bounds. The extension to double-family quadrature is shown to be beneficial, resulting in novel optimal anisotropic quadrature schemes. The new methods are applied to strongly anisotropic full-tensor field problems and yield results with sharp resolution, with only minor or practically zero spurious oscillations.     

Anharmonic phonons and structural phase transitions

Exact results for the order parameter and the meansquare displacement as functions of temperature are given for a quantum interacting phonon Hamiltonian with quartic anharmonicities. Upper bounds for the transition temperature are also derived. Approximate theories including the mean field approximation, the random phase approximation (or quasiharmonic approximation) and the self consistent approach (using Blume-Hubbard scheme) are compared with our exact results. The mean field approximation for the meansquare displacement is found to violate our bounds.The classical value is shown to form a lower bound for the kinetic energy. Upper bounds for the kinetic energy are obtained showing the region of temperature in which the use of the high temperature expansion of the fluctuation-dissipation theorem is justified. Comparison of the Hamiltonian and our results with electron-paramagnetic-resonance measurements are discussed.     

New upper bounds for the magnetization in ferromagnetic one-component systems

A collection of upper bounds on the magnetization in ferromagnetic one-component spin systems is derived from generalizations of the mean-field approximation. In particular, it contains sequences of bounds that converge to the true magnetization. The spin systems in question satisfy the GHS-inequality.Institut für Theoretische Physik, Universität Zürich, Zürich, Switzerland     

Renormalization group in the local potential approximation

In the local potential approximation, renormalization group equations reduce to a semilinear parabolic partial differential equation. We derive this equation and show the relation with the hierarchical model. We construct a family of non-trivial fixed points u _2n ^* ,n=2,3,4,..., which have the form ofn-well potentials and exist in the ranges of dimensions 2<d<d _n=2+2/(n?1). Asd←d _n u _2n ^* tends to zero. For the Wilson fixed pointu*₄, we give bounds on critical exponents. In the case of dipole gas in this approximation we show that no non-trivial fixed points exist.     

Bounds on natural frequencies of composite circular membranes: Integral equation methods

This paper is concerned with finding upper and lower bounds for the natural frequencies of vibration of a circular membrane with stepped radial density. Such problems, involving discontinuous coefficients in the differential equation, may be treated by using classical variational methods. However, it is shown here that eigenvalue estimation techniques based on an integral equation formulation are more effective. Integral equation methods provide more accurate upper bounds, for a comparable amount of effort, and supply improvable lower bounds as well.     

Upper bounds to the susceptibility and the spin pair correlation function of the Ising ferromagnet

Improved upper bounds are presented to the susceptibility and the spin pair correlation function given in the Bethe approximation, for the Ising ferromagnet under zero external field above the critical temperature of the Bethe approximation.     

Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Bénard convection

The upper bound on the minimum number of modes for a qualitatively correct Galerkin approximation to the two-dimensionl Bénard convection is discussed. For that case and the ext case, selection rules are introduced insuring that the approximation satisfies the same energy balance conditions as the exact solution. These rules together with the proof of boundedness of the Galerkin approximation help establish bounds on the amplitudes of the modes. Finally, results of some mathematical (numerical) experiments carried out in this context are discussed.     

Entropy production estimates for Boltzmann equations with physically realistic collision kernels

We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL ¹ convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL ¹ neighborhood of equilibrium.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

Poissonian Behavior of Ising Spin Systems in an External Field

We apply the Stein–Chen method for Poisson approximation to spin-half Ising-type models in positive external field which satisfy the FKG inequality. In particular, we show that, provided the density of minus spins is low and can be expanded as a convergent power series in the activity (fugacity) variable, the distribution of minus spins is approximately Poisson. The error of the approximation is inversely proportional to the number of lattice sites (we obtain upper and lower bounds on the total variation distance between the exact distribution and its Poisson approximation). We illustrate these results by application to specific models, including the mean-field and nearest neighbor ferromagnetic Ising models.     

The role of the long-wave approximation in describing the propagation of weakly non-linear waves under the influence of both dispersion and dissipation

Quantitative comparison between an elementary stationary shock wave solution to the Korteweg-de Vries-Burgers equation and a corresponding elementary solution to Whitham's equation where no long-wave approximation is made shows that this approximation introduces negligible error.     

A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations

In this paper, we extend the earlier work [M. Barrault, Y. Maday, N. C. Nguyen, A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris, Série I 339 (2004) 667–672; M.A. Grepl, Y. Maday, N.C. Nguyen, A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, M2AN Math. Model. Numer. Anal. 41 (3) (2007) 575–605.] to provide a posteriori error estimation and basis adaptivity for reduced-basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence. The essential components are (i) rapidly convergent reduced-basis approximations – (Galerkin) projection onto a space spanned by N global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space; (ii) stable and inexpensive interpolation procedures – methods which allow us to replace nonaffine parameter functions with a coefficient-function expansion as a sum of M products of parameter-dependent coefficients and parameter-independent functions; (iii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest; (iv) optimal basis construction – processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set; and (v) offline/online computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the online stage – in which, given a new parameter value, we calculate the output of interest and associated error bounds – depends only on N, M, and the affine parametric complexity of the problem; the method is thus ideally suited for repeated and reliable evaluation of input–output relationships in the many-query or real-time contexts.     

Master Equation and Fokker–Planck Equation: Comparison of Entropy and of Rate Constants

A usual approximation of the master equation is provided by the Fokker–Planck equation. For chemical systems with one species, we prove generally that the prediction of the rate constant of the metastable state given by the Master equation and the Fokker–Planck approximation differ exponentially with respect to the size of the system. We show that this is related to the fact that the entropy of the metastable state is not described correctly by the Fokker–Planck equation. We prove that the rate given by the Fokker–Planck equation overestimates that rate given by the Master equation.     

Replica Bounds for Optimization Problems and Diluted Spin Systems

In this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by Guerra (cond-mat/0205123) to prove that the replica method generates variational bounds for disordered systems. We analyze a family of models that includes the Viana–Bray model, the diluted p-spin model or random XOR-SAT problem, and the random K-SAT problem, showing that the replica/cavity method, at the various levels of approximation, provides systematic schemes to obtain lower bounds of the free-energy at all temperatures and of the ground state energy. In the case of K-SAT and XOR-SAT it thus gives upper bounds of the satisfiability threshold. Our analysis underlines deep connections with the cavity method which are not evident in the long range case.     

THE TOTAL DERIVATIVE OPERATOR AND ITS INFLUENCE OVER THE QCD SUM RULES FOR BARYON MASS

We investigate two-point function of baryon operator including the total derivative operator. The upper bounds of baryon mass are improved by introducing the total derivative operator, and we explained further the reason why the improvemenfs for the mass bounds are obtained by means of the zero width approximation. The analysis for baryon A(1405) is made.