Exact Integral Equation Research Articles

Some integral equations occuring in the classical theory of fluids

The problem of calculating the pair distribution function in classical statistical mechanics is approached through an expression of this function in a new topological parameter. The expansion is shown to lead to an infinite set of integral equations, one for each order of the parameter. The first of these equations is examined in detail and is used to derive a new exact integral equation for the pair distribution function which incorporates the series rearrangements of Meeron and Rodemich. A brief discussion of numerical calculations is given.

A New Approach to the Theory of Classical Fluids. III

A classical system enclosed in a finite volume and exerted by external forces is treated in quite a general way. The main results, which may be applicable to solids as well as to fluids, are as follows. Exact integral equations are found for the one- and two-particle distribution functions. Some thermodynamic functions are expressed in terms of these distribution functions. It is shown that there exist variational principles saying that the grand partition function is to be maximum with respect to the variations in the one- and two-particle distribution functions. Variational principles are found also for the Helmholtz free energy. It is suggested that Mayer's theory of condensation may in fact give the end point of the metastable gaseous state. It is pointed out that the hyper-netted chain approximation, proposed previously by one of the present auhors, has a meaning in solids as well as in fluids.

Statistical thermodynamics of plasmas

A systematic method for the computation, with arbitrary accuracy, of the thermodynamic functions of a gaseous plasma is developed from first principles. The theory of the grand partition function is used to derive a set of exact integral equations, which determine the electronic and ionic distribution functions. An approximate solution of these equations yields results similar to those of the Debye-Hueckel theory of electrolytes. More exact solutions are developed and used to compute the equation of state and other thermodynamic functions for plasmas.

Theory of Classical Fluids and the Convolution Approximation (Note on Papers by Tohru Morita)

A method employed by Morita for the approximate calculation of the free energy and pair distribution function of classical fluids is seen to be identical with the nodal expansion method. The limit of the nodal expansion sequence results in an approximate integral equation for the potential of average force. This convolution approximation is seen to be consistent with the Ornstein-Zernike theory of fluids, and thus provides a convenient means for investigating the behavior of fluids near the critical point. The convolution approximation results also from neglecting the non-convolutory terms in an exact integral equation for the pair distribution function.

Quantum-mechanical many-particle treatment of sound propagation

A quantum-mechanical treatment of the problem of sound propagation is discussed. The relevant features of a field theoretic many-particle formalism applicable to the problem are summarized. The special formal relations which apply in the treatment of transport properties and sound propagation are presented. Exact and approximate integral equations for the density correlation function or sound propagator are discussed. In developing approximate equations, the time-invariance properties that result from the fact that many relevant macroscopic operators commute with the Hamiltonian were taken into account. It is shown that the mechanism for sound propagation can vary from system to system. It was observed than in "perfect" gases sound propagation is related to single- particle damping, a second-order effect. A second-order Boltzmann-like equation was obtained by the general approximation technique and solved for the perfect- gas case. (M.C.G.)

A New Approach to the Theory of Classical Fluids. I

An exact integral equation is found for the pair distribution function. The integral equation is of somewhat different nature from the usual ones known in the theory of classical fluids, in the point that it involves an infinite series. The Helmholtz free energy is expressed as a series expansion which may be more rapidly convergent than the usual one. It is shown that the integral equation can be derived also by means of a variational principle from the expression for the free energy. It is pointed out that the theory of classical fluids may be constructed with the knowledge of the pair distribution function alone, even if a form of the pair interaction potential is not known.

Exact treatment of particle correlation functions and free energy

The exact integral equation for the pair distribution functions, published recently by J. M. J. van Leeuwen, J. Groeneveld and J. de Boer 1) is readily derived by previously published methods 2) 3). The relation of the new equation to the Ornstein-Zernike formula for the direct correlation function is discussed, and the importance of this relation for treatment of phase transitions is noted. Exact formulas for the Helmholtz free energy are presented. The convolution approximation, obtained by neglecting prototype graphs, and its relation to the Kirkwood superposition approximation and to the nodal expansion sequance are discussed.

Exact integral equation solutions and synthesis for a large class of optimum time variable linear filters

This paper presents the exact integral equation solution and synthesis for a large class of optimum time variable linear filters characterizing many physical problems. The signal random process is expressed in nonstationary Fourier series ensemble form, with certain statistical information assumed about its coefficients. The noise perturbation is represented by a damped exponential-cosine autocorrelation function, which is of major importance in fields of physics and engineering, such as radar, meteorology, and automatic control. For any finite operating period from 0 to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> , the optimum time variable weighting function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(\tau, t)</tex> is found to be of a separable form, consisting of functions of parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> multiplied by functions of parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> , plus two delta function contributions at the beginning and end. Valid synthesis designs are developed for such separable weighting functions. Asymptotic synthesis techniques are formulated which cover special situations of long-time or short-time operation. The results are applied to two examples of practical interest.

Problem of Ghost States in Field Theories

It is shown that the ghost problem in the Lee model cannot be circumvented by using a renormalized integral equation as a basis instead of the Hamiltonian formulation. This equation (the Low equation) possesses no solutions for coupling constants larger than the critical value, and an infinite family of solutions for coupling constants smaller than the critical value. The Low equation for the Lee model is also mathematically equivalent to an exact integral equation for the boson propagator in relativistic field theory, and permits one therefore to describe the properties of possible ghosts in a relativistic theory in terms of already known properties of Lee-model ghosts. There exists an interesting symmetry between this integral equation and its solution, both of which have the form of dispersion relations. The excellent experimental tests of quantum electrodynamics provide no evidence that the theory is free of the ghost problem.

A new interpretation of the integral equation formulation of cylindrical antennas

The validity of the conventional integral equation formulation of cylindrical antennas is often criticized because of an approximation of the kernel of the integral equation. If the antenna is of the form of a cylindrical tube there is an exact integral equation formulation. By means of the variational technique the integral equation for the cylindrical tube can be solved approximately. When the length of the tube is large compared to its diameters the input impedance of the structure is found to be the same as if one had used the approximate integral equation at the very beginning of the analysis. This new procedure therefore clarifies one main ambiguity involved in the early works of the theory of cylindrical antennas based upon the approximate integral equation. It also enables us to treat the problem of relatively thick cylindrical antennas.

Covariant integral equations for Heisenberg operators

ABSTRACTSets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews &amp; Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.

Two-Group Perturbation Theory in Neutron Transport Theory

This paper describes the application of perturbation theory to two-velocity-group neutron problems. We consider the exact integral equations satisfied by the neutron densities. The treatment is, therefore, not restricted to systems whose dimensions are large compared with thè relevant mean free paths, and is more general than that reported by Glasstone and Edlund. The latter describe a theory, due to Wigner, in which it is assumed that the neutron densities satisfy the diffusion equations, i.e. the theory is applicable only to large systems.

Ionosphere reflection coefficients by variational technique

One of the important problems of ionosphere work is to predict the nature of the electromagnetic wave reflected by the ionosphere when a wave of known form is incident upon it. In the present paper, this problem is treated by considering a plane ionosphere model for which the electron density and earth's magnetic field are continuous but otherwise arbitrary functions of the height. It is shown that if a plane wave impinges on such a model at an arbitrary angle of incidence, then four complex reflection coefficients suffice to characterize the reflected wave as to its intensity, phase, and polarization. Thus the problem becomes one of the calculating of these four numbers. The attack on this problem is based on an adaptation of a method due to the physicist, J. Schwinger, wherein, with the aid of an exact integral equation for the electric field in the ionosphere, one derives variational formulas for the four reflection coefficients. These formulas may then be used either to calculate the coefficients numerically or to obtain approximate expressions for them in terms of the various physical parameters of the problem. Finally, as a consequence of a certain symmetry property of the variational formulas, a reciprocity theorem for ionospheric propagation is deduced.