Lawrence R. Mead

Maximum entropy in the problem of moments

The maximum‐entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem. In important special cases, such as the Hausdorff moment problem, we establish necessary and sufficient conditions for the existence of a maximum‐entropy solution and examine the convergence of the resulting sequence of approximations. A number of explicit illustrations are presented. In addition to some elementary examples, we analyze the maximum‐entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems. We also briefly indicate possible applications to the Lee–Yang theory of Ising models, to the summation of divergent series, and so on. The general conclusion is that maximum entropy provides a valuable approximation scheme, a serious competitor of traditional Pade‐like procedures.

Approximate solution of Fredholm integral equations by the maximum‐entropy method

An approximate means of solving Fredholm integral equations by the maximum‐entropy method is developed. The Fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum‐entropy methods has been successfully implemented in a previous paper by Mead and Papanicolaou [L. R. Mead and N. Papanicolaou, J. Math. Phys. 25, 2404 (1984)]. Several explicit examples are given of approximate maximum‐entropy solutions of Fredholm integral equations of the first and second kinds and of the Wiener–Hopf type. Both the weaknesses and strengths of the method are discussed.

Continuous Hahn polynomials and the Heisenberg algebra

Continuous Hahn polynomials have surfaced in a number of somewhat obscure physical applications. For example, they have emerged in the description of two‐photon processes in hydrogen, hard‐hexagon statistical mechanical models, and Clebsch–Gordan expansions for unitary representations of the Lorentz group SO(3,1). In this paper it is shown that there is a simple and elegant way to construct these polynomials using the Heisenberg algebra.

An analytical example of renormalization in two‐dimensional quantum mechanics

As a concrete example of the idea of renormalization, quantum mechanical scattering of particles by a two‐dimensional delta‐function potential is considered. The renormalization of the scattering cross section is carried out exactly and analytically. The calculation, free from obscuring mathematical details required for realistic field theories, may aid in making the idea of renormalization more accessible.

A new formula describing the scaffold structure of spiral galaxies

We describe a new formula capable of quantitatively characterizing the Hubble sequence of spiral galaxies including grand design and barred spirals. Special shapes such as ring galaxies with inward and outward arms are also described by the analytic continuation of the same formula. The formula is r(φ) =A/log [B tan (φ/2N)]. This function intrinsically generates a bar in a continuous, fixed relationship relative to an arm of arbitrary winding sweep. A is simply a scale parameter while B, together with N, determines the spiral pitch. Roughly, greater N results in tighter winding. Greater B results in greater arm sweep and smaller bar/bulge, while smaller B fits larger bar/bulge with a sharper bar/arm junction. Thus B controls the ‘bar/bulge-to-arm’ size, while N controls the tightness much like the Hubble scheme. The formula can be recast in a form dependent only on a unique point of turnover angle of pitch – essentially a one-parameter fit, aside from a scalefactor. The recast formula is remarkable and unique in that a single parameter can define a spiral shape with either constant or variable pitch capable of tightly fitting Hubble types from grand design spirals to late-type large barred galaxies. We compare the correlation of our pitch parameter to Hubble type with that of the traditional logarithmic spiral for 21 well-shaped galaxies. The pitch parameter of our formula produces a very tight correlation with ideal Hubble type suggesting it is a good discriminator compared to logarithmic pitch, which shows poor correlation here similar to previous works. Representative examples of fitted galaxies are shown.

Maximum entropy summation of divergent perturbation series

In this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients. The perturbation expansion for the ground‐state energy E(g) of the octic oscillator defined by H=p2/2+x2/2+gx8 is a series of the form E(g)∼ 1/2 +∑(−1)n+1 Angn. This series is terribly divergent because for large n the perturbation coefficients An grow like (3n)!. This growth is so rapid that the solution to the moment problem is not unique and ordinary Pade summation of the divergent series fails. A completely different kind of procedure based on the principle of maximum entropy for reconstructing the function E(g) from its perturbation coefficients is presented. Very good numerical results are obtained.

Discrete time quantum mechanics

Abstract This paper summarizes a research program that has been underway for a decade. The objective is to find a fast and accurate scheme for solving quantum problems which does not involve a Monte Carlo algorithm. We use an alternative strategy based on the method of finite elements. We are able to formulate fully consistent quantum-mechanical systems directly on a lattice in terms of operator difference equations. One advantage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated. Furthermore, the scheme provides an easy way in which to obtain the energy levels of the theory numerically. A generalized version of this discretization scheme can be applied to quantum field theory problems. The difficulties normally associated with fermion doubling are eliminated. Also, one can incorporate local gauge invariance in the finite-element formulation. Results for some field theory models are summarized. In particular, we review the calculation of the anomaly in two-dimensional quantum electrodynamics (the Schwinger model). Finally, we discuss nonabelian gauge theories.

Lyapunov exponents and the natural invariant density determination of chaotic maps: an iterative maximum entropy ansatz

We apply the maximum entropy principle to construct the natural invariant density and the Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique, that is based on the solution of the Hausdorff moment problem via maximizing Shannon entropy, we estimate the invariant density and the Lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments. The accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available. Furthermore, we also consider a very complex example for which no exact analytical result for the invariant density is available. A comparison of our results to those available in the literature is also discussed.

Maximum‐entropy approach to classical hard‐sphere and hard‐disk equations of state

The maximum‐entropy procedure for extrapolating series is applied to sum the known virial series for the hard‐disk and hard‐sphere fluids. This procedure is found to produce numerical results more accurate than Pade approximants, and it agrees with Monte Carlo simulations of these systems. The maximum‐entropy results extrapolate the series smoothly to densities arbitrarily near to maximum close packing, and predict no singularities whatsoever.

Error Bounds in Maximum Entropy Approximations

A useful technique in underdetermined inverse problems is that of maximum entropy. A simple error bound for averages over a distribution approximated by the maximum entropy method in the case of the undetermined Hausdorff moment problem was devised. Under the conditions specified, the error bound for averages over such an approximate distribution can be very tight. Numerical examples to illustrate are presented.