Extracts from Mary Tsingou's papers

We present below extracts from two Los Alamos Reports and two journal papers which involved Mary Tsingou Menzel.

E Fermi, J Pasta, S Ulam and M Tsingou, Studies of nonlinear problems I, Los Alamos Scientific Laboratory (May 1955).

1.1. Abstract.

A one-dimensional dynamical system of 64 particles with forces between neighbours containing nonlinear terms has been studied on the Los Alamos computer MANIAC I. The nonlinear terms considered are quadratic, cubic, and broken linear types. The results are analysed into Fourier components and plotted as a function of time.

The results show very little, if any, tendency toward equipartition of energy among the degrees of freedom.

The last few examples were calculated in 1955. After the untimely death of Professor E Fermi in November, 1954, the calculations were continued in Los Alamos.

1.2. Extracts from the report.

This report is intended to be the first one of a series dealing with the behaviour of certain nonlinear physical systems where the nonlinearity is introduced as a perturbation to a primarily linear problem. The behaviour of the systems is to be studied for times which are long compared to the characteristic periods of the corresponding linear problems.

The problems in question do not seem to admit of analytic solutions in closed form, and heuristic work was performed numerically on a fast electronic computing machine (MANIAC I at Los Alamos). We thank Miss Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine. The ergodic behaviour of such systems was studied with the primary aim of establishing, experimentally, the rate of approach to the equipartition of energy among the various degrees of freedom of the system. Several problems will be considered in order of increasing complexity. This paper is devoted to the first one only.
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The solution to the corresponding linear problem is a periodic vibration of the string. If the initial position of the string is, say, a single sine wave, the string will oscillate in this mode indefinitely. Starting with the string in a simple configuration, for example in the first mode (or in other problems, starting with a combination of a few low modes), the purpose of our computations was to see how, due to nonlinear forces perturbing the periodic linear solution, the string would assume more and more complicated shapes, and, for t tending to infinity, would get into states where all the Fourier modes acquire increasing importance.
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Let us say here that the results of our computations show features which were, from the beginning, surprising to us. Instead of a gradual, continuous flow of energy from the first mode to the higher modes, all of the problems show an entirely different behaviour. Starting in one problem with a quadratic force and a pure sine wave as the initial position of the string, we indeed observe initially a gradual increase of energy in the higher modes as predicted (e. g., by Rayleigh in an infinitesimal analysis). Mode 2 starts increasing first, followed by mode 3, and so on. Later on, however, this gradual sharing of energy among successive modes ceases. Instead, it is one or the other mode that predominates. For example, mode 2 decides, as it were, to increase rather rapidly at the cost of all other modes and becomes predominant. At one time, it has more energy than all the others put together! Then mode 3 undertakes this role. It is only the first few modes which exchange energy among themselves and they do this in a rather regular fashion. Finally, at a later time mode 1 comes back to within one per cent of its initial value so that the system seems to be almost periodic. All our problems have at least this one feature in common. Instead of gradual increase of all the higher modes, the energy is exchanged, essentially, among only a certain few. It is, therefore, very hard to observe the rate of "thermalization" or mixing in our problem, and this was the initial purpose of the calculation.
A Blair, N Metropolis, J von Neumann, A H Taub and M Tsingou, A study of a numerical solution to a two-dimensional hydrodynamical problem, Los Alamos Scientific Laboratory (28 September 1957).

2.1. Abstract.

This report contains results obtained on Maniac I when that machine was used to solve numerically a set of difference equations representing the two-dimensional motion of two incompressible fluids subject only to gravitational and hydrodynamical forces. Eulerian coordinates were used. The report contains three appendices taken from handwritten notes of John von Neumann and from a letter by him to Stanislaw Ulam. John von Neumann suggested the study and these notes outline the mathematical approach used on the problem.

2.2. Preface.

Sometime ago John von Neumann discussed with one of us the general question of the numerical integration of the hydrodynamical equation in two-space dimensions. In the one-dimensional model, the method of Lagrange has been practical. This stems from the fact that the mass points that are used in the mathematical description maintain their spatial ordering, i.e., nearest neighbours initially remain so throughout the motion. In contrast, the nearest-neighbour relation in a two-dimensional array may, by virtue of the extra degree of freedom, be altered in the course of time. Thus the Lagrangian scheme implies a reshuffling of the mass points to determine the current nearest neighbours. This sorting is a complication and may add considerably to the computing time.

In contradistinction, the Eulerian scheme is not concerned with nearest-neighbour considerations, but material boundary specifications suffer from inexactness. It was not clear how this impreciseness may vary with time, or whether material interfaces could be treated separately in a tractable manner. To this end von Neumann suggested the study of a relatively simple two-space dimensional flow of an inviscid incompressible fluid in a gravitational field. A system of two liquids with different densities was to be the first step. With the characteristic brilliance and clarity of style, he soon produced a detailed discussion of the mathematical attack in the form of handwritten notes and a letter to Stanislaw Ulam; these are given as Appendices I, II, and III. The conversion of these methods into a computational form has gone through several phases and has occupied several of us over a much longer period.

He never saw the completion of this, one of his last efforts.
A Blair, N Metropolis, J von Neumann, A H Taub and M Tsingou, A study of a numerical solution to a two-dimensional hydrodynamical problem, Math. Tables Aids Comput.13 (1959), 145-184.

3.1. Introduction.

The purpose of this paper is to report the results obtained on Maniac I when that machine was used to solve numerically a set of difference equations approximating the equations of two-dimensional motion of an incompressible fluid in Eulerian coordinates. More precisely, the problem was concerned with the two-dimensional motion of two incompressible fluids subject only to gravitational and hydrodynamical forces which at time $t = 0$ were distributed as illustrated in Fig. 1.

This problem was discussed and formulated for machine computation by John von Neumann and others. His own original draft of a discussion of the differential and difference equations is given in Appendix I, and an iteration scheme for solving systems of linear equations is given in Appendix II. In the main body of this paper we shall outline the derivation of the equations employed by the computer and refer to these appendices for detailed discussions concerning them where necessary. Some of von Neumann's difference equations were modified in the course of the work. The reasons for these modifications and their nature will be enlarged upon in the course of the discussion.
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3.2. Computation Time.

The results reported in this paper were run on Maniac I with $I = 15$ and $J = 38$ , that is, with 624 lattice points (518 interior points). The program required 300 words (600 orders of code exclusive of print routines and exclusive of orders necessary for moving information to and from the magnetic drum because of the limited electrostatic storage capacity of Maniac I). About 3750 words of dynamic storage were required. The time required for running one time cycle of the program on Maniac I was 18 seconds for each iteration cycle plus 100 seconds for all the rest of the program. The iteration process converges so as to give accuracy in an additional decimal place every 10 minutes, so that one time cycle requires about an hour for six-place accuracy (about 200 iterations) or a half hour for three-place accuracy (about 100 iterations). About 40 per cent of this time, however, is used in transfers to and from the magnetic drum, so that this much time is to be charged to the fact that a 4000-word problem was being run on a machine with 1000-word random-access memory capacity.
J L Tuck and M T Menzel, The superperiod of the nonlinear weighted string (FPU) problem, Advances in Mathematics 9 (1972), 399-407.

4.1. Extracts from the paper.

The Maniac I computer (N Metropolis) started working at Los Alamos early in 1952. E Fermi, who was visiting the Laboratory, J R Pasta, and S M Ulam entertained themselves by considering what new problems it opened up for study. One such, in classical fluid theory in its simplest approximation, considers a linear array of atoms linked by nonlinear forces. The results of the first calculations, which were coded by one of us (Mary Tsingou Menzel), were so surprising that the investigators were enticed into a study of nonlinear systems generally. The first system to be examined consisted of a one-dimensional array of mass points linked by light Hookean springs made nonlinear by addition of a small term ...
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As Fermi observed, nobody at that time could state, without calculating it out first, what such a string, started out with a pure sinusoidal displacement, would look like after a few thousand oscillations at its fundamental frequency, although statistical mechanics makes general statements about the diffusion of energy and equipartition among modes, and Poincaré tells us that all accessible points in phase space will be approached arbitrarily closely.
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At first the problem behaved as expected and energy appeared and grew in harmonics of the fundamental, but soon the process became strangely selective and not diffusion-like. At 25 oscillations, the string configuration began to retrace its steps, passing through previous complications in reverse order. By about 50 oscillations, the complications were unscrambled and the string was back - all but a discrepancy of a few percent - to its half sinusoid starting configuration. Other force laws, cubic and discontinuous, showed complications differing in detail but with similar recurrences. This behaviour was not at all according to statistical-mechanical expectations. Nor was it Poincaré-like ... Fermi became quite excited and thought that something new and important might be at hand. This happened in 1953; Fermi's untimely death occurred in November 1954. In the following year, J Pasta, who had left the laboratory, returned to make a few more computer runs and assemble the material which appeared as Los Alamos Report LA-1940 (Physics) dated November 2, 1955.

This attracted the attention that might have been expected for such a quiet publication in a noisy world. Few, if any, references to it appeared in the literature over the next 5 years, but the results circulated as a piece of mathematical curiosa by word of mouth.

Last Updated June 2024