2 Fermi, Pasta, Ulam and Tsingou :
The FPUT experiments on MANIAC

To begin this section, let us quote Ulam :

“Computers were brand-new ; in fact the Los Alamos Maniac was barely finished. The Princeton von Neumann machine had met with technical and engineering difficulties that had prolonged its perfection. [...] As soon as the machines were finished, Fermi, with his great common sense and intuition, recognized immediately their importance for the study of problems in theoretical physics, astrophysics, and classical physics. We discussed this at length and decided to attempt to formulate a problem simple to state, but such that a solution would require lengthy computation which could not be done with pencil and paper or with existing mechanical computers. [...] we found a typical one requiring long-range prediction and long-time behavior of a dynamical system. It was the consideration of an elastic string with two fixed ends, subject not only to the usual elastic force of strain proportional to strain, but having, in addition, a physically correct small non- linear term. The question was to find out how this non-nonlinearity after very many periods of vibrations would gradually alter the wellknown periodic behavior of back and forth oscillation in one mode ; how other modes of the string would become more important ; and how, we thought, the entire motion would ultimately thermalize [...] John Pasta, a recently arrived physicist, assisted us in the task of flow diagramming, programming, and running the problem on the Maniac. Fermi had decided to try to learn how to code the machine by himself. [...] Our problem turned out to have been felicitously chosen. The results were entirely different qualitatively from what even Fermi, with his great knowledge of wave motions, had expected. [1]

More precisely, the model, obtained as the discretization of a partial differential equation model of a string, consists in a one-dimensional array of identical masses coupled to their nearest neighbors by springs, with fixed end-points. When a mass is moved away from its position at rest, it undergoes a force pulling it back to equilibrium. [2]
They considered 16, 32 and 64 masses. To their great surprise, the system did not exhibit “thermalization”, that is, equipartition of energy, but instead a complex quasi-periodic behavior. This was in contradiction with the so-called “ergodic hypothesis” which was assumed to hold true in this case.

A technical report appeared in 1955 [12] but Mary Tsingou (born in 1928) who wrote the algorithm and programmed MANIAC I is not on the list of authors. In a footnote it is written : “We thank Miss Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine.” Notice that Tsingou is not mentioned in Ulam’s quote above (and in fact nowhere in his book). Nowadays, this model is named the FPUT model instead of the FPU model. We refer to [8] for more informations. There are rumors throughout the literature that the MANIAC I was “accidently” left on one night and that the scientists came back after realizing their mistake to find astonishing results. According to Tsingou, the computers were run at night because the numeric iteration computations for long time scales were very slow and the computer was being used for weapons design during the day.
We cannot resist quoting what Pasta says about MANIAC I at a conference held in 1977 :

“The program was of course punched on cards. A DO loop was executed by the operator feeding in the deck of cards over and over again until the loop was completed !”.

Let us notice that there were people who believed that thermal relaxation times for the FPUT system were much too long to be observed during the short integration runs made by Tsingou, in other words they believed that the observed behavior was a transient effect. But Tsingou and Tuck [3] made this conjecture very unlikely by extensive numerical experiments in 1972 [4].