## Numerical calculation and the Fermi Pasta Ulam Tsingou paradox

**First experiments with computers and the discovery of chaos **

Around **1947 John von Neumann** developed one of the **first programmable electronic calculators** which raised a lot of interest over at Los Alamos. One of the “young veterans” of the Manhattan Project, **Nicholas Metropolis,** used von Neumann’s machine** to study how to program physics equations in a computer so as to simulate physical phenomena** and predict experimental results. With von Neumann and Stanislaw Ulam, Metropolis introduced **the modern “Monte Carlo” data-processing method** to simulate **stocastic physical processes.** Fermi jumped at the chance to insert his own equations into the computer and worked with Ulam and another computational physicist, **John Pasta,** to study the problem he had come across during his application for the Normal School in 1918. They inserted the **equations of a vibrating string** **into the Los Alamos computer** and simulated its behavior.

Numerical simulations were done by Mary Tsingou. The paper they wrote was the first fundamental contribution to Chaos Theory.

The idea was to study a crystalline solid, portraying it as **a long chain of particles connected by springs.** These springs exerted an elastic force not simply proportional to deformation as is what generally occurs but instead are non-linear. To their great surprise, they discovered the next day that, after thousands of oscillations, the system’s behavior was very different from what intuition had led them to expect. Fermi, in fact, thought that, after many iterations, the system would have shown a behavior in which the influence of the initial conditions would have diminished and the system would have become more or less random.

As they wrote in their report of May, 1955 after Fermi’s death: **“the results show very little, if any, tendency towards equal division of energy between the bodies of system”.** Some years later, Ulam affirmed tha “the results of the calculations done by the MANIAC computer were interesting and rather surprising to Fermi”.

T**he Fermi-Pasta-Ulam-Tsingou paradox** thus signalled the birth of a new field: non-linear physics in which there is a significant dependence on initial conditions.

The problem had fundamental importance **in chaos and** **soliton theories.** The unexpected result also made evident for the first time in history the great potential of **numerical** **computational simulations;** in fact, by many it’s considered **the beginning of the era o**f **computer simulation.**

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## Fermiac

**Enrico Fermi’s analog simulations**

Fermiac is a **small brass chassis** about 30 centimeters long, **invented by Enrico Fermi** and **built by Percy King in 1947.** It’s an **analog computer,** activated by hand and conceived for the study of the change over time of the **number of neutrons present in a nuclear device.** The Fermiac was developed while the supercomputer of the time was being rebuilt and moved from Philadelphia to its final destination in Aberdeen. In the summer of 1947, Enrico Fermi visited the Los Alamos labs and realized that the scientists there were having difficulties because they didn’t have a computer capable of using the Monte Carlo method recently developed by Ulam and Metropolis.

This method consisted of taking random samples to obtain numerical solutions and the use of computers was vital to be able to take the **largest number of samples possible** and get accurate results. **Fermi suggesting building a simple mechanical device** able to show on paper the tracks of hypothetical neutrons as they move through a model of the cells of a reactor or a nuclear weapon. **King built the device and the T-Division** group **at Los Alamos** **used it successfully until 1949** when availability of electronic calculators became common for American scientists.

As you can see from t**he exact replica in the Fermi Museum,** the Fermiac is composed of three parts:

**A Plexiglas platform**that serves to select the direction of the neutrons. The pointer at its center has a pencil lead in the lower part to trace the neutrons’ movements on paper.**A cylinder in the back**that measures the time passed based on the speed of the specific neutron being studied. Two possible energies are available for the neutrons: slow and fast.**A cylinder in the front**that measures the distance covered by the neutron between succeeding collisions based on the speed of the neutron and on the properties of the material it’s passing through. Ten different materials are available for the composition of the nuclear device.

To use the Fermiac, a **two-dimensional scale design** was created of the nuclear device being studied. Then an initial neutron population was established (at Los Alamos, 100 was the usual number) and for each of these a direction, energy, first collision position or possible escape route was determined. These were statistically determined based also on the characteristics of the target material.

It was also necessary to establish **the nature of the collision of each neutron:** elastic impact, inelastic impact or fission (if the material allowed for it) and the distance until the next collision. In case of fission, **the history of all new neutrons generated by the process was traced.** The Fermiac was also used to plot and draw directly on the map of the nuclear device, the trajectory of every neutron plus all those generated by it. The time of each step of the simulation was measured thanks to notches on a click wheel placed on the back cylinder. In this way, scientists could **simulate the behavior of chain reactions in a nuclear device** and, according to whether the neutron population thus generated grew, diminshed or remained stable over time, it was possible to determine if that system was, respectively, **supercritical, undercritical or critical.**