The Fermi Pasta Ulam (FPU) Problem and The Birth of Nonlinear Science

In 1953, Enrico Fermi, John Pasta, and Stan Ulam initiated a series of computer studies aimed at exploring how simple, multi-degree of freedom nonlinear mechanical systems obeying reversible deterministic dynamics evolve in time to an equilibrium state describable by statistical mechanics. Their expectation was that this would occur by mixing behavior among the many linear modes. Their intention was then to study more complex nonlinear systems, with the hope of modeling turbulence computationally.

The results of this first study of the so-called Fermi-Pasta-Ulam (FPU) problem, which were published in 1955 and characterized by Fermi as a “little discovery, ” showed instead of the expected mixing of linear modes a striking series of (near) recurrences of the initial state and no evidence of equipartition. This work heralded the beginning of both computational physics and (modern) nonlinear science. In particular, the work marked the first systematic study of a nonlinear system by digital computers (“experimental mathematics”) and led directly to the discovery of “solitons,” as well as to deep insights into deterministic chaos and statistical mechanics.

In this talk, I will review the original FPU problem and trace several distinct lines of research that arose from it. Specifically, I will show how a continuum approximation to the original discrete system led to the discovery of “solitions” and how recent treatments of the FPU and related spatially extended discrete systems reveal the presence of “Intrinsic Localized Modes” (ILMs)” and of “q-breathers.”

I will then describe briefly the basic mechanism that allows the existence of ILMs, discuss some of their essential features, and illustrate a few of the wide range of physical systems in which they have recently been observed. I will show how “q-breathers” can give a plausible quantitative explanation for the recurrence phenomenon observed by behavior by FPU and how these results can be reconciled with mixing, equipartition, and statistical mechanics.

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