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Project 10: - The Fermi–Pasta–Ulam–Tsingou Problem

Published online by Cambridge University Press:  01 February 2024

Pawel Scharoch
Affiliation:
Wrocław University of Science and Technology
Maciej P. Polak
Affiliation:
University of Wisconsin, Madison
Radosław Szymon
Affiliation:
Wrocław University of Science and Technology
Katarzyna Holodnik-Malecka
Affiliation:
Wrocław University of Science and Technology
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Summary

This chapter exploresthe Fermi–Pasta–Ulam–Tsingou (FPUT) problem in the context of a one-dimensional chain of interacting point masses. We start by modelling the system as a chain of masses interacting through a force dependent on their relative displacements. Next, we simplify this system to harmonic oscillators under a linear force dependency, further developing it to describe a wave-like behaviour. The chapter discusses dispersion relations and the impact of boundary conditions leading to discretisation of allowed wave modes. A non-linear, second-order interaction is then included, complicating the system’s dynamics and necessitating the use of numerical methods for its solution. We then track the system’s evolution in a multidimensional phase space, leading to observations of seemingly chaotic motion with emerging periodicity. Energy conservation and its flow through the system are crucial aspects of the analysis. A detailed numerical procedure is provided, involving solution of initial value problems, mode projections, and energy computations to explore the complex behaviour inherent in the FPUT problem.

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Publisher: Cambridge University Press
Print publication year: 2024

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