The Tsingou Recurrence: FPUT Lattice Simulator

👩‍💻 The Hidden Pioneer: Mary Tsingou Menzel (1928–2026)

In the summer of 1953 at Los Alamos National Laboratory, Enrico Fermi, John Pasta, and Stanislaw Ulam posed what seemed like an intuitive physics question: if a 1D crystal lattice with non-linear spring forces is excited in a single fundamental mode, how quickly will thermal equilibrium (equipartition of energy) be reached across all degrees of freedom?

Expectation was that energy would rapidly scatter irreversibly into high-frequency modes (ergodicity). Instead, Mary Tsingou’s code revealed an astonishing mathematical surprise: after energy cascaded into modes 2, 3, and 4, it mysteriously coalesced back into the initial fundamental mode with over 97% recovery. This non-ergodic periodicity became known as Fermi-Pasta-Ulam recurrence, and more justly in recent decades as the Fermi-Pasta-Ulam-Tsingou (FPUT) problem.

Tsingou's code directly inspired Martin Kruskal and Norman Zabusky in 1965 to discover solitons (solitary wave solutions to the Korteweg-de Vries equation) and laid the foundation for modern nonlinear physics and deterministic chaos theory.

📐 Mathematical Formulation

For $N$ masses coupled with non-linear springs and fixed endpoints ($y_0 = y_{N+1} = 0$), the governing equation of motion is:

The normal mode amplitudes $A_k$ and modal energies $E_k$ are evaluated via the discrete sine transform:

• 1952 Mary Tsingou joins Los Alamos as a mathematician and one of the first programmers of MANIAC I.
• 1955 LA-1940 report published posthumously for Fermi; Tsingou credited only in a footnote.
• 2008 Physicists led by Thierry Dauxois publish papers advocating for renaming the phenomenon to FPUT.