The Tsingou Recurrence: Interactive FPUT Lattice & Computational Physics Simulator

The Tsingou Recurrence: Interactive FPUT Lattice

In 1955, physicist Mary Tsingou Menzel programmed the MANIAC I computer to simulate a nonlinear string, expecting thermal chaos. Instead, the system surprised Enrico Fermi, John Pasta, and Stanislaw Ulam by returning nearly 100% of its energy back into mode 1. Explore her original symplectic numerical experiment below.

Coupled Nonlinear Lattice Displacement (y_i) STATE: MODE 1 DOMINANT (RECURRING)

Click & drag beads to pluck • Symplectic Velocity-Verlet Integrator

PRESETS:

Fourier Harmonic Energy Spectrum E_k (k = 1 ... 8) Mode 1: 94.2% • Mode 2: 3.8%

Simulation Telemetry 32 Masses

Recurrence Cycle

1 (96%)

Mode 1 Energy

94.2%

Higher Modes (2-8)

5.8%

Step Count

Lattice & Nonlinearity Parameters

Quadratic Nonlinearity (α) 0.25

F = -k·Δy - α·(Δy)² (Tsingou original formulation)

Cubic Nonlinearity (β) 0.00

F_cubic = -β·(Δy)³ (Symmetric FPU-β model)

Mass Count (N) 32

Simulation Time Step (dt) 0.05

The Footnote Attribution

In the official 1955 Los Alamos report LA-1940, Enrico Fermi, John Pasta, and Stanislaw Ulam were listed as primary authors. Mary Tsingou wrote the assembly and numerical code for the MANIAC I computer that performed every calculation, yet she was recognized only in a footnote:

"We wish to express our thanks to Miss Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine."

In 2008, physicists Thierry Dauxois and colleagues officially advocated renaming the phenomenon the Fermi-Pasta-Ulam-Tsingou (FPUT) problem.

The Paradox That Birth of Solitons and Chaos

Before Tsingou's simulation in 1953-1955, the prevailing ergodic hypothesis predicted that any nonlinear coupling would quickly spread energy equally among all degrees of freedom (equipartition / thermalization). Fermi, Pasta, and Ulam believed they were observing an intuitive thermal relaxation test.

Instead, after hundreds of thousands of steps on MANIAC I, Tsingou noticed the energy flowed cleanly into mode 2, mode 3, mode 4, and then concentrated almost entirely back into mode 1. This lack of thermalization became known as the FPU paradox, directly spurring Zabusky and Kruskal (1965) to discover solitons and launching modern computational nonlinear science.

Symplectic MANIAC I Implementation

The simulator implements the exact 1D discretized wave equation with fixed boundary conditions ($y_0 = y_N = 0$). Equations of motion:

d²y_i/dt² = (y_{i+1} + y_{i-1} - 2y_i) + α[(y_{i+1}-y_i)² - (y_i-y_{i-1})²] + β[(y_{i+1}-y_i)³ - (y_i-y_{i-1})³]

Using modern symplectic Velocity-Verlet, Hamiltonian energy is conserved to within 0.01% over thousands of cycles, exactly mirroring the digital precision pioneered by Tsingou on 1,024 vacuum tubes.