djvu The Fermi–Pasta–Ulam–Tsingou Paradox: The Geometry of Trojan Coherence

The Fermi–Pasta–Ulam–Tsingou Paradox and the Geometry of Trojan Coherence by Doug Doucette offers a structural reinterpretation of one of nonlinear science’s most enduring puzzles. Since Fermi, Pasta, Ulam, and Tsingou’s 1955 numerical experiment, the FPUT paradox has been summarized as a weakly nonlinear oscillator chain that refuses to thermalize. Initialized with energy in low-frequency normal modes, the system instead returns close to its initial state and exhibits long-lived metastability before eventual equipartition. This book argues that the surprise is not the presence of nonlinearity but the fact that nonlinear coupling initially fails to produce effective global transport.

The central claim is that recurrent and metastable FPUT regimes satisfy the admission criteria of the Trojan framework—a local Hamiltonian architecture abstracted from the stability of Trojan asteroids at the triangular Lagrange points. In this framework, long-lived coherence arises from a finite-order normal-form structure in which dominant actions are preserved while transport enters only through higher-order remainders, resonant channels, or controlled leakage. The book demonstrates that the FPUT chain, near low-mode recurrent states and q-breather periodic orbits, exhibits precisely this architecture: a coherent modal backbone, persistent spectral separation or finite-order nonresonance, near-integrable organization, suppressed action drift, and structural exit through resonance.

Two independent pillars support the argument. The optical recurrence experiment of Pierangeli et al. (2018) shows that FPUT-type recurrence can be experimentally prepared, parameter-controlled, and lost exactly when integrability is degraded—evidence for an integrable skeleton governing high-dimensional nonlinear evolution over finite windows. Inside the discrete FPUT Hamiltonian itself, the q-breather theory of Karve, Rose, and Campbell (2024) supplies the internal skeleton: localized periodic orbits in normal-mode space whose Floquet stability and resonant bifurcations mark the protected regime and its controlled breakdown. Resonances of the form ωq≈pωq′ \omega_q \approx p \omega_{q'} ωq≈pωq′ produce composite periodic orbits that enlarge the coherent backbone rather than immediately destroying it.

The thin/thick chaos distinction supplies the diagnostic language. Local instability or modal exchange is not failure; failure occurs only when instability acquires transport reach to the modal failure region. Spectral entropy, recurrence fidelity, resonance distance, leakage flux, and transport-graph connectivity become precise, falsifiable measures. The book converts recurrence from a paradox into a model case of protected nonlinear coherence and thermalization into resonant thickening—the organized enlargement of transport networks once protective divisors collapse.

Written for researchers in nonlinear dynamics, Hamiltonian systems, and complexity science, the book combines rigorous finite-time estimates, normal-form arguments, and explicit falsification criteria with a unifying geometric vision. It shows that stability is not the absence of perturbation but the geometric containment of transport, and that the path from recurrence to equilibrium is a sequence of identifiable structural exits rather than a collapse from order into disorder.