Recursive behavior in a diatomic FPUT lattice

Technical Summary: Recursive Behavior in a Diatomic FPUT Lattice

Problem Statement
The paper investigates the long-time dynamics and energy recurrence phenomena in a diatomic Fermi–Pasta–Ulam–Tsingou (FPUT) lattice with cubic anharmonic potential. While the classical FPUT recurrence in monatomic chains is a well-established phenomenon driven by nonlinear interactions, the authors seek to understand how the presence of two distinct mass types (light and heavy) alters these dynamics. Specifically, they address whether the diatomic dispersion relation, which features both acoustic and optical branches, enables new types of resonant interactions and recurrence mechanisms that are impossible in monatomic systems.

Methodology
The study employs a combination of analytical derivation and numerical simulation:

1. Model Formulation: The authors define a diatomic chain with alternating masses $m_1$ and $m_2$ and derive the Hamiltonian, including linear and cubic nonlinear terms. They perform a transformation to diagonalize the linear operator, projecting the system onto optical ($+$) and acoustic ($-$) normal modes.
2. Resonance Analysis: Theoretical conditions for three-wave resonant interactions are derived. The authors specifically analyze the "acoustic-acoustic-optical" combination, determining the specific mass ratios required for the resonance condition $\omega_{-,k_1} + \omega_{-,k_2} = \omega_{+,k_3}$ to be satisfied.
3. Numerical Simulations: Using a symplectic integrator (velocity Verlet algorithm), the authors simulate the full diatomic FPUT lattice. They test two initial conditions: (a) exciting the fundamental acoustic mode to observe classical FPUT recurrence, and (b) exciting a single optical mode (or two acoustic modes) to test for resonance-induced recurrence.
4. Reduced Modeling: A reduced dynamical model consisting of three coupled Ordinary Differential Equations (ODEs) is derived in Fourier space, isolating the three resonant modes. This model is compared against the full lattice simulations.
5. Robustness and Limits: The study examines the linear limit (vanishing nonlinearity), the effect of higher-order nonlinear terms (using Toda and granular chain potentials), and the impact of random mass fluctuations (disorder) on the recurrence. Finally, the continuous limit ($N \to \infty$) is taken to derive a system of three coupled Partial Differential Equations (PDEs).

Key Contributions and Results

• Two Distinct Recurrence Mechanisms: The authors identify two types of recurrence in the diatomic lattice:

  1. Classical FPUT Recurrence: Observed when the fundamental acoustic mode is excited. The period of recurrence scales as $1/\sqrt{\alpha}$ (where $\alpha$ is the nonlinear strength), consistent with the monatomic case. This mechanism relies on non-resonant three-wave interactions and occurs for arbitrary mass ratios.
  2. Resonance-Induced Recurrence: A novel mechanism unique to the diatomic lattice. It arises from exact three-wave resonant interactions between two acoustic modes and one optical mode. This recurrence is only possible when the mass ratio $m_2/m_1$ satisfies specific conditions (specifically $1 < m_2/m_1 \leq 3$).

• Characteristics of Resonance-Induced Recurrence:

  • Mode Confinement: Unlike the classical recurrence which involves energy exchange among many modes, this recurrence is confined to a resonant triad. The vast majority of the system's energy remains within these three modes.
  • Robustness to Nonlinearity: In the weakly nonlinear limit (small $\alpha$), the classical recurrence weakens significantly (energy exchange decreases). In contrast, the resonance-induced recurrence remains robust; the energy exchange magnitude stays near unity, with only the time scale rescaling as $\alpha$ changes.
  • Initial Conditions: The recurrence can be triggered by initially exciting the optical mode (with a small perturbation to acoustic modes) or by directly exciting the two acoustic modes. The latter case results in a faster recurrence time scale.

• Reduced Model and Integrability:

  • The authors successfully derive a reduced three-mode ODE system that accurately reproduces the dynamics of the full lattice for the resonant case.
  • This reduced system is identified as a classical integrable system with Manley-Rowe invariants.
  • In the thermodynamic limit, the model reduces to a system of three coupled PDEs (the classical three-wave resonant interaction system), which is known to be integrable.

• Robustness Tests:

  • Higher-Order Terms: Simulations using Toda and granular chain potentials (which include higher-order nonlinearities) show that the inclusion of these terms has a negligible effect on the recurrence dynamics.
  • Mass Disorder: The study investigates the effect of Gaussian fluctuations in particle masses. While the resonance condition is strict, the authors find that complete recurrence persists in a significant portion of trials even with mass variances up to 10%, though incomplete or absent recurrence occurs in some realizations.

Significance
The paper demonstrates that the diatomic FPUT lattice supports a distinct, robust form of energy recurrence driven by exact three-wave resonances between acoustic and optical branches. This phenomenon is fundamentally different from the classical FPUT recurrence, as it relies on the specific dispersion properties of the diatomic chain and persists strongly even in the weakly nonlinear regime. By deriving a reduced integrable model and showing its correspondence to the full system, the work provides a theoretical framework for understanding energy localization and exchange in diatomic nonlinear lattices. The findings suggest that such resonant mechanisms could be a dominant factor in the thermalization (or lack thereof) of diatomic systems, distinguishing them from their monatomic counterparts.