Measuring FPUT thermalization with Toda integrals

Area of Science:

• Nonlinear dynamics
• Statistical mechanics
• Computational physics

Background:

• The Fermi-Pasta-Ulam-Tsingou-α (FPUT-α) model is a fundamental system for studying nonlinear dynamics and the transition to chaos.
• Understanding the ergodic properties of such systems is crucial for validating statistical mechanics predictions.

Purpose of the Study:

• To investigate the ergodic properties of the FPUT-α model for generic initial conditions.
• To determine the characteristic time scales associated with the system's approach to equilibrium.
• To explore the influence of system size and energy density on ergodization.

Main Methods:

• Utilizing a Toda integral as an adiabatic invariant and observable for equilibrium time.
• Comparing the Toda integral ergodization time with the inverse of the maximum Lyapunov exponent and its saturation time.
• Numerically measuring the dependence of energy density on critical system size.

Main Results:

• The Toda integral ergodization time is system size independent for large chains but grows dramatically for smaller sizes below a critical threshold.
• This critical size depends on the energy density, suggesting a link to the Kolmogorov-Arnold-Moser (KAM) regime.
• The critical energy density for the KAM regime approximately follows a 1/N² decay with the number of particles N.

Conclusions:

• Action diffusion leads to ergodic temporal fluctuations in the FPUT-α model over its equilibrium time scale.
• The system size and energy density play critical roles in the emergence of ergodicity and the potential breakdown of the KAM regime.
• The observed 1/N² scaling provides insights into the transition to chaos in many-body systems.