Resonance overlap, secular effects, and nonintegrability: an approach from ensemble theory

Area of Science:

• Classical mechanics
• Statistical physics
• Nonlinear dynamics

Background:

• Analyzing the time evolution of classical multiresonance nonintegrable Hamiltonian systems is crucial for understanding complex dynamics.
• Existing methods like the lambda(2)t expansion are established for thermodynamic systems in nonequilibrium statistical physics.

Purpose of the Study:

• To analyze the ensemble-level time evolution of classical multiresonance nonintegrable Hamiltonian systems with few degrees of freedom.
• To determine the most secular series for the time evolution of expectation values of physical observables.

Main Methods:

• Application of time-dependent perturbation analysis to the Liouville equation.
• Investigation of asymptotic expansions for ensemble-level dynamics.

Main Results:

• Discovery of a square root of lambda t expansion, distinct from the lambda(2)t expansion found in thermodynamic systems.
• The asymptotic expansion is valid only at the ensemble level, not for individual trajectories.
• The expansion exhibits broken time symmetry, consistent with nonequilibrium statistical mechanics.

Conclusions:

• The study reveals a novel asymptotic expansion for small nonintegrable systems with few degrees of freedom.
• The findings highlight differences between ensemble and trajectory descriptions in chaotic systems.
• The Chirikov overlapping criterion's relation to this approach is discussed, providing a link to chaos theory.