Analytic lyapunov exponents in a classical nonlinear field equation

Area of Science:

• Nonlinear dynamics
• Mathematical physics
• Chaos theory

Background:

• The Fermi-Pasta-Ulam (FPU) beta model is a classical field system.
• Understanding the chaotic behavior of such systems is crucial in physics.

Purpose of the Study:

• To investigate the nonlinear wave equation, the continuum limit of the FPU beta model.
• To determine the Lyapunov exponent and its energy dependence for this field equation.

Main Methods:

• Evaluating the lattice-spacing dependence of the Lyapunov exponent for the FPU model.
• Utilizing a Riemannian description of Hamiltonian chaos.
• Analyzing the continuum limit of the FPU beta model.

Main Results:

• The nonlinear wave equation exhibits a positive Lyapunov exponent (lambda(1)).
• An analytic expression for the energy dependence of lambda(1) was derived.
• This provides the first example for field equations demonstrating such a property.

Conclusions:

• The study establishes a positive Lyapunov exponent for the nonlinear wave equation, a key finding for field equations.
• It highlights challenges in statistical mechanical treatments of classical field systems compared to dynamical descriptions.