Random Initial Data and Average Shock Time in the Fermi-Pasta-Ulam-Tsingou Chain

Area of Science:

• Nonlinear dynamics
• Statistical physics
• Condensed matter physics

Background:

• The Fermi-Pasta-Ulam-Tsingou (FPU) chain is a fundamental model for studying nonlinear lattice dynamics and the approach to thermal equilibrium.
• Understanding prethermalization is crucial for systems that do not rapidly reach thermal equilibrium.
• Previous studies have explored FPU dynamics, but the robustness of prethermalization under generic random initial conditions requires further investigation.

Purpose of the Study:

• To investigate the dynamics of the FPU chain with long-wavelength random initial data.
• To characterize the formation and robustness of the prethermal state.
• To derive a precise mathematical expression for the average shock time.

Main Methods:

• Analysis of the FPU chain dynamics under long-wavelength random initial conditions.
• Identification of Burgers-type shock formation and turbulentlike spectrum.
• Application of advanced probabilistic techniques (Dudley, Talagrand) to derive asymptotic expressions.
• Thermodynamic limit analysis for large systems.

Main Results:

• Demonstration that prethermalization, Burgers-type shock, and turbulentlike spectrum are robust under generic long-wavelength random initial conditions.
• Derivation of a sharp asymptotic expression for the average shock time in the thermodynamic limit.
• Proof that the shock time scales as (psqrt[logp])^{-1} for large p (number of excited modes), indicating it is an intensive quantity with logarithmic corrections.

Conclusions:

• The prethermal state in the FPU chain is a robust phenomenon under generic random initial conditions.
• The derived shock time scaling provides a quantitative understanding of prethermalization dynamics.
• These findings advance the understanding of thermalization processes in nonlinear systems.