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Time scale to ergodicity in the Fermi-Pasta-Ulam system
J. De Luca1, A. J. Lichtenberg, M. A. Lieberman
1Department of Physics, University of California, Berkeley, California 94720Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720.
Chaos (Woodbury, N.Y.)
|March 1, 1995
Summary
Energy levels determine how quickly the Fermi-Pasta-Ulam system reaches equipartition. Above a critical energy, equipartition is reached in N(2) time, while below it, the time is exponentially longer.
Area of Science:
- Nonlinear dynamics
- Statistical mechanics
- Computational physics
Background:
- The Fermi-Pasta-Ulam (FPU) system is a fundamental model for studying energy equipartition in nonlinear systems.
- Understanding energy flow and thermalization in conservative systems is crucial for statistical mechanics.
Purpose of the Study:
- To investigate the approach to energy equipartition in the N-dimensional FPU Hamiltonian with quartic nonlinearity.
- To determine the critical energy threshold and time scales for equipartition.
- To develop a theoretical framework for predicting the system's behavior.
Main Methods:
- Numerical simulations of orbits with initial energy in low-frequency modes.
- Resonant normal form treatment to analyze nonlinear mode excitation.
- Theoretical modeling of time evolution and mode excitation.
Main Results:
- A critical energy, independent of system dimension N, was identified for equipartition.
- Above critical energy, equipartition time scales as N(2).
- Below critical energy, equipartition time becomes exponentially long.
Conclusions:
- The study provides a theoretical explanation for the observed equipartition behavior in the FPU system.
- The developed theory accurately predicts critical energy, time scales, and nonlinear mode behavior.
- Results offer insights into energy dynamics in nonlinear conservative systems.