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Slow algebraic relaxation in quartic potentials and related results
R S Sinkovits1, S Sen, J C Phillips
1San Diego Supercomputer Center, San Diego, CA 92186-5608, USA.
Summary
Classical particle relaxation in anharmonic potentials exhibits a 1/t decay at all temperatures. Increasing temperature shifts power spectra from a sharp peak to a broad band, revealing complex dynamics in strongly anharmonic systems.
Area of Science:
- Statistical Mechanics
- Nonlinear Dynamics
- Computational Physics
Background:
- Understanding particle relaxation dynamics is crucial in statistical mechanics.
- Anharmonic potentials present significant challenges compared to harmonic systems.
- Previous studies often focused on specific temperature regimes or simpler potentials.
Purpose of the Study:
- To investigate the asymptotic relaxation behavior of a classical particle in specific anharmonic potentials.
- To analyze the temperature dependence of relaxation functions and their power spectra.
- To explore analytical and numerical methods for studying strongly anharmonic systems.
Main Methods:
- Numerical simulations of particle dynamics.
- Analytical solutions of the equation of motion.
- Application of the continued fraction formalism.
- Calculation of velocity and position relaxation functions.
- Power spectrum analysis of relaxation functions.
Main Results:
- Confirmed asymptotic decay of relaxation functions as sin(omega(0)t)/t for V(x)=+/-x(2)/2+x(4)/4 potentials.
- Observed a temperature-dependent transition in power spectra from a sharp peak to a broad band.
- Identified a generalized 1/t(phi) decay for potentials with leading terms x(2n), where phi = 1/(n-1).
Conclusions:
- The 1/t relaxation is a robust feature in these anharmonic systems across temperatures.
- Strong anharmonicity introduces unique challenges for analytical approaches like continued fractions.
- The findings offer insights into the dynamics of various physical processes involving anharmonic oscillators.