R.S. Sinkovits, S. Sen, J.C. Phillips, S. Chakravarti. Slow algebraic relaxation in quartic potentials and related results. Physical Review E 59: (6) 6497-6512, 1999.

We present a detailed report [see S. Sen et al., Phys. Rev. Lett. 77, 4855 (1996)] of our numerical and analytical studies on the relaxation of a classical particle in the potentials V(x) = +/-x(2)/2+x(4)/4. Both of the approaches confirm that at all temperatures, the relaxation functions (e.g., Velocity relaxation function and position relaxation function) decay asymptotically in time t as sin(omega(0)t)lt. Numerically calculated power spectra of the relaxation functions show a gradual transition with increasing temperature from a single sharp peak located at the harmonic frequency omega(0) to a broad continuous band. The 1/t relaxation is also found when V(x) is a polynomial in powers of x(2) with a nonvanishing coefficient accompanying the x(4) term in V(x). Numerical calculations show that in the cases in which the leading term in V(x) behaves as x(2n) With integer n, the asymptotic relaxation exhibits 1/t(phi) decay where phi = 1/(n - 1). We briefly discuss the analytical approaches to relaxation studies in these strongly anharmonic systems using direct solution of the equation of motion and using the continued fraction formalism approach for relaxation studies. We show that the study of the dynamics of strongly anharmonic oscillators poses unique difficulties when studied via the continued fraction or any other time-series construction based approaches. We close with comments on the physical processes in which the insights presented in this work may be applicable.