Slow algebraic relaxation in quartic potentials and related results

We present a detailed report [see S. Sen ct al., Phys. Rev. Lett. 77, 4855 (1996)] of our numerical and analytical studies on the relaxation of a clas sical particle in the potentials V(x) = +/-x(2)/2+x(4)/4. Both of the appro aches confirm that at all temperatures, the relaxation functions (e.g., Vel ocity relaxation function and position relaxation function) decay asymptoti cally in time t as sin(omega(0)t)lt. Numerically calculated power spectra o f the relaxation functions show a gradual transition with increasing temper ature from a single sharp peak located at the harmonic frequency omega(0) t o 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 t he x(4) term in V(x). Numerical calculations show that in the cases in whic h 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 discus s the analytical approaches to relaxation studies in these strongly anharmo nic systems using direct solution of the equation of motion and using the c ontinued fraction formalism approach for relaxation studies. We show that t he study of the dynamics of strongly anharmonic oscillators poses unique di fficulties when studied via the continued fraction or any other time-series construction based approaches. We close with comments on the physical proc esses in which the insights presented in this work may be applicable. [S106 3-651X(99)09306-X].