Relaxation in a Duffing potential

The solution of the equation of motion for a particle in a Duffing potential, V(x) = α1x2/2 + α2x4/4 (α1, α2 > 0) for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies. It is, however, non-trivial to determine whether such “discrete” frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by Fronzoni et al. (J. Stat. Phys. 41 (1985) 553). We recover and extend upon the results of Fronzoni et al. to show analytically, via Mori-Lee theory, that “essentially discrete” (i.e. well-defined peaks with finite but “small” width) temperature-dependent frequencies characterize the autocorrelation functions in a canonical ensemble