Algebraic Relaxation Laws for Classical Particles in 1D Anharmonic Potentials

Abstract

Using extensive numerical analysis and exact calculations we show that the relaxation of a classical particle in 1D anharmonic potential landscapes with a leading quartic term follows a $1/t$ decay law at all temperatures, leading to a logarithmically increasing mean square displacement. For leading anharmonic terms of form ${x}^{2n}$ we find that the asymptotic relaxation is consistent with $1/{t}^{\ensuremath{\varphi}}$, where $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/(n\ensuremath{-}1)$ at all temperatures. We briefly comment on the possible implications of this result in the study of displacive structural transitions and in complex systems.