Algebraic relaxation laws for classical particles in 1D anharmonic potentials

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 x²ⁿ we find that the asymptotic relaxation is consistent with 1/t^φ, where φ = 1⋀(n-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.