Brownian localization: A generalized coupling model yielding a nonergodic Langevin equation description

A minimal system-plus-reservoir model yielding a nonergodic Langevin equation is proposed, which origins from cubic-spectral density of environmental oscillators and momentum-dependent coupling. This model allows the ballistic diffusion and classical localization simultaneously, in which the fluctuation-dissipation relation is still satisfied but the Khinchin theorem is broken. The asymptotical equilibrium for nonergodic system requires the initial thermal equilibrium, however, when the system starts from nonthermal conditions, it does not approach the equilibration even a nonlinear potential is used to bound particle, this can be confirmed by the zeroth law of thermodynamics. In the dynamics of Brownian localization, due to the memory damping function inducing a constant term, our results show that the stationary distribution of the system depends on its initial preparation of coordinate rather than momentum. The coupled oscillator chain with a fixed end boundary acts as such heat bath, which has long been used in studies of collinear atom/solid-surface scattering and lattice vibration, we investigate this problem from the viewpoint of nonergodicity.