Long-tailed dissipationless hydromechanics: weak thermalization and ergodicity breaking

We analyze the dynamic properties of dissipationless Generalized Langevin Equations in the presence of fluid inertial kernels possessing power-law tails, $k(t)\sim t^{-\kappa }$. While for $\kappa >1$ the dynamics is manifestly non ergodic, no thermalization occurs, and particle motion is ballistic, new phenomena arise for $0<\kappa <1$. In this case, a form of weak thermalization appears in the presence of thermal/hydrodynamic fluctuations and attractive potentials. However, the absence of dissipation clearly emerges once an external constant force is applied: an asymptotic settling velocity cannot be achieved as the expected value of the particle velocity diverges.