Long-tailed dissipationless hydromechanics: weak thermalization and ergodicity breaking

TL;DR

This paper investigates the complex dynamics of dissipationless hydromechanical systems with power-law fluid kernels, revealing non-ergodic behavior, weak thermalization, and the absence of dissipation under external forces.

Contribution

It introduces a detailed analysis of generalized Langevin equations with power-law kernels, highlighting novel phenomena like weak thermalization and ergodicity breaking in dissipationless systems.

Findings

For $$, dynamics are non-ergodic with no thermalization and ballistic motion.

For $0 < $, weak thermalization occurs due to fluctuations and potentials.

External constant forces do not lead to steady velocities, indicating absence of dissipation.

Abstract

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.