Incomplete thermalization from trap-induced integrability breaking: lessons from classical hard rods

The familiar Newton’s cradle can be modeled as a one-dimensional gas of hard rods trapped in a harmonic potential, which breaks integrability of the hard-rod interaction in a non-uniform way. We explore the consequences of such broken integrability for the dynamics of a large number of particles and find three distinct regimes: initial, chaotic, and stationary. The initial regime is captured by an evolution equation for the phase-space distribution function. However, for any finite number of particles, this hydrodynamics breaks down due to a “complexity crisis” and the dynamics become chaotic after a characteristic time scale determined by the inter-particle distance and scattering length. At long times, the system fails to thermalize and the time-averaged ensemble of individual trajectories is not micro-canonical, but it is a stationary state of the hydrodynamic evolution. We close by discussing logical extensions of the results to similar systems of quantum particles.