Incomplete Thermalization from Trap-Induced Integrability Breaking: Lessons from Classical Hard Rods

We study a one-dimensional gas of hard rods trapped in a harmonic potential, which breaks integrability of the hard-rod interaction in a nonuniform 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. For any finite number of particles, this hydrodynamics breaks down and the dynamics becomes chaotic after a characteristic timescale determined by the interparticle distance and scattering length. The system fails to thermalize over the timescale studied (10^{4} natural units), but the time-averaged ensemble is a stationary state of the hydrodynamic evolution. We close by discussing logical extensions of the results to similar systems of quantum particles.