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.

Figure 1

(a) An illustration of three-rod dynamics. The Poincaré sector is defined as the set of configurations just after a 1–2 collision. They are indicated by dashed lines. The Poincaré map sends the left one to the right one. (b) Orbits of the Poincaré recurrence map, with $H=4$ and vanishing center-of-mass energy. The sector is bijectively parametrized by $p_2-p_1$ and $p_3$. Different colors distinguish distinct orbits.

Figure 2

Comparing hard-rod dynamics and THRE in the initial regime. (a) Comparing the density at origin [$n(0)$]. The hard-rod data (circles) are obtained by averaging over 200 realizations with $N=1024$ rods, representing a circular IC profile with ${\sigma }_x={\sigma }_p$, ${\rho }_{m}a=0.4$ [Eq. (6)]. The THRE data (curve) are obtained from its numerical integration [19, 20]. (b) The density profile evolution obtained from the rod simulation.

Figure 3

(a) Evolution of phase-space distribution $\rho (x,p)$ for the squeezed IC (${\rho }_m=1/2$, ${\sigma }_x=1/2$), with different $N$. At $t=64$, the $N=2048$ system preserves a complex structure, which is completely smeared out for $N=128$. (b) Entropy increase during time evolution, starting from different ICs. The solid curves represent entropy growth estimated using the entropy production term in [14].

Figure 4

Dynamical chaos measured as separation of trajectories. (a) Demonstrating the scaling law of time to chaos [Eq. (1)]. $N=64$ for all data. (Main plot) Scaling collapse, initial regime, and the crossover to chaos. (i) Raw data. (ii) Scaling collapse of the exponential chaos regime. The initial-regime oscillation has a period independent of ${\rho }_{m}a$ and close to that of a single harmonic oscillator. (b) Suppression of the chaos in larger systems, as illustrated by the decrease of the Lyapunov exponent. (Main:) Raw data for ${\rho }_{m}a=1/2$. (Inset) Data collapse, suggesting the scaling $\gamma \propto N^{-0.25}$.

Figure 5

(a) (Non)-Gaussianity of the velocity distribution of the time-averaged ensemble, as revealed by the moment-ratio test. (b) Comparing the velocity distribution with the reconstructed one from the density, assuming that the late-time ensemble solves the stationary THRE [Eq. (8)]. The two long-time ensembles are obtained from two squeezed (red) and circular (black) ICs, both with ${\rho }_{m}a=1/2$.