Boundary-induced classical Generalized Gibbs Ensemble with angular momentum

We investigate the impact of the boundary shape on the thermalization behavior of a confined system of classical hard disks at low packing fraction and thus in the gas regime. We use both analytical calculations and numerical simulations, and leveraging on the insights from the maximum entropy principle, we explore how the geometry of the boundary influences the thermal equilibration process in such systems. Our simulations involve hard disks confined within varying boundary shapes, using both event-driven and time-driven simulations, ranging from conventional square boundaries to circular boundaries, showing that the two converge to different ensembles. The former converges to the Gibbs Ensemble, while the latter converges to the Generalized Gibbs Ensemble (GGE), with angular momentum as the extra conserved quantity. We introduce an order parameter to characterize the deviations from the Gibbs ensemble, and show that the GGE is not time-reversal invariant, it violates ergodicity and leads to a near-boundary condensation phenomenon. Because of this, we argue that Monte Carlo methods should include angular momentum in this situation. We conclude by discussing how these results lead to peculiar violations of the Bohr-van Leeuwen theorem.

💡 Research Summary

The paper investigates how the shape of a confining boundary influences the thermalization of a low‑density hard‑disk gas in two dimensions. Using both analytical calculations based on the maximum‑entropy principle and extensive event‑driven and time‑driven molecular dynamics simulations, the authors demonstrate that the nature of the asymptotic statistical ensemble depends critically on whether the boundary conserves angular momentum.

In a square (or periodic) box, collisions with the walls do not preserve the total angular momentum about the centre, so the only conserved extensive quantity is the total kinetic energy. Consequently the system relaxes to the standard Gibbs (canonical) ensemble, with factorized Gaussian distributions for positions and momenta.

In contrast, a circular boundary with perfectly elastic, specular reflections conserves the z‑component of angular momentum (L_z) in addition to energy. The presence of two independent conserved quantities forces the equilibrium distribution to be a Generalized Gibbs Ensemble (GGE):

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