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The Quantizer problem is a tessellation optimisation problem where pointconfigurations are identified such that the Voronoi cells minimise the secondmoment of the volume distribution. While the ground state (optimal state) in 3Dis almost certainly the body-centered cubic lattice, disordered and effectivelyhyperuniform states with energies very close to the ground state exist thatresult as stable states in an evolution through the geometric Lloyd's algorithm[Klatt et al. Nat. Commun., 10, 811 (2019)]. When considered as a statisticalmechanics problem at finite temperature, the same system has been termed the'Voronoi Liquid' by [Ruscher et al. EPL 112, 66003 (2015)]. Here we investigatethe cooling behaviour of the Voronoi liquid with a particular view to thestability of the effectively hyperuniform disordered state. As a confirmationof the results by Ruscher et al., we observe, by both molecular dynamics andMonte Carlo simulations, that upon slow quasi-static equilibrium cooling, theVoronoi liquid crystallises from a disordered configuration into thebody-centered cubic configuration. By contrast, upon sufficiently fastnon-equilibrium cooling (and not just in the limit of a maximally fast quench)the Voronoi liquid adopts similar states as the effectively hyperuniforminherent structures identified by Klatt et al. and prevents the orderingtransition into a BCC ordered structure. This result is in line with thegeometric intuition that the geometric Lloyd's algorithm corresponds to a typeof fast quench.

Low-temperature statistical mechanics of the QuanTizer problem: fast quenching and equilibrium cooling of the three-dimensional Voronoi Liquid