  Artificial ice systems have been designed to replicate paradigmatic phenomena
observed in frustrated spin systems. Here we present a detailed theoretical
analysis based on Monte-Carlo simulations of the low energy phases in an
artificial colloidal ice system, a recently introduced ice system where an
ensemble of repulsive colloids are two-dimensionally confined by gravity to a
lattice of double wells at a one-to-one filling. Triggered by recent results
obtained by Brownian dynamics simulations [A. Lib\'al et al, Phys. Rev. Lett.
120, 027204 (2018)], we analyze the energetics and the phase transitions that
occur in the honeycomb geometry (realizing the analogue of a spin ice system on
a kagome lattice) when decreasing the temperature. When the particles are
restricted to occupy the two minima of the potential well, we recover the same
phase diagram as the dipolar spin ice system, with a long range ordered chiral
ground state. In contrast, when considering the particle motion and their
relaxation within the traps, we observe ferromagnetic ordering at low
temperature. This observation highlights the fundamental role played by the
continuous motion of the colloids in artificial ice systems.
