  The statistical mechanical behavior of weakly nonlinear multimoded optical
settings is attracting increased interest during the last few years. The main
purpose of this work is to numerically investigate the main factors that affect
the thermalization process in photonic lattices. In particular, we find that
lattices with identically selected properties (such as temperature, coupling
coefficient, lattice size, and excitation conditions) can exhibit very
different thermalization dynamics and thus thermalization distances. Our
investigation is focused on two different two-dimensional lattices: the
honeycomb lattice and the triangular lattice. Our numerical results show that,
independently of the excitation conditions, the honeycomb lattice always
thermalizes faster than the triangular lattice. We mainly explain this behavior
to the quasilinear spectrum that promotes wave-mixing in the honeycomb lattice
in comparison to the power-like spectrum of the triangular lattice. In
addition, we investigate the combined effects of temperature as well as the
sign and magnitude of the nonlinearity. Switching either the sign of the Kerr
nonlinear coefficient or the sign of the temperature can lead to significant
differences in the thermalization dynamics, a phenomenon that can be physically
explained in terms of wave instabilities. Larger absolute values of the
temperature |T| result in more uniform distributions for the power occupation
numbers and faster thermalization speeds. Finally, as expected, increasing the
magnitude of the nonlinearity results in accelerated thermalization. Our
findings provide valuable insights into optical thermalization in discrete
systems where experimental realization may bring about new possibilities for
light manipulation and applications.
