  The thermodynamic non-equilibrium (TNE) effects and the relationships between
various TNE effects and entropy production rate, morphology, kinematics, and
dynamics during two initially static droplet coalescence are studied in detail
via the discrete Boltzmann method. The temporal evolutions of the total TNE
strength ($D^*$) and the total entropy production rate ($\dot S$) can both
provide concise, effective and consistent physical criteria to distinguish the
stages of droplet coalescence. Specifically, when $\bar D^*$ and $\dot S$ reach
their maxima, it corresponds to the time when the liquid-vapor interface length
changes the fastest; when $D^*$ and $\dot S$ reach their valleys, it
corresponds to the moment of the droplet being the longest elliptical shape.
During the merging process, the force contributed by surface tension in the
coalescence direction acts as the primary promoting force for droplet
coalescence and reaches its maximum concurrently with coalescent acceleration.
In contrast, the force contributed by non-organized momentum fluxes (NOMFs) in
the coalescing direction inhibits the merging process and reaches its maximum
at the same time as $D^*$. For the coalescence of two unequal size droplets,
the smaller droplet exhibits larger values for TNE intensity, merging velocity,
driving force contributed by surface tension, and resistance contributed by
NOMFs. Moreover, these values gradually increase with the initial radius ratio
of the large and small droplets due to larger curvature. However,
non-equilibrium components and forces related to shear velocity in the small
droplet, are all smaller than those in the larger droplet and gradually
decrease with the radius ratio.
