  This article presents an entropy stability analysis of smoothed dissipative
particle dynamics (SDPD) to review the validity of particle discretization of
entropy equations. First, we consider the simplest SDPD system: a simulation of
incompressible flows using an explicit time integration scheme, assuming a
quasi-static scenario with constant volume, constant number of particles, and
infinitesimal time shift. Next, we derive a form of entropy from the
discretized entropy equation of SDPD by integrating it with respect to time. We
then examine the properties of a two-particle system for a constant temperature
gradient. Interestingly, our theoretical analysis suggests that there exist
eight different types of entropy stability conditions, which depend on the
types of kernel functions. It is found that the Lucy kernel, poly6 kernel, and
spiky kernel produce the same types of entropy stability conditions, whereas
the spline kernel produces different types of entropy stability conditions. Our
results contribute to a deeper understanding of particle discretization.
