Testing an entropy estimator related to the dynamical state of galaxy clusters

We propose the entropy estimator $H_Z$, calculated from global dynamical parameters, in an attempt to capture the degree of evolution of galaxy systems. We assume that the observed (spatial and velocity) distributions of member galaxies in these systems evolve over time towards states of higher dynamical relaxation (higher entropy), becoming more random and homogeneous in virial equilibrium. Thus, the $H_Z$-entropy should correspond to the gravitacional assembly state of the systems. This was tested in a sample of 70 well sampled clusters in the Local Universe whose gravitational assembly state, classified from optical and X-ray analysis of substructures, shows clear statistical correlation with $H_Z$. This estimator was also tested on a sample of clusters (halos) from the IllustrisTNG simulations, obtaining results in agreement with the observational ones.