Exploring the Evolution of the Entropy Related to Mass Ordering in Multi-planet Systems

We define an entropy related to mass ordering in multi-planetary systems. Through Monte Carlo simulations and N-body dynamical numerical simulations, we explore the impact of dynamical evolution on the entropy of planetary mass ordering. Our findings suggest that collisions and positional swaps during the dynamical evolution of planetary systems can modify the entropy of mass ordering within the system. Positional swaps tend to gradually increase the entropy of mass ordering, whereas collisions may lead to a decrease. Despite the potential decrease in the system's entropy value, the ratio of the current system's mass ordering entropy to the maximum attainable mass ordering entropy consistently rises, indicating the system's progression toward equilibrium. Observations on Kepler multi-planetary systems reveal that about 16.9\%\pm4.7\% of them still maintain an ordered mass distribution, implying that these systems may not have experienced significant dynamical evolution.