Modeling Entropy Evolution

This study embarks on a two-fold exploration to understand entropy's temporal evolution. Firstly, we sought a general relationship between the rate of change of entropy and entropy itself. Guided by fundamental principles such as the second law of thermodynamics, the entropy of a system having an upper limit, and the dependence of entropy's rate of change on its state relative to equilibrium, we formulated a mathematical model that illustrates entropy's dependence on time. The second part of this study involves the free expansion of an ideal gas. Through kinematic modeling, we discern how entropy evolves over time. Our simulation introduces a new variable which is a function of particles' positions and momenta in phase space. By correlating this variable with entropy at equilibrium, we validate our general model, revealing a consistent alignment between our theoretical predictions and simulated outcomes.