Thermalization and irreversibility of an isolated quantum system II

The irreversible entropy increase described by the second law of thermodynamics is fundamentally tied to thermalization and the emergence of equilibrium. In the first part of our work (Ref: arXiv.2503.04152), we constructed an isolated gas system model and numerically demonstrated irreversible growth of entanglement entropy caused by erasure of spread non-equilibrium state information. Here, we mathematically prove that for a typical macroscopic system in a non-equilibrium state $|\phi_0\rangle$, the quantum state $|\phi'_0\rangle = \hat{O}(t)|\phi_0\rangle$ will inevitably evolve toward equilibrium. Our work demonstrates that the second law of thermodynamics, and consequently the ergodic hypothesis in statistical physics, can be understood and proven from a quantum information perspective. From this perspective, the second law can be stated as: In typical macroscopic physical systems, the spreading and erasure of non-equilibrium information is inevitable.