Deriving the second law of thermodynamics and exploring its boundaries

Abstract

The second law of thermodynamics still lacks a general proof applicable to both classical and quantum systems across broad ranges of time scales, interactions, and degrees of nonequilibrium. In this paper, we show that when the macrostate-level probability $f$ of an isolated system increases monotonically with the number of possible microstates $Ω$ (i.e., $\partial f/\partial Ω> 0$), the second law emerges naturally from the continuity equation of $f$ in phase space; a special case of $\partial f/\partial Ω> 0$ is Boltzmann's assumption of equal a priori equilibrium probabilities. Based on this finding, the second law can be readily derived for fully chaotic systems. The derivation does not rely on dynamical details, highlighting the statistical nature of entropy increase. In contrast, for a locally nonchaotic system, the positive correlation between $f$ and $Ω$ may break down (i.e., $\partial f/\partial Ω\leq 0$). Consequently, the conventional framework of thermodynamics does not apply, and entropy can decrease spontaneously without any energetic penalty.