From Boltzmann's problem to Kac's chaos

This thesis examines the classical problem of explaining how macroscopic irreversibility emerges from reversible microscopic laws, commonly referred to as Boltzmann’s problem. The analysis begins with the second law of thermodynamics and Boltzmann’s statistical interpretation, according to which entropy increases because equilibrium states overwhelmingly dominate the phase space. Particular attention is devoted to the main objections and paradoxes, such as Poincaré’s recurrence theorem, and to simplified models like the Ehrenfest urn model, which provide an intuitive framework to address these challenges. The second part focuses on the contribution of Mark Kac, who introduced a rigorous probabilistic framework for the notion of molecular chaos. By employing exchangeable random variables, empirical measures, and scaling limits of many-particle systems, Kac formalized the mechanism through which statistical independence arises. This perspective establishes a precise mathematical bridge between Boltzmann’s heuristic ideas and modern probability theory, offering a deeper understanding of the statistical foundations of irreversibility.