Dynamical and Probabilistic Approaches to Irreversibility

Boltzmann's H-theorem is considered a great triumph of science. Though some modifications are necessary to adapt it to modern dynamical theories, it is well established that one of its main tenets remains widely accepted: the introduction of probability is a key element in achieving a transition from time-reversible, deterministic dynamical laws at the microscopic level to irreversible laws describing the approach to equilibrium of isolated macroscopic systems. Thus, it is somehow surprising that we still find instances where this subject is labeled as paradoxical and elusive. More remarkable is the fact that this often happens in texts that succeed in presenting Boltzmann's ideas with clarity. In order to shed light on how probability allows us to go form microscopic reversibility to macroscopic irreversibility, we use numerical results from a two-dimensional lattice gas composed of distinguishable particles. We discuss the roles played by noise, coarse graining, and probability. The simplicity of our model might help the newcomer to this area in better grasping Boltzmann's fundamental breakthrough.