Ergodic theory is a branch of mathematics that deals with the study of dynamical systems and their behaviors over time. There are many synonyms for ergodic theory such as mathematical physics, phase space geometry, statistical mechanics, chaotic dynamics, and probability theory. In mathematical physics, ergodic theory is used to study the behavior of physical systems such as atoms, molecules, and gases. Phase space geometry is a term used to describe the study of the evolution of a system over time, while statistical mechanics is the study of the statistical behavior of many-particle systems. Chaotic dynamics is the study of chaotic systems, while probability theory is the study of random variables and their behaviors.