Ergodic theory, entropy and application to statistical mechanics

This is an expository paper written after eight weeks at the Mathematics Research Experience for Undergraduates at the University of Chicago.

Ergodic theory originated in statistical mechanics. This paper first introduces the ergodic hypothesis, a fundamental problem in statistical mechanics. In order to come up with a solution, this paper explores some basic ideas in ergodic theory. Next, the paper defines measure-theoretical entropy and shows its connection to physical entropy. Lastly, these results are used to construct Gibbs ensembles, a useful tool in statistical mechanics.