Abstract: The Kubo-Martin-Schwinger (KMS) condition was introduced in the late 1950s and was initially concerned with physical systems. Since then, the basic mathematical theory has been established and works like those of Bratteli and Robinson or Pedersen have illuminated the scenery. Nowadays, the KMS states of systems of both physical and mathematical origin are studied. In these talks, we concern ourselves with finite quantum mechanical systems in thermal equilibrium, in which the KMS states are proven to be exactly the Gibbs states of statistical ensembles. The KMS condition will arise naturally, matching C*-algebraic states with density matrices. Using minimal prerequisites, we will discuss certain foundational matters on our way to introducing both a quantum mechanical reality and the statistical ideas relevant to us. We will mostly refrain from proving statements, as this would take much time.

Part I: Density Matrices. In the first talk, we aim to introduce the relevant quantum mechanics in matrix form. This will be done axiomatically, while a few relevant perspectives and outlooks will be noted. Lastly, we will discuss the density matrix of Gibbs states and the notion of entropy.

Part II: Gibbs and KMS States In the second and final talk we will establish a connection between C*-algebraic states and density matrices. The aim is to introduce the KMS condition as an equilibrium condition. To this end, we will show that the state corresponding to the Gibbs density matrix represents thermal equilibrium.