Statistical Mechanics usually consists in calculating averages and variances in some ensemble. In quantum statistical mechanics, the ensemble is represented by a state, i.e., a density matrix. However, it is also very important to consider ensembles of states. This allows to understand what are the typical properties of quantum systems. Probabilistic methods are the only way of probing the space of quantum many-body mechanical states with N particles, as its dimension is exponential in N, and therefore absolutely untreatable otherwise. To this end, one has to take averages over the whole Hilbert space. In Part I, we shall see how use Haar measure to compute group averages and variances in the Hilbert space and discuss the notion of typicality of Entanglement. In Part II, we will see how this can be exploited in the foundations of statistical mechanics. We will then switch to some research topics, namely, how to incorporate locality and discuss recent applications and open problems.

Coffee: 12:45 pm, 241 Compton