MARK KAC SEMINAR

I will first illustrate in which sense the macroscopic laws (e.g. Euler Equation, Heat equation, ...) are expected to be a consequence of the Hamiltonian dynamics. Then I will discuss the technical obstacles to the rigorous implementation of such a research program and some related recent results.

A basic postulate of equilibrium statistical mechanics is that the macroscopic state of a system at thermal equilibrium is appropriately described by a probability distribution that maximize the pressure (the Boltzmann distribution). For lattice models in which microscopic states are configurations of symbols on an infinite lattice, Dobrushin, Lanford and Ruelle (DLR) showed that under broad conditions, local and global maximization of the pressure lead to the same class of measures — namely (shift-invariant) Gibbs measures are the same as equilibrium measures.

I will discuss some variants and generalizations of this theorem. The strongest generalization is a "relative" version of the DLR theorem for systems in contact with a random environment, in which the pressure maximization is relative to the environment. The environment affects the interaction energies and determines the set of allowed configurations. This theorem covers classic examples of disordered systems (e.g., the Ising model with random external field) as well as examples in which hard constraints are present (e.g., Ising model on percolation clusters). The underlying lattice can be any countable amenable group. The role of hard constraints (i.e., the underlying subshift of allowed configurations) will be emphasized.

This is joint work with Sebastián Barbieri, Ricardo Gómez-Aíza and Brian Marcus [arXiv:1809.00078].