MARK KAC SEMINAR

In quantum mechanics, the result of a measurement is intrinsically random and induces a back-action on the measured system. The resulting law of a sequence of (indirect) measurements is a generalisation of hidden markov models. They form a weakly dense set of probability measures distinct from the set of weak Gibbs measures while still accessible through thermodynamic formalism. Motivated by physics, I am interested in understanding the large time properties of the related dynamical system. Depending on time, in this presentation I will define the probability measures involved and review some results on limit theorems (law of large numbers, central limit theorem, large deviation principle…). I will present the two main standard strategy of proof. I will then present an original approach we developed with some collaborators using sub-additivity to study the entropy production associated with the repeated measurement dynamical system. I will explain how these result can be used to formalise the hypothesis testing of the arrow of time and how it relates to out of equilibrium thermodynamics.

We study a lattice gas subject to Kawasaki dynamics at inverse temperature $\beta>0$ in a large finite box $\Lambda_\beta \subset \mathbb{Z}^2$ of size $|\Lambda_\beta| = e^{\Theta\beta}$, with $\Theta>0$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $\Delta>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta \in (U,2U)$ and in the limit as $\beta\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $\Lambda_\beta$ that subsequently grows by absorbing particles from the surrounding gas. We will see that in a very large volume ($\Theta> 2\Delta-U$) critical droplets appear more or less independently in boxes of moderate volume ($\Theta< 2\Delta-U$), a phenomenon referred to as homogeneous nucleation. This is a joint work with Alexandre Gaudillière, Frank den Hollander, Francesca Romana Nardi, Enzo Olivieri and Elisabetta Scoppola.