Self-organized criticality and thermodynamic formalism

We develop a thermodynamic formalism for a dissipative version of the Zhang model of Self-Organized Criticality, where a parameter allows us to tune the local energy dissipation. By constructing a suitable Markov partition we define Gibbs measures (in the sense of Sinai, Ruelle, and Bowen), partition functions, and topological pressure allowing the analysis of probability distributions of avalanches. We discuss the infinite-size limit in this setting. In particular, we show that a Lee-Yang phenomenon occurs in the conservative case. This suggests new connections to classical critical phenomena.