Self-Organized Criticality and Thermodynamic formalism

We introduce a dissipative version of the Zhang's model of Self-Organized Criticality, where a parameter allows to tune the local energy dissipation. We analyze the main dynamical features of the model and relate in particular the Lyapunov spectrum with the transport properties in the stationary regime. We develop a thermodynamic formalism where we define formal Gibbs measure, partition function and pressure characterizing the avalanche distributions. We discuss the infinite size limit in this setting. We show in particular that a Lee-Yang phenomenon occurs in this model, for the only conservative case. This suggests new connexions to classical critical phenomena.