Thermodynamic formalism in the thermodynamic limit: Diffusive systems with static disorder

The chaotic properties of diffusive systems with static disorder can be calculated from a free-energy-type function, the Ruelle pressure, $\psi \left(\beta \right)$ depending on an inverse temperaturelike variable, $\beta$. For a typical system of physical interest, we show that, in the thermodynamic limit, the Ruelle pressure has a singularity and two branches (a high and low temperature "phase"), corresponding to transitions between different localized states, with an extended state possible at the transition point. More generally, for all systems with static disorder in any number of dimensions, the Ruelle pressure depends sensitively on rare atypical fluctuations in the static disorder, and is independent of the global structure of the disorder that determines the transport coefficients.