Thermodynamic formalism and localization in Lorentz gases and hopping models

The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure $\psi (\beta )$. The inverse-temperature like variable $\beta$ allows one to scan the structure of the probability distribution in the dynamic phase space. This formalism is applied here to a Lorentz Lattice Gas, where a particle moving on a lattice of size $L^d$ collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinit e systems has two branches joining with a slope discontinuity at $\beta =1$. The low and high $\beta$–branches correspond to localization of trajectories on respectively the most chaotic'' (highest density) region, and the most deterministic’’ (lowest density) region, i.e. $\psi (\beta )$ is completely controlled by rare fluctuations in the distribution of scatterers on the lattice, and it does not carry any information on the global structure of the static disorder. As $\beta$ approaches unity from either side, a localization-delocalization transition leads to a state where trajectories are extended and carry information on transport properties. At finite $L$ the narrow region around $\beta =1$ where the trajectories are extended scales as $(\ln L{)}^{-\alpha }$, where $\alpha$ depends on the sign of $1-\beta$, if $d>1$, and as $(L\ln L)^{-1}$if$d=1$. This result appears to be general for diffusive systems with static disorder, such as random walks in random environments or for the continuous Lorentz gas. Other models of random walks on disordered lattices, showing the same phenomenon, are discussed.