The exact solutions of simple models allow us to obtain the large deviation functions of density profiles and of the current through simple systems in contact with two reservoirs at different densities. These simple models show that non-equilibrium systems have a number of properties which contrast with equilibrium systems: phase transitions in one dimension, non local free energy functional, violation of the Einstein relation between the compressibility and the density fluctuation, non-Gaussian density fluctuations. They also lead to a general expression for the current fluctuations through a diffusive system in contact with two reservoirs.
B Derrida, J L Lebowitz, E R Speer Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case Phys. Rev. Lett. 87, 150601 (2001)
B Derrida, B Doucot, P-E Roche, Current fluctuations in the one dimensional Symmetric Exclusion Process with open boundaries J. Stat. Phys. 115, 717-748 (2004)
T. Bodineau, B Derrida Current fluctuations in non-equilibrium diffusive systems: an additivity principle Phys. Rev. Lett. 92, 180601 (2004)