Phase Transitions to Nonlinear Heat Transport - From Equilibrium Statistical Mechanics to Systems Far from Equilibrium

One of the great accomplishments of physics in the 20th century, pioneered by Boltzmann, Maxwell, Gibbs and others, was the development of statistical mechanics. Perhaps the crowning achievement of this theory was the elucidation of critical phenomena and phase transitions, aided by the development of renormalization group theory by Kadanoff, Fisher, Widom and Wilson. One essential concept in statistical mechanisms is that equilibrium is a state of minimum free energy. Many real world phenomena, however, are not in thermal or mechanical equilibrium. If the perturbations are small, one can extend statistical mechanics to weakly non-equilibrium systems and describe transport phenomena such as the transport of heat, e.g., Fourier heat transport. As the forcing away from equilibrium is further increased, perturbative approaches fail and one is confronted by the topic of systems far from equilibrium. I will present examples at each stage of this transition from equilibrium statistical mechanics to highly non-equilibrium phenomena. An important point is that for the latter systems, there is no general minimization principle that determines the realized state of the system.