Ergodicity: How Can It Be Broken?

The introduction of the ergodic hypothesis can be traced back to the contributions by Boltzmann to the foundations of Statistical Mechanics. The formulation of this hypothesis was at the origin of a long standing debate between supporters and opponents of the Boltzmann mechanistic formulation of thermodynamics. The great intuition of the Austrian physicist nevertheless inspired the following contributions that aimed at establishing rigorous mathematical basis for ergodicity. The first part of this chapter will be devoted to reconstructing the evolution of the concept of ergodicity, going through the basic contributions by Birkhoff, Khinchin, Kolmogorov, Sinai etc. The second part will be focused on more recent case studies, associated with the phenomenon known as “ergodicity breaking” and its relations with physical systems. In particular, we describe how it can be related to the presence of exceedingly large relaxation time scales that emerge in nonlinear systems (e.g., the Fermi-Pasta-Ulam model and the Discrete Nonlinear Schrödinger Equation) and to the coexistence of more than one equilibrium phase in disordered systems (spins and structural glasses).