TABLE OF CONTENTS A lot of phase transitions in condensed matter are associated with broken symmetries. However, there are many other phase transitions in which symmetry breaking cannot be seen, but there is what we call broken ergodicity. Broken symmetry is seen to be a part of broken ergodicity.
In the late 19 century, Boltzmann introduce the hypothesis of ergodicity as the basis of statistical mechanics. [a] Afterwards Gibbs introduced ensemble theory to substitute for the ergodic hypothesis, but the difference is not large. We will use the weaker expression, i.e., quasi-ergodicity. This is the statistical assumption that for a many-particle system in thermal equilibrium, if the experimental time is long enough, the phase space trajectory which describes the time evolution of the system will come arbitrarily close to any specified point in the phase space accessible to the system. As a result, observed quantities are given by the average taken over all of the allowed phase space. In essence, ergodicity means that the ensemble average, i.e., the phase space average, can be used to replace the time average of the variables evolving from any single initial condition.
A condensed material includes a great number of particles composed of electrons and ions. From a dynamical point of view, we can define a microscopic state by specifying all of the dynamical variables of the system. However, only a few physical quantities, say the temperature, the pressure and the density, are usually taken to specify macroscopic state...
