The Kauzmann Transition to an Ideal Glass Phase

The idea that a thermodynamic glass transition of some sort underlies the observed glass formation has been highly debated since Kauzmann first stressed the hypothetical entropy crisis that could take place if one were able to equilibrate supercooled liquids below the experimental glass transition temperature $T_g$. This a priori unreachable transition at some $T_K<T_g$ has since received a firm theoretical basis as a key feature predicted by the mean-field theory of the glass transition. In this chapter, we assess whether, and in which form, such a transition can survive in finite dimensions, and we review some of the recent computer simulation work addressing the issue in $2$- and $3$-dimensional glass-forming liquid models. We also discuss theoretical reasons to focus on an apparently inaccessible singularity.