A key concept in glasses: the entropy crisis

The figure below introduces the idea of an "entropy crisis" and the Kauzmann temperature in glasses. It also leads to profound and controversial questions about the intimate connection between thermodynamics and kinetics in glasses.

Each solid curve shows the temperature dependence of the entropy of a supercooled liquid, relative to that of the crystal, above T_g, the glass transition temperature. T_m is the melting temperature of the crystal. The dashed curves are entropy in the glassy state.

The figure is taken from a very helpful review and adapted from Walter Kauzmann's classic 1948 paper.

What is going on?
The entropy of a liquid is greater than a solid [think latent heat of melting] so Delta S is positive. But, the specific heat capacity of a liquid is also greater than that of a solid [the vibrational, translational, and rotational degrees of freedom are all "softer" and less constrained]. Hence, the slope of Delta S vs. T must be positive.
Now, suppose that the liquid is supercooled so incredibly slowly that the glass does not form and you keep lowering the temperature, then at some temperature Delta S becomes negative. This extrapolated temperature [see the light blue straight line] is known as the Kauzmann temperature.

Why does this matter?
By the third law of thermodynamics, the entropy of the crystal goes to zero as the temperature goes to zero. Thus the supercooled liquid, could have negative entropy, which is physically nonsense.
Formation of the glass prevents this possibility. But, formation of the glass involves kinetics. So is there some deep connection between thermodynamics and kinetics? The review  discusses some possible connections. The extent of that connection is one of the controversial questions in glasses.