Strong experimental evidences and computer simulations suggest a liquid-liquid transition in the supercooled confined water in two differentamorphous states, low-density amorphous ice and high-density amorphous ice, connected by a first-order phase transition and foreseen as amorphous counterparts of low-density liquid and high-density liquid, respectively, being the main evidence to support the existence of a liquid-liquid transition and a second critical point of water. The theoretical description of the extensions beyond the critical point of water can be performed by two different approaches: the multiple Widom lines and the Frenkel line. The order parameter of a liquid-liquid transition is the boson peak, upon which in supercooled confined water tracks the Widom line determined by a dynamiccrossover transition, which can also be described by the Frenkel line. A similar phenomenon occurs in the dynamic transition of macromolecules or protein glass transition. The theoretical model for the specific heat of disordered solids based on a noncommutative space conceives the origin of the boson peak as the van Hove singularity and reaffirms the experimental suggestion that the increase in specific heat of glassy materials arises from new optical modes of very low frequency. This proposal intends to establish a theoretical connection between the multiple Widom lines and Frenkel line in the liquid-liquid phase transition of supercooled confined water in biological macromolecules, and the parallel development of the phonon theory of liquid thermodynamics in a noncommutative space.
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