The glass transition as a boundary layer problem of the energy landscape kinetic master equations and its relationship with quantum path integrals

The glass transition is a process in which a non-periodic solid is obtained by fast cooling a melt. In this work, an analysis is made of the energy landscape kinetic equations that allow to describe such phenomena. In particular, it is shown that the glass transition occurs at the boundary layer of the energy landscape kinetic master equations, as solutions always have steep gradients and are very sensitive to the initial conditions. Moreover, it is discussed that the evolution operator is given by a path integral where the cooling rate plays a role akin to the Planck constant in a quantum mechanical problem. As an example, the kinetic equation properties of a two-level fully connected energy landscape system, capable of exhibiting both a glass and a thermodynamic phase transition, both of which are analytically solvable, are analysed in detail.