Solving the initial condition of the string relaxation equation of the string model for glass transition: part-I

The string model for the glass transition canquantitatively describe the universal $alpha $-relaxation inglassformers, including the average relaxation time, thedistribution function of the relaxation time, and the relaxationstrength as functions of temperature. The string relaxation equation(SRE) of the model, at high enough temperatures, simplifies to thewell-known single particle mean-field Debye relaxation equation, andat low enough temperatures to the well-known Rouse--Zimm relaxationequation that describes the relaxation dynamics of linearmacromolecules. However, its initial condition, necessary to thefurther model predictions of glassy dynamics, has not been solved.In this paper, the special initial condition (SIC) of the SRE, i.e.for straight strings and the dielectric spectrum technique that isone of the most common methods to measure the glassy dynamics, wassolved exactly. It should be expected that the obtained SIC wouldbenefit the solution of the general initial condition of the SRE ofthe string model, i.e. for stochastically spatially configuratingstrings, as will be described in separate publications.