Fictive temperature and the glassy state

Within the glass science community, it is common to describe the state of a glass in terms of a fictive temperature or a distribution of fictive temperatures. However, a number of different definitions of fictive temperature are being practiced in the field. Based on the previous literature, at least three definitions are possible: (a) microscopic, which entails a mapping of the nonequilibrium glassy structure to comparable equilibrium liquid structures; (b) macroscopic, by representing the property values of a glass in terms of equilibrium states with equivalent configurational property values; and (c) kinetic, where the fictive temperatures are used to represent the various relaxation modes within the glass. Of these, the first, microscopic definition offers the possibility of writing a simplified statistical mechanical model of the nonequilibrium glassy state in terms of a linear combination of equilibrium liquid states. However, in this paper, we show that the microscopic physics of a glass cannot, in general, be described in this manner. The fictive temperature description of the glassy state is rigorous only under special circumstances, such as the case of an infinitely fast quench through the glass transition regime. Nevertheless, the microscopic definition of fictive temperature can provide a reasonable description of ensemble-averaged state properties such as enthalpy and molar volume, where there is a cancellation of errors. The concept of fictive temperature distribution cannot, however, capture accurately the fluctuations in enthalpy and molar volume in the glassy state. We also show that fictive temperature mapping does not provide an accurate description of the low-temperature dynamics of glass. An alternative description of the glassy state in terms of an enthalpy landscape can account for the true statistical mechanics of the glassy state without relying on any notion of fictive temperature.