Fictive temperature of a glassy system in terms of frequency-dependent specific heat: a memory function approach

Upon rapid quenching of the temperature of a glass forming liquid, the system falls out of equilibrium due to its finite relaxation time. Additionally, the relaxation becomes progressively slower with time. The created nonequilibrium state of the glassy system is conveniently described by introducing a fictive temperature, which describes the properties of the instantaneous state of the nonequilibrium system. The fictive temperature T_f(t) is time dependent. During cooling, the fictive temperature is higher than the actual temperature. After the cooling or quenching has ceased, the fictive temperature approaches the final temperature at a rate that depends on the relaxation properties of the liquid. In this work, we use linear response theory to connect the time dependence of the fictive temperature to the memory function, which is shown to be related to the frequency-dependent specific heat, which itself depends on the fictive temperature T_f(t). Thus, one requires a self-consistent calculation to capture the interdependence of the relaxation rate and the structural response function. We present a numerical calculation where we apply our relations to silica, where the relaxation function that describes the frequency-dependent specific heat is modeled as a stretched exponential William-Watts (WW) function, while the relaxation time is modeled as a Vogel-Fulcher-Tammann (VFT) function. We calculate the fictive temperature self-consistently. T_f(t) exhibits the fall out from the actual temperature as time (t) progresses.