Nonlinear Energy Response of Glass Forming Materials

A theory for the nonlinear energy response of a system subjected to a heat bath is developed when the temperature of the heat bath is modulated sinusoidally. The theory is applied to a model glass forming system, where the landscape is assumed to have 20 basins and transition rates between basins obey a power law distribution. It is shown that the statistics of eigenvalues of the transition rate matrix, the glass transition temperature $T_g$, the Vogel-Fulcher temperature $T_0$ and the crossover temperature $T_x$ can be determined from the 1st- and 2nd-order ac specific heats, which are defined as coefficients of the 1st- and 2nd-order energy responses. The imaginary part of the 1st-order ac specific heat has a broad peak corresponding to the distribution of the eigenvalues. When the temperature is decreased below $T_g$, the frequency of the peak decreases and the width increases. Furthermore, the statistics of eigenvalues can be obtained from the frequency dependence of the 1st-order ac specific heat. The 2nd-order ac specific heat shows extrema as a function of the frequency. The extrema diverge at the Vogel-Fulcher temperature $T_0$. The temperature dependence of the extrema changes significantly near $T_g$ and some extrema vanish near $T_x$.