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A theory for the nonlinear energy response of a system subjected to a heatbath is developed when the temperature of the heat bath is modulatedsinusoidally. The theory is applied to a model glass forming system, where thelandscape is assumed to have 20 basins and transition rates between basins obeya power law distribution. It is shown that the statistics of eigenvalues of thetransition rate matrix, the glass transition temperature $T_g$, theVogel-Fulcher temperature $T_0$ and the crossover temperature $T_x$ can bedetermined from the 1st- and 2nd-order ac specific heats, which are defined ascoefficients of the 1st- and 2nd-order energy responses. The imaginary part ofthe 1st-order ac specific heat has a broad peak corresponding to thedistribution of the eigenvalues. When the temperature is decreased below $T_g$,the frequency of the peak decreases and the width increases. Furthermore, thestatistics of eigenvalues can be obtained from the frequency dependence of the1st-order ac specific heat. The 2nd-order ac specific heat shows extrema as afunction 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 $T_x$.Comment: 20 pages, 10 figure

Nonlinear energy response of glass forming materials