Temperature dependence nonergodicity

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

In the previous section, at temperatures above 7g, the Johari-Goldstein relaxation time has been shown to correspond well to the primitive relaxation time, and both are related to the structural a-relaxation time by Eq. (10). This equation should continue to hold at temperatures below T,. However, testing this relation in the glassy state is difficult because of either the scarcity or the unspecified thermal history of the data on the a-relaxation time xa. In fact, a reliable characterization of the structural relaxation can be acquired only at equilibrium, and such condition is rarely satisfied below Tg. Glassy systems are nonergodic, and their properties can depend on aging time and thermal history. Anyway, for glasses in isostructural state with a constant Active temperature Tf, both ra and To as well as Xjg should have Arrhenius dependences with activation enthalpies Ea, Eq, and Ejg respectively. Eq. (10) leads us to the relation... 

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