Thermodynamic picture of the glassy state gained from exactly solvable models

A picture for thermodynamics of the glassy state was introduced recently by us [Phys. Rev. Lett. 79, 1317 (1997); 80, 5580 (1998)]. It starts by assum ing that one extra parameter, the effective temperature, is needed to descr ibe the glassy state. This approach connects responses of macroscopic obser vables to a field change with their temporal fluctuations, and with the flu ctuation-dissipation relation, in a generalized, nonequilibrium way. Simila r universal relations do not hold between energy fluctuations and the speci fic heat. In the present paper, the underlying arguments are discussed in g reater length. The main part of the paper involves details of the exact dyn amical solution of two simple models introduced recently: uncoupled harmoni c oscillators subject to parallel Monte Carlo dynamics, and independent sph erical spins in a random field with such dynamics. At low temperature, the relaxation time of both models diverges as an Arrhenius law, which causes g lassy behavior in typical situations. In the glassy regime, we are able to verify the above-mentioned relations for the thermodynamics of the glassy s tate. In the course of the analysis, it is argued that stretched exponentia l behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime.