Thermodynamic picture of the glassy state gained from exactly solvable models

TL;DR

This paper presents an exact analysis of simple models to understand the thermodynamics of the glassy state, emphasizing the role of an effective temperature and verifying related fluctuation relations.

Contribution

It provides an exact dynamical solution for two models, demonstrating the thermodynamic relations in the glassy regime and discussing the concept of effective temperature.

Findings

Relaxation time diverges as an Arrhenius law at low temperature.

Verified fluctuation relations for the glassy state.

Stretched exponential behavior is not fundamental to glasses.

Abstract

A picture for thermodynamics of the glassy state was introduced recently by us (Phys. Rev. Lett. {\bf 79} (1997) 1317; {\bf 80} (1998) 5580). It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, non-equilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper the underlying arguments are discussed in greater length. The main part of the paper involves details of the exact dynamical solution of two simple models introduced recently: uncoupled harmonic oscillators subject to parallel Monte Carlo dynamics, and independent spherical spins in a random field with such dynamics. At low temperature the relaxation time of both models diverges as an Arrhenius law, which causes glassy behavior in typical situations. In the glassy regime we are able to verify the above mentioned relations for the thermodynamics of the glassy state. In the course of the analysis it is argued that stretched exponential behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime.