Glasses have nonzero zero-temperature entropies. Because they are out of equilibrium, the ‘‘thermodynamic’’ entropy, determined by heat flow, is not equal to the ‘‘statistical’’ entropy, which measures volumes in phase space. We discuss the relationship between the two kinds of entropy in nonequilibrium systems and show that the thermodynamic entropies measured by cooling and heating form lower and upper bounds to the statistical entropy. In a computer simulation of a glass, the distribution of thermodynamic entropies measured by repeated fast coolings provides information about the dynamics of the glass. Entropy distributions are presented for a spin glass and a simple two-level system, and the distributions are used as a tool to compare the dynamics of the two models.

• Received 24 August 1989

DOI:https://doi.org/10.1103/PhysRevB.41.2261