Non-equilibrium thermodynamics. IV: Generalization of Maxwell, Claussius-Clapeyron and Response Functions Relations, and the Prigogine-Defay Ratio for Systems in Internal Equilibrium

TL;DR

This paper generalizes classical thermodynamic relations for non-equilibrium systems in internal equilibrium, clarifies the nature of glass transitions, and evaluates the Prigogine-Defay ratio, revealing its typical deviation from unity except at certain transitions.

Contribution

It introduces a generalized framework for Maxwell, Clausius-Clapeyron, and response function relations in non-equilibrium systems with internal variables, extending classical thermodynamics.

Findings

Generalized Maxwell and Clausius-Clapeyron relations for systems with internal variables.

Identification of multiple types of glass transitions, including continuous and zeroth order.

Prigogine-Defay ratio equals 1 only at the conventional transition L→gL.

Abstract

We follow the consequences of internal equilibrium in non-equilibrium systems that has been introduced recently [Phys. Rev. E 81, 051130 (2010)] to obtain the generalization of Maxwell's relation and the Clausius-Clapeyron relation that are normally given for equilibrium systems. The use of Jacobians allow for a more compact way to address the generalized Maxwell relations; the latter are available for any number of internal variables. The Clausius-Clapeyron relation in the subspace of observables show not only the non-equilibrium modification but also the modification due to internal variables that play a dominant role in glasses. Real systems do not directly turn into glasses (GL) that are frozen structures from the supercooled liquid state L; there is an intermediate state (gL) where the internal variables are not frozen. Thus, there is no single glass transition. A system possess several kinds of glass transitions, some conventional (L \rightarrow gL; gL\rightarrow GL) in which the state change continuously and the transition mimics a continuous or second order transition, and some apparent (L\rightarrow gL; L\rightarrow GL) in which the free energies are discontinuous so that the transition appears as a zeroth order transition, as discussed in the text. We evaluate the Prigogine-Defay ratio {\Pi} in the subspace of the observables at these transitions. We find that it is normally different from 1, except at the conventional transition L\rightarrow gL, where {\Pi}=1 regardless of the number of internal variables.