The Kovacs effect: a master equation analysis

TL;DR

This paper analyzes the Kovacs effect, a non-monotonic relaxation phenomenon in glasses, using master equations and linear response theory, providing analytical insights and a simple model illustration.

Contribution

It offers a general master equation framework for understanding the Kovacs effect and derives explicit analytical results for a two-level system with disorder.

Findings

Reproduces experimental features of the Kovacs effect

Derives an explicit relaxation function showing resonance behavior

Validates the approach with a simple two-level model

Abstract

The Kovacs or crossover effect is one of the peculiar behaviours exhibited by glasses and other complex, slowly relaxing systems. Roughly it consists in the non-monotonic relaxation to its equilibrium value of a macroscopic property of a system evolving at constant temperature, when starting from a non-equilibrium state. Here, this effect is investigated for general systems whose dynamics is described by a master equation. To carry out a detailed analysis, the limit of small perturbations in which linear response theory applies is considered. It is shown that, under very general conditions, the observed experimental features of the Kovacs effect are recovered. The results are particularized for a very simple model, a two-level system with dynamical disorder. An explicit analytical expression for its non-monotonic relaxation function is obtained, showing a resonant-like behaviour when the dependence on the temperature is investigated.