In statistical mechanics and condensed matter physics, the Kovacs effect is a kind of memory effect in glassy systems below the glass-transition temperature. A.J. Kovacs observed that a system's state out of equilibrium is defined not only by its macro thermodynamical variables, but also by the inner parameters of the system. In the original effect, in response to a temperature change, under constant pressure, the isobaric volume and free energy of the system experienced a recovery characterized by non-monotonic departure from equilibrium, whereas all other thermodynamical variables were in their equilibrium values. It is considered a memory effect since the relaxation dynamics of the system depend on its thermal and mechanical history. The effect was discovered by Kovacs in the 1960s in polyvinyl acetate. Since then, the Kovacs effect has been established as a very general phenomenon that comes about in a large variety of systems, model glasses,
tapped dense granular matter, spin-glasses, molecular liquids, granular gases, active matter, disordered mechanical systems, protein molecules, and more.
The effect in Kovacs' experiments Kovacs' experimental procedure on polyvinyl acetate consisted of two main stages. In the first step, the sample is instantaneously quenched from a high initial temperature T 0 {\displaystyle T_{0}} to a low reference temperature T r {\displaystyle T_{r}} , under constant pressure. The time-dependent volume of the system in T r {\displaystyle T_{r}} , V ( t ) | T r {\displaystyle V(t)|_{T_{r}}} , is recorded, until the time t e q {\displaystyle t_{eq}} when the system is considered to be at equilibrium. The volume at t e q {\displaystyle t_{eq}} is defined as the equilibrium volume of the system at temperature T r {\displaystyle T_{r}} :
V ( t e q ) | T r ≡ V e q ( T r ) {\displaystyle V(t_{eq})|_{T_{r}}\equiv V_{eq}(T_{r})}
In the second step, the sample is quenched again from T 0 {\displaystyle T_{0}} to a temperature T 1 {\displaystyle T_{1}} that is lower than T r {\displaystyle T_{r}} , so that T 0 > T r > T 1 {\displaystyle T_{0}>T_{r}>T_{1}} . But now, the system is held at temperature T 1 {\displaystyle T_{1}} only until the time t 1 {\displaystyle t_{1}} when its volume reaches the equilibrium value of T r {\displaystyle T_{r}} , meaning V ( t 1 ) | T 1 = V e q ( T r ) {\displaystyle V(t_{1})|_{T_{1}}=V_{eq}(T_{r})} . Then, the temperature is raised instantaneously to T r {\displaystyle T_{r}} , so both the temperature and the volume agree with the same equilibrium state. Naively, one expects that nothing should happen when the system is at V = V e q ( T r ) {\displaystyle V=V_{eq}(T_{r})} and T = T r {\displaystyle T=T_{r}} . But instead, the volume of the system first increases and then relaxes back to V e q ( T r ) {\displaystyle V_{eq}(T_{r})} , while the temperature is held constant at T r {\displaystyle T_{r}} . This non-monotonic behavior in time of the volume V ( t ) {\displaystyle V(t)} after the jump in the temperature can be simply captured by:
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