A magnet that is magnetised then demagnetised does not return to the same state as before. The coercive field required to cancel the magnetisation differs from the one that created it. The ascending curve and the descending curve do not overlap. Ewing (1881) names this loop: hysteresis. From the Greek ὑστέρησις, delay. The system's present lags behind its equilibrium state. It carries the trace of its past.
The phenomenon is not magnetic. It is structural. A metal bent then straightened is no longer the same metal. The dislocations introduced by plastic deformation do not disappear when the load is removed. The Bauschinger effect (1886) shows that the elastic limit in compression after tension is lower than the initial elastic limit. The material remembers the direction in which it was forced.
The glass transition is hysteretic. The cooling path and the heating path do not coincide. The transition temperature $T_g$ depends on the rate, not only on the direction. A glass cooled quickly and a glass cooled slowly do not have the same volume, the same enthalpy, or the same residual stress state. The thermal history is inscribed in the structure.
Baldwin and Krugman (1989) identify hysteresis in international economics. An exchange rate that depreciates drives exporters out of the market. When the rate returns to its initial value, the exporters do not come back. Lost market shares are not restored. Entry and exit have different thresholds. The system has a memory that the instantaneous state variables do not contain.
In each case, the same structure: the present state of the system is not sufficient to describe it. One needs the path that brought it there. Two systems in the same macroscopic state, with the same temperature, the same pressure, the same composition, can behave differently if their histories differ. The state is the same. The system is not.
Doctrine
A system with hysteresis is a system whose present does not contain enough information to predict the future. The past acts on the future through the present without being visible in the present. Memory is not a recording. It is a deformation that persists.
We work with materials that remember. Each operation leaves a trace that the next operation cannot erase. The process has a direction that is not that of time. It is that of irreversibility inscribed in matter.
Vecteur ouvert
If the path matters as much as the state, then the complete description of a system requires its history. A system with hysteresis is a system with multiple boundary conditions, not only the beginning and the end, but every intermediate turning point. The question is how much past must be retained to predict the future, and whether that quantity is finite.
