Hysteresis

Hysteresis is the property of a system whose present state depends on its history of inputs, not merely on the current values of those inputs. The same external conditions can produce qualitatively different internal states depending on the path taken to reach them. The classical example is a ferromagnet: the magnetization at a given temperature and field strength differs depending on whether the field was increased from zero or decreased from saturation — the magnet remembers its past.

The memory is encoded in the system's internal structure, not in an external record. In ferromagnets, magnetic domains are trapped in local free-energy minima separated by barriers; reversing the magnetization requires surmounting these barriers, which the applied field may not be strong enough to overcome even when the equilibrium state has changed. This makes hysteresis a special case of metastability: the system occupies a local minimum that is stable against the present input but would not have been reached from a different history.

Hysteresis appears across domains far beyond magnetism. In ecology, predator-prey systems can exhibit hysteretic recovery: a lake eutrophied by nutrient runoff may not return to its clear state even when nutrient loading is reduced below the threshold that caused the bloom, because the algal community has shifted to a stable alternative state. In climate science, ice-albedo feedback creates hysteresis in the Earth's climate: once polar ice sheets collapse, they may not regrow at the same CO₂ concentration that previously sustained them. In social systems, institutional norms exhibit hysteresis — trust, once lost, requires more evidence to restore than was needed to maintain it.

The mathematical signature of hysteresis is a loop in the input-output plane rather than a single-valued function. The area of the loop represents dissipated energy (in physical systems) or accumulated historical cost (in social systems). Hysteresis is therefore not merely a path dependence but a thermodynamic or organizational cost of history — the system pays a price for every cycle through its state space.