Hysteresis induced by incomplete relaxation under finite-rate disturbance

Hysteresis, a widespread occurrence in both natural and artificial systems, is typically discussed under the quasistatic limit. However, studying the hysteresis behavior of complex systems in real world is highly challenging due to the involvement of nonequilibrium processes. In this work, we report that a stochastic model based on competing firms in a market, which exhibits a first-order phase transition, may also show hysteresis effects when subjected to a disturbance applied at a finite rate. We find that the cause of such phenomena is the incomplete relaxation, which continuously delays the system's responding dynamics. The strength of hysteresis effects can be measured by the ratio of characteristic timescales of relaxation and disturbance. This study provides an additional perspective for understanding hysteresis behavior and offers assistance in studying how complex systems respond to environmental changes.