Under certain conditions, a counterintuitive behavior—an initially hotter sample freezes faster when quenched to a cold bath than an identical system initialled at a lower temperature—is known as the Mpemba effect (ME). Here we identify the existence of the ME in mean-field systems (MFS). Specifically, the thermal contact between MFS and a large thermal reservoir is built up using the microcanonical Monte Carlo algorithm. The simulation results unambiguously demonstrate that an initial hotter system undergoes the paramagnetic-ferromagnetic phase transition faster than the initial cooler one. The ME here originates from the back-reaction of the MFS system on the reservoir, which is thus an embodiment of non-Markovianness in relaxation. In addition, we confirm that the ME survives in the thermodynamic limit. And the significance of reservoir size is also explored: A smaller heat reservoir facilitates the overall relaxation process. In general, this work establishes a theoretical non-Markovian ME framework, which may shed light on widening the understanding of the mechanism behind the ME in other substances, including water.

The Mpemba effect is the phenomenon in which the system with high initial temperature cools faster than the system with low initial temperature when all other conditions are the same. A theoretical model of the Mpemba effect through the canonical first-order phase transition is proposed in this paper, which shows that in the cooling processes, the path of the first-order phase transition of the system with the high initial temperature does not pass through any metastable state, while the path of the first-order phase transition of the system with the low initial temperature passes through a metastable state, which leads to the occurrence of the Mpemba effect. Then an example of the theoretical model is given in the Blume-Emery-Griffiths model. The Monte Carlo algorithm is adopted to calculate the estimated times for both systems with different initial temperature to cool down and undergo a first-order phase transition. The simulation results demonstrate a Mpemba effect in the system. Moreover, the evolution paths of the first-order phase transitions of the systems with high and low initial temperatures are given, respectively. The theoretical model presented here may help explain the Mpemba effect in water.

The counterintuitive phenomenon—that an initially hotter water freezes faster than initially cooler water—is named the “Mpemba effect.” Although it has been known for centuries, the underlying mechanism remains unclear. Recently, the Mpemba effect rekindled the interest of researchers since several studies identified that it might occur in some Markovian systems, and a general statistical-physical Mpemba effect framework was correspondingly proposed. In our previous study [Z.-Y. Yang and J.-X. Hou, Phys. Rev. E 101, 052106 (2020)], we observed the non-Markovian Mpemba effect in a mean-field system (MFS), where the Mpemba effect originates from the back-reaction of the thermal reservoir. Naturally, the phase transition time is the key to the occurrence of the Mpemba effect, which, however, has not been quantitatively described. Following the direction of previous work, this study rigorously derives the phase transition time under different conditions, and quantitatively describes the mechanism of the non-Markovian Mpemba effect in a MFS. In addition, the validation of our theory was further verified via the microcanonical Monte Carlo simulation. An accurate description of the underlying mechanism of our proposed MFS facilitates the generalization of the Mpemba effect framework in statistical physics and may benefit in answering the riddle of the century, the original Mpemba effect in water.

For a long-range interacting spin chain model, the microcanonical ensemble predicts a region of negative specific heat and a temperature jump at the transition energy. After two similar long-range interacting subsystems of different size at different temperatures are weakly coupled, they exchange energy and the total microcanonical entropy of the full system increases irreversibly. The hot subsystem could spontaneously absorb heat from the cold subsystem via the thermal contact and the final equilibrium temperature could be lower than the initial temperatures of the cold subsystem. This result is confirmed by numerical simulations using the microcanonical Monte Carlo algorithm.