Abstract:
The Mpemba effect is an anomalous relaxation process where an initially hotter system is quenched into a cold bath and reaches equilibrium faster than an initially cooler system.
Mpemba and Obsborne observed that a quenched liquid starts freezing all the quicker as its initial temperature increases [1].
Although there are debates about the existence of such an anomalous relaxation, experimental evidence supports the Mpemba effect for both classical [2] and quantum systems [3].
Then, the physics of the Mpemba effect attracts many researchers (see reviews [4,5]).
I have also published several papers on the Mpemba effect for both classical and quantum systems [6-9].
In this talk, I briefly review the current status of the Mpemba effect and introduce a simple, exactly solvable model using the Fokker-Planck equation in a one-dimensional bistable continuous potential system with an explanation of how to solve it.
I also try to extend this analysis to a two-dimensional system.

[1] A. Kumar and J. Bechhoefer, Nature 584, 64 (2020).
[2] L. Kh. Joshi et al., PRL 133, 010402 (2024); S. A. Shapira et al, PRL 133, 010403 (2024).
[4] G. Teza et al., arXiv:2502.01758.
[5] F. Ares et al., Nat. Rev. Phys. 7, 451 (2025).
[6] S. Takada, H. Hayakawa, and A. Santos, PRE 103, 032901 (2021).
[7] A. K. Chatterjee, S. Takada, and H. Hayakawa, PRL 131, 080402 (2023).
[8] A. K. Chatterjee, S. Takada, and H. Hayakawa, PRA  110, 022213 (2024) [9] T. Van Vu, and H. Hayakawa, PRL 134, 107101 (2025).

Prof. Hisao Hayakawa from Yukawa Institute for Theoretical Physics, Kyoto University