Most of our intuition about the behavior of physical systems is shaped by observations at or near thermal equilibrium. However, even a basic phenomenon such as a thermalquench can lead to states far from thermal equilibrium, where counterintuitive,anomalous effects can occur. A prime example of anomalous thermal relaxation is the Mpemba effect, in which a system prepared at a hot temperature cools down to the temperature of the cold environment faster than an identical system prepared at a warm temperature. Although originally reported for water more than 2000 years ago by Aristotle, the recent observation of analogous relaxation speedups in a variety of systems has motivated the search for generic explanations from the point of view of nonequilibrium statistical mechanics. Here, we review anomalous relaxation effects, which all share a nonmonotonic dependence of relaxation time versus initial “distance” from the final state or from the phase transition. We review the early water experiments and classify the zoology of anomalous relaxation phenomena related to the Mpemba effect. We then introduce general concepts and provide a modern definition of the effect, focusing on the theoretical frameworks of stochastic thermodynamics, kinetic theory, Markovian dynamics, and phase transitions. We discuss the recent experimental and numerical developments that followed these theoretical advances. These developments paved the way for the prediction and observation of novel phenomena, such as the inverse Mpemba effect. The review is self-contained and introduces anomalous relaxation phenomena in single- and many-body systems, both classical and quantum. We also discuss the broader relevance of the Mpemba effect, including its relation with equilibrium and dynamical phase transitions and its experimental implications. We end with perspectives that connect anomalous speedups to new ideas for designing optimal heating/cooling protocols, heat engines, and efficient samplers.
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