Speedups in nonequilibrium thermal relaxation: Mpemba and related effects

TL;DR

This review surveys anomalous nonequilibrium thermal relaxations, centering on the Mpemba effect as a nonmonotonic approach to equilibrium or steady states. It links three theoretical frameworks—Markovian dynamics, phase-transition theory, and kinetic theory of granular/molecular gases—showing how spectral properties, metastability, and energy-transport couplings yield faster relaxation from hotter initial conditions under various circumstances. The authors compile experimental demonstrations (notably in colloids and quantum ion traps) and numerical observations, and they discuss strong, inverse, and boundary-coupling variants, as well as statistical aspects across model ensembles. Beyond fundamental interest, the work outlines practical implications for optimal heating/cooling protocols, heat-engine efficiency, state preparation, and computational sampling, highlighting the broad relevance of anomalous relaxation phenomena in both classical and quantum settings.

Abstract

Most of our intuition about the behavior of physical systems is shaped by observations at or near thermal equilibrium. However, even a thermal quench 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 reported for water more than 2000 years ago by Aristotle, the recent observations of analogous relaxation speedups in a variety of systems have motivated the search for general explanations. 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. The final state can be an equilibrium or a nonequilibrium steady state. We first review the water experiments and classify the anomalous relaxation phenomena related to the Mpemba effect. We then provide a modern definition of the Mpemba 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 phase transitions and its experimental implications. We end with perspectives that connect anomalous speedups to ideas for designing optimal heating/cooling protocols, heat engines, and efficient samplers.

Figures (55)

• Figure 1: Time to start freezing as a function of initial sample temperature. Source: Reprinted with permission from mpemba1969cool.
• Figure 2: Sketch of an equilibrium locus in a high-dimensional distribution space. Each point along the blue-to-red line represents a Boltzmann equilibrium $\pi(T)$ at a different bath temperature $T\in[T_0,T_b]$. A quasistatic protocol brings the system from an equilibrium configuration $T_0$ to an equilibrium temperature $T_b$ via a path closely following the equilibrium locus. A thermal quench generally leads the evolution through a path (black line), along which the system explores configurations that do not correspond to any Boltzmann equilibrium.
• Figure 3: Sketch of the evolution of a distance measure $D$ in a thermal quench towards a bath temperature $T_b$. The red line describes the evolution of a system prepared at an initial Boltzmann equilibrium $\pi(\vec{x},T_h)$ at a hot temperature $T_h$, while the blue line results from a quench from a Boltzmann equilibrium $\pi(\vec{x},T_c)$ at a cold temperature $T_c$. The Mpemba effect is indicated by the crossing of the distance measures at time $t^*$. The small insets illustrate the different shapes of Boltzmann equilibria in a system described by the dashed-line potential, $U(\vec{x})$.
• Figure 4: (a) Sketch of an energy landscape with three metastable states. Each state's energy is denoted by $E_i$, and the barrier between two states is denoted by $B_{i\!j}$. (b) An effective description of the energy landscape in (a) as a three-state system, where $R_{i\!j}$ denotes the transition rate from $j$ to $i$ and is given by Eq. \ref{['eq:arrhenius']}. Source: Reprinted with permission from lu2017nonequilibrium.
• Figure 5: (a) The probability distribution among the three states can be described by the vector $\vec{p} = (p_1 , p_2 , p_3 )$, and all possible values of $\vec{p}$ form a simplex in $(p_1 , p_2 , p_3)$ (shaded triangle). The curved line is the quasistatic locus, namely the set of Boltzmann distributions $\vec{\pi}(T)$ corresponding to different temperatures from $0$ (blue end) to $\infty$ (red end). (b) The projection coefficient $a_2(T_0,T_b)$ at $T_b=0.1$ exhibits a nonmonotonic dependence on $T_0$, with $\partial_{T_0}a_2(T_0,T_b)=0$ at $T_0\simeq 0.42$, which is a sufficient condition to prove the existence of a Mpemba effect in this system. (c) The set of all normalized probability distributions forms a simplex, the dashed blue triangle. The red solid curve is the quasistatic locus $\vec{\pi}(T)$. The dashed arrows are the two right eigenvectors of the rate matrix, $\mathbf{W}$, $\vec{v} _2$, and $\vec{v} _3$, associated with slow (green) and fast (blue) relaxation modes. The dotted lines represent the relaxation process, $\vec{p} ^{\, c} (t)$ starting at $T_c =0.42$ (orange) and $\vec{p} ^{\, h} (t)$ starting at $T_h = 1.3$ (purple). In this example with $T_b=0.1$, the ratio $\frac{a_2(T_0,T_b) \lambda_2(T_b)}{a_3(T_0,T_b) \lambda_3(T_b)}\lesssim 10^{-3}$ for every $T_0>0$, implying that, in the first stages of the relaxation, the out-of-equilibrium relaxation is along $\vec{v}_3$. The inset shows that the distances ${D} [\vec{p} ^{\, h}(t),\vec{\pi}(T_b) ]$ and ${D} [\vec{p} ^{\, c} (t), \vec{\pi}(T_b) ]$ both decrease with time. The initially hot system starts at a larger distance, but after some time, its distance from equilibrium is smaller than that of the initially cold system. Source: Panels (a) and (c) are reprinted with permission from lu2017nonequilibrium.