Mpemba Index and Anomalous Relaxation

The Mpemba effect is a counterintuitive relaxation phenomenon, where a system prepared at a hot temperature cools down faster than an identical system initiated at a cold temperature when both are quenched to an even colder bath. Such nonmonotonic relaxations are observed in various systems, including water, magnetic alloys, polymers, and driven granular gases. We analyze the Mpemba effect in Markovian dynamics and discover that a stronger version of the effect often exists for a carefully chosen set of initial temperatures. In this strong Mpemba effect, the relaxation time jumps to a smaller value leading to exponentially faster equilibration dynamics. The number of such special initial temperatures defines the Mpemba index, whose parity is a topological property of the system. To demonstrate these concepts, we first analyze the different types of Mpemba relaxations in the mean-field antiferromagnetic Ising model, which demonstrates a surprisingly rich Mpemba-phase diagram. Moreover, we show that the strong effect survives the thermodynamic limit and that it is tightly connected with thermal overshoot; in the relaxation process, the temperature of the relaxing system can decay nonmonotonically as a function of time. Using the parity of the Mpemba index, we then study the occurrence of the strong Mpemba effect in a large class of thermal quench processes and show that it happens with nonzero probability even in the thermodynamic limit. This study is done by introducing the isotropic model for which we obtain analytical lower bound estimates for the probability of the strong Mpemba effects. Consequently, we expect that such exponentially faster relaxations can be observed experimentally in a wide variety of systems.

Popular Summary

In the Mpemba effect, a hot system cools down faster than an identical system starting at a cold temperature. First observed in water, the effect has recently been seen in other liquids, magnetic alloys, and granular gases. This raises the question as to whether a generic mechanism can explain all these effects. Using a mathematical analysis, we find that the Mpemba effect is a nonequilibrium shortcut in the relaxation process and that there are two strengths for the Mpemba effect, which are associated with different shortcut directions.

We reveal a phenomenon called the strong Mpemba effect, in which cooling from specific temperatures is exponentially faster and where the relaxation shortcut is in a different direction than for any other temperature. This is different from the weak Mpemba effect, where the system equilibrates to the environment from the same direction as if there were no effect, just somewhat faster.

The strong Mpemba effect is not only an interesting observation but also an important theoretical tool: It is robust against small changes in the system and is a common occurrence. Using a model system, we find a rich Mpemba phase diagram. We observe that the strong effect survives in the thermodynamic limit, which implies that it may be relevant for real-world, macroscopic systems.

The Mpemba effect is an example of a nonintuitive “shortcut” in the thermal relaxation of physical systems. Anomalous relaxations of physical systems are not well understood and potentially useful in optimization protocols (such as cooling, heating, and efficient sampling algorithms). Our goal is to have a fundamental theoretical understanding of anomalous relaxations, as well as to develop applications that exploit fast relaxations.

Figure 1

The geometry of the strong Mpemba effect in a three-state system. The probability simplex is illustrated by the blue triangle. The set of all equilibrium distributions form the red curve. The blue and red points on the curve correspond to $T=0$ and $T=\infty$. The Mpemba index is nonzero if the equilibrium curve crosses the $a_2=0$ plane (green) not only at the bath temperature (illustrated here by the pink point) but also at some other temperature (the green point).

Figure 2

The mean-field antiferromagnetic Ising model Mpemba-phase diagram. Upper panel: The phase diagram calculated for $N=400$ spins. There are eight different Mpemba phases in this system: (i) white; no direct or inverse Mpemba effects, (ii) blue; weak direct and no inverse effects, (iii) green; weak inverse and no direct effects, (iv) burgundy; strong inverse with ${\mathcal{I}}_M^{\mathrm{inv}}=1$ and no direct effects, (v) violet; strong inverse with ${\mathcal{I}}_M^{\mathrm{inv}}=2$ and no direct effects, (vi) light blue; strong inverse with ${\mathcal{I}}_M^{\mathrm{inv}}=1$ and weak direct effects, (vii) gray; strong direct with ${\mathcal{I}}_M^{\mathrm{dir}}=1$ and strong inverse with ${\mathcal{I}}_M^{\mathrm{inv}}=1$, and (viii) yellow; strong direct with ${\mathcal{I}}_M^{\mathrm{inv}}=1$ and weak inverse effects. The blue line is the antiferromagnetic-to-paramagnetic phase-transition line of this model calculated in Ref. [25]. Lower panel: The same calculation for $N=70$ spins. Although the exact location of the boundaries between the different phases is not identical in the two computations, the overall structure is the same. The three dashed lines are lines of constant magnetic field $H=1.03$, $H=1$, and $H=0.97$, and their corresponding equilibrium loci in the thermodynamic limit are given in Fig. 3. Note that the phase diagram has an abrupt jump at $H=1$, which corresponds to the jump in the equilibrium locus in Fig. 3.

Figure 3

The “equilibrium locus” of the mean-field antiferromagnetic Ising model at $H=0.97$, $H=1$, and $H=1.03$ plotted in the plane of average magnetization densities in each subgraph $\overline{x_1}$, $\overline{x_2}$. Note the sharp transition in the curve’s shape around $H=1$. This sharp transition corresponds to an abrupt transition in $a_2(T)$, which is clearly seen in the Mpemba-phase diagram in Fig. 2: The three equilibrium lines here correspond to the three dashed lines in the lower panel of Fig. 2.

Figure 4

Upper panel: The thick green dashed line is the equilibrium locus, and the colored lines are the relaxation trajectories toward the equilibrium point (the red circle). All relaxation trajectories except the dashed red one approach equilibrium from the slow direction. The red dashed trajectory approaches equilibrium from the fast direction, and it corresponds to the strong inverse Mpemba trajectory (with bath temperature $T_b=0.5$ and initial temperature $T_M^{\infty }=0.1377$). Its relaxation rate is 14.7 times faster than the other trajectories; see Fig. 5. Lower panel: A comparison between the thermodynamic limit and finite system with $N=400$. The dash-dot thick green line is the thermodynamic limit equilibrium line, and the dashed black line is the projection into $(\overline{x_1},\overline{x_2})$ of the equilibrium line in the finite system. The projections of the relaxation trajectories for several initial temperatures are shown as thin lines, and the relaxation which corresponds to the strong Mpemba [calculated by $a(T_M)=0$] is the thick blue dashed line. The inset shows an enlargement of the relaxation trajectories near the equilibrium. The strong Mpemba initial condition approaches equilibrium from a different direction.

Figure 5

The distance from equilibrium [defined in Eq. (27)] in logarithmic scale as a function of time for several initial conditions in the thermodynamic limit. The initial condition corresponding to the strong effect is the dashed line.

Figure 6

Comparison between the strong effect in the $N$ spin system and the strong effect in the thermodynamic limit. The temperature from which there is a strong effect on the finite-$N$ system $T_M(N)$ converges to the temperature from which there is a strong effect in the thermodynamic limit $T_M^{\infty }$.

Figure 7

$T_{\mathrm{eq}}-T_{\mathrm{bath}}$ as a function of time for various initial temperatures along the equilibrium line. The inverse Mpemba effect is shown here as a crossing of two curves—initially, the colder system heats up faster. The figure also shows that the temperature can overshoot the environment temperature—the system can reach equilibrium temperatures which are higher than that of the environment.

Figure 8

The direct Mpemba index ${\mathcal{I}}_M^{\mathrm{dir}}$ is odd in the double-wedge shaded region of the $f_{\parallel }{f}_{\perp }$ plane if ${\mathit{u}}_2^{\mathrm{dir}}·\mathit{w}>0$, and while if ${\mathit{u}}_2^{\mathrm{dir}}·\mathit{w}<0$, the direct Mpemba index is odd for the complementary region (white). See Eq. (43).

Figure 9

The probability for the odd direct Mpemba index for a particular random draw of $L=15$ energy levels obtained two independent ways: analytically by averaging over the isotropic ensemble (solid line) and numerically by averaging over random barriers (points). The energy level realization is drawn from a Gaussian distribution with zero mean and standard deviation 1.5. Each point corresponds to the average of 4000 barrier realizations taken from a truncated Gaussian distribution with zero mean and standard deviation 15 (the Gaussian is truncated to have only positive $B_{ij}$ values). The solid line represents the analytical result for the isotropic ensemble Eq. (50).

Figure 10

Lower bound for the probability of the strong Mpemba effect—the probability of the parity being positive for the direct Mpemba index [see Eq. (15)] for the case of REM with random barriers. The number of energy levels is $L=10$ and the bath temperatures are $k_B{T}_b=0.1$ (left) and $k_B{T}_b=1.0$ (right). The energies are IIDs from Eq. (59) with variance ${\sigma }_E^2{\log }_{2}L$, and the barriers are IIDs from Eq. (60) with variance ${\sigma }_B^2{\log }_{2}L$. Each point on the density plot is averaged over $10^5$ realizations. The condensation phase transition is at $k_B{T}_{\text{critical}}={\sigma }_E/\sqrt{2\ln 2}\approx 0.84{\sigma }_E$. We notice that the probability of having a direct strong Mpemba effect is finite and even high for certain regions of the ${\sigma }_E{\sigma }_B-$ plane for $T_b<T_{\text{critical}}$ (left panel).

Figure 11

Lower bound for the probability of the strong Mpemba effect—the probability of the parity being positive for the direct Mpemba index [see Eq. (15)] for the case of the REM with random barriers and different system sizes $L\in [4,20]$. The bath temperature is $k_B{T}_b=0.1$. The energies are IIDs from Eq. (59) with variance ${\sigma }_E^2{\log }_{2}L$ and ${\sigma }_E=1.0$, and the barriers are IIDs from Eq. (60) with variance ${\sigma }_B^2{\log }_{2}L$. Each point on the density plot is averaged over $2\times {10}^5$ realizations. We notice that the probability of the parity being positive for the direct Mpemba index seems to be converging to a finite limiting value with increasing system size.