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# Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 4, 2017 (received for review January 23, 2017)

## Significance

It is commonly expected that cooling a hot system takes a longer time than cooling an identical system initiated at a lower temperature. Surprisingly, this is not always the case; in various systems, including water and magnetic alloys, it has been observed that a hot system can be cooled faster. These anomalous cooling effects are referred to as “the Mpemba effect”, and so far they lack a generic details-independent explanation. Based on recent developments in the theory of nonequilibrium thermodynamics, we propose a generic mechanism for similar effects, demonstrate it in various systems, and predict a similar anomalous behavior in heating.

## Abstract

Under certain conditions, it takes a shorter time to cool a hot system than to cool the same system initiated at a lower temperature. This phenomenon—the “Mpemba effect”—was first observed in water and has recently been reported in other systems. Whereas several detail-dependent explanations were suggested for some of these observations, no common underlying mechanism is known. Using the theoretical framework of nonequilibrium thermodynamics, we present a widely applicable mechanism for a similar effect, the Markovian Mpemba effect, derive a sufficient condition for its appearance, and demonstrate it explicitly in three paradigmatic systems: the Ising model, diffusion dynamics, and a three-state system. In addition, we predict an inverse Markovian Mpemba effect in heating: Under proper conditions, a cold system can heat up faster than the same system initiated at a higher temperature. We numerically demonstrate that this inverse effect is expected in a 1D antiferromagnet nearest-neighbors interacting Ising chain in the presence of an external magnetic field. Our results shed light on the mechanism behind anomalous heating and cooling and suggest that it should be possible to observe these in a variety of systems.

Consider cooling a system initiated at a hot temperature by coupling it to a cold bath. Intuitively, we expect that the cooling time should increase with the system’s initial temperature. Surprisingly, this is not always the case: More than 2,300 years ago Aristotle documented that the inhabitants of Pontus were using hot water, instead of cold water, to prepare ice rapidly (1). Nowadays, anomalous cooling in water or other substances is referred to as the Mpemba effect: When two samples of the same substance identical in all macroscopic parameters except for their initial temperatures are simultaneously cooled by the same cold bath, it can take less time to cool the sample initiated at a higher temperature.

The exact mechanism behind anomalous cooling in water and even its existence (2, 3) are still under debate. Clearly, this effect contradicts Newton’s heat law, where the rate of change of the system’s temperature is simply proportional to the temperature difference between the system and the bath. When the cooling process is quasistatic, namely when the system’s temperature follows Newton’s heat law, the hot system necessarily lags behind the cold system; i.e., the Mpemba effect cannot occur. On the other hand, this effect has been observed in water (4) and more recently in several other substances, e.g., nanotube resonators (5), magneto-resistance alloys (6), clathrate hydrates (7), and granular systems (8).

The cooling process in the Mpemba effect, i.e., quenching, is in general not quasistatic, but rather a genuine far-from-equilibrium process. Several aspects of nonequilibrium cooling have been considered to partially explain the Mpemba effect in water: hydrodynamic effects (9), supercooling (10, 11), evaporation (12, 13), impurities in the water sample (14, 15), and even the microscopic structure of the hydrogen bond (16, 17). Whereas these mechanisms explain some of the observations in water and clathrate hydrates, they are all substance specific and thus cannot explain the Mpemba effect observed in other substances, e.g., in magneto-resistance alloys or granular systems.

In this paper, we consider anomalous cooling processes in the general framework of nonequilibrium statistical mechanics. For systems undergoing Markovian dynamics, we provide a sufficient condition, accompanied with heuristic intuition for its appearance. We stress in advance that it is not clear that the proposed theory is the dominant mechanism in the specific phenomenon observed in water, and to distinguish between the two we refer to the former as the Markovian Mpemba effect. To illustrate the Markovian Mpemba effect, we first demonstrate it in a minimal three-state system and then in one of the most important models in statistical mechanics—the Ising model. We show that the Markovian Mpemba effect can be found in an antiferromagnetic nearest-neighbor interacting Ising spin chain, in the presence of an external magnetic field. We then predict the existence of an inverse Markovian Mpemba effect that occurs during heating: When two systems are heated by the same hot bath, it can take a shorter time to heat the initially colder system. This inverse effect is demonstrated on both the three-state system and the antiferromagnetic Ising chain.

This paper is structured in the following order. In *Theory*, we set up the framework to describe cooling processes in Markovian dynamics, introduce the distance-from-equilibrium function, discuss its properties, and use it to precisely define the Markovian Mpemba effect. These ideas are then illustrated by a minimal model of the Markovian Mpemba effect—a three-state Markov system. In *Results* we analytically obtain a sufficient condition for the Markovian Mpemba effect and demonstrate it for the three-state system as well as for the antiferromagnetic 1D Ising chain. An additional result is the prediction of an inverse Markovian Mpemba effect, which is demonstrated using similar systems. In *Discussion: Energy Landscape and the Markovian Mpemba Effect*, we provide some physical intuition for the Markovian Mpemba effect based on certain geometric properties of the system’s energy landscape. This intuition is numerically demonstrated by an example of diffusive dynamics. And finally, we present *Conclusions*.

## Theory

### Markovian Dynamics.

In the absence of a thermal bath, any isolated classical system—be it a glass of water or a magnetic alloy—evolves deterministically in its phase space in accordance with its Hamiltonian. (We neglect in this discussion any quantum source of nondeterminism.) However, when coupled to a thermal bath, the dynamic of the system is no longer deterministic due to the random thermal fluctuations. It is instructive to describe systems coupled to a thermal bath by a probability distribution **1** are widely used to model a variety of thermal processes, e.g., the Kramer–Fokker–Planck dynamic for a Brownian particle, the Glauber dynamics of a classical spin system, and the Lindbladian dynamics of a quantum system.

To simplify the presentation, we first focus on systems with a finite number of states, although a similar analysis can be carried out for systems with a continuous state space (example in *Discussion: Energy Landscape and the Markovian Mpemba Effect*). Let us denote the probability to find a system at the

### The Distance Function.

In most previous descriptions of the Mpemba effect, the cooling rate was characterized by measuring the time it takes for the water to freeze (this is a very subtle definition, discussion in ref. 4) or by tracking the readout of a thermometer (19). However, the former cannot be applied to systems without a phase transition, although anomalous cooling was observed in such systems, e.g., in granular materials far from the jamming transition. The readout of a thermometer, on the other hand, might depend on specific details such as the sensor’s exact location and its working principle.

To generalize the discussion on anomalous cooling and avoid these issues, we quantify the rate of cooling by constructing a distance-from-equilibrium function, **4**. Given a distance-from-equilibrium function we can define the Markovian Mpemba effect as follows: If there exist three temperatures, **2**, and if there exists some finite time

There are many reasonable choices for a distance-from-equilibrium function. However, the Markovian Mpemba effect may be falsely reported due to a poor choice of the distance function *i*) When the system relaxes toward its thermal equilibrium (from any initial distribution), the value of *ii*) The distance from equilibrium of a system initiated at temperature *iii*) the distance from equilibrium is a continuous, convex function of *SI Appendix*, section I for a derivation),*SI Appendix*, section II,

### Example: A Three-State Model.

To illuminate the discussion, we next introduce a minimal model that shows the Markovian Mpemba effect—a three-state system (Fig. 1*B*) that is coarse grained from an overdamped system with a three-well energy landscape (Fig. 1*A*). In this specific example, the energies of the three states are given by *C*, this space forms a 2D simplex whose vertices are located at

For each temperature **4** is a point in the triangle. The set of Boltzmann distributions for all temperatures *C*). In quasistatic cooling, both the hot and the cold systems evolve along the quasistatic locus toward the equilibrium, and hence the hot system lags behind the cold system. However, this is not the case in a nonequilibrium cooling process. By solving the master equation, we show in Fig. 2 that the hot system and the cold system evolve according to two nonoverlapping paths [*Inset*.

## Results

### Sufficient Condition for the Markovian Mpemba Effect.

The Markovian Mpemba effect occurs when the distance from equilibrium of the initially hot system is larger than that of the initially cold system at *SI Appendix*, section III for a proof).

Using the condition

Let us demonstrate this condition for the three-state system example. A sufficient condition for the Markovian Mpemba effect is that the tangent to the quasistatic locus is parallel to *Inset*. After

### The Markovian Mpemba Effect in the Ising Model.

To show that the discussed effect is not the result of a careful tuning of parameters, we next demonstrate the Markovian Mpemba effect in the Ising model, in which the number of parameters is significantly smaller than the number of microstates (*SI Appendix*, section V). We consider a 1D chain of

The transition rate between state

In other words, only a single spin flip is allowed at each instance of time, and the rate of any spin flip is proportional to the exponent of the energy differences between the two corresponding microstates.

Specifically, we used *Discussion: Energy Landscape and the Markovian Mpemba Effect*). The temperature of the bath is *A* shows the coefficient of the slowest decay rate of systems starting from a point on the quasistatic locus as a function of the temperature, *T*, reaches the maximum at *A*, *Inset* shows, the distance from equilibrium of the initially hot system drops below that of the initially cold system after time

### The Inverse Markovian Mpemba Effect.

So far we have considered anomalous cooling processes. Next, we predict the existence of a similar effect for heating processes, where an initially cold system (

Interestingly, this inverse effect can be found in a similar 1D antiferromagnetic Ising chain, for different parameters. The *B*). This prediction is verified by choosing the two initial temperatures to be *B*, *Inset*).

## Discussion: Energy Landscape and the Markovian Mpemba Effect

To gain some physical intuition into the effect, we next discuss its relationship with certain geometric properties of the system’s energy landscape. A common feature for systems that demonstrate the Markovian Mpemba effect is that the energy landscape defined on their microstates has multiple local minima or metastable energy wells (therefore we used the antiferromagnetic, rather than the ferromagnetic 1D Ising model). These are schematically illustrated in Fig. 5*A*. The relaxation rate from a metastable state into the global minimum is slow, whereas the relaxation rates from the high-energy states into the global minimum are fast (arrows in Fig. 5*A*). The equilibrium distribution corresponding to

To illustrate this argument, we construct an example of Fokker–Planck dynamics in the 1D potential landscape shown in Fig. 5*B*. At high temperature *C*). When coupled to the cold bath, the high-energy configurations in the initially hot system rapidly flow into the lowest-energy well, whereas in the initially cold system the probability in the left metastable energy well takes a long time to relax into the lowest-energy well. In other words, the relaxation of the cold initial condition is slow because it requires some probability to transfer across a barrier from the metastable well into the lowest well, whereas in the initially high-temperature system only a small amount of probability has to cross the barrier when relaxing toward the final equilibrium, and the relaxation is thus faster. This argument is numerically verified by calculating the distance from equilibrium function of the hot and the cold systems—these are plotted in Fig. 5*C*, *Inset*. See *SI Appendix*, section IV for a detailed discussion of the example.

## Conclusions

In this paper we have discussed a Markovian Mpemba effect in cooling and predicted a similar effect in heating (the inverse Markovian Mpemba effect). We have found the sufficient condition for a system to have these effects. Such anomalous cooling and heating effects were numerically demonstrated in a minimal three-state model, the Ising model, and Fokker–Planck dynamics. Similar analyses can be used to predict and identify these effects in many other systems.

By generalizing the Mpemba effect, we demonstrate that anomalous cooling and heating effects are expected not only in water, but also in well-studied systems such as the Ising model, and that these counterintuitive effects can be reconciled with nonequilibrium stochastic thermodynamics. Finding the explicit cause of the Mpemba effect in freezing water is still an interesting open problem, which may involve detailed knowledge of the hydrogen bond formation and hydrodynamic currents during cooling. Among the existing explanations of the Mpemba effect in water, convection flow and anomalous relaxation of the hydrogen bonds are manifestly Markovian out-of-equilibrium processes, and hence they might be related to the general mechanism described in this paper.

## Acknowledgments

We thank C. Jarzynski, A. Dinner, J. Weeks, S. Deffner, J. Horowitz, R. Remsing, Zhixin Lu, O. Hirschberg, J. Katz, and T. Witten for useful discussions and Y. Subasi, R. Pugatch, and G. Ariel for careful reading of the manuscript. O.R. acknowledges financial support from the James S. McDonnell Foundation. Z.L. acknowledges financial support from the NSF under Grant DMR-1206971 and the NSF Materials Research Science and Engineering Center at the University of Chicago under Grant DMR-1420709.

## Footnotes

↵

^{1}Z.L. and O.R. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: orenraz{at}gmail.com or zhiyuelu{at}gmail.com.

Author contributions: Z.L. and O.R. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1701264114/-/DCSupplemental.

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