The Mpemba effect is a phenomenon whereby a hot liquid can freeze faster than a cold one, under certain conditions.

This counterintuitive fact has been known at least since the time of Aristotle, some 2,300 years ago. It was rediscovered by Erasto Mpemba, a Tanzanian high school student, who, together with physicist Denis Osborne, first investigated it in the 1960s. *Physics Education*was aptly titled “cool?” (cold?, in English). A few months ago she became a viral trend on social networks, which consisted of throwing boiling water into the air to see how it quickly freezes and turns into snow.

Has been some controversy regarding this phenomenon, since it has been difficult to consistently replicate the result in the laboratory. The smallest details matter a great deal, such as the size, shape, and material of the container, or even where the thermometer is placed. Various explanations have been proposed for the Mpemba effect: convection, evaporation, overcooling, impurities in the water sample, dissolved gases, etc. There is no commonly accepted argument, all of the above phenomena seem to play a role.

Furthermore, while these explanations may be partially true for water, they fail to explain the phenomenon for other substances, such as magnetoresistance alloys, polymer alloys, and granular systems, where it is also observed. In a 2017 study A general theoretical explanation of this phenomenon is offered, using the theory of nonequilibrium thermodynamics.

According to this theory, any point in the phase space of an equilibrium fluid can be described by three numbers: its temperature, volume, and number of particles. However, when the fluid is in the process of cooling, it is not in equilibrium and the number of states required to describe the system increases to infinite dimensions, so infinite numbers are needed to precisely quantify what the state of the fluid is.

When a hot liquid is placed in a cold environment, it tries to reach its lowest energy state. However, the energy landscape defined in its states has multiple local minima —that is, points that take lower values than those around them—, called metastable energy wells. If a hot liquid enters these metastable energy wells, it has more energy to more easily escape from it and find the global minimum—the cooling temperature—whereas if a cooler liquid enters one of the metastable energy wells , you will spend more time on it.

With more precision, the state of the system can be modeled by means of a probability distribution function, which describes the probabilities of all the possible states that it could have, whose evolution is governed by a linear differential equation. For a system with finite states, this evolution is determined by the properties of the so-called transition matrix. Now, for this type of system, whatever the initial probability distribution is, it will, at some point, converge to the equilibrium state. However, due to the shape of the transition matrix, there is a special probability distribution function that will converge to the steady state at the slowest possible rate, compared to all other initial probability distributions.

Thus, to observe the Mpemba effect, we need the distance from the probability distribution function to the steady state to be smaller for the hotter liquid, compared to the colder liquid, after some time. This can happen if, initially, the pdf for the colder liquid is closer to this special pdf than for the warmer liquid. This will result in the cooler liquid converging to the equilibrium state more slowly than the warmer liquid, after some time.

Scientists predicted a reverse Mpemba effect: that when two liquids heat up, the colder one can heat up faster. Phenomenon now observed in experiments

Using this analysis, the paper’s authors predicted a reverse Mpemba effect, in which when two liquids heat up, the colder one may heat up faster. This phenomenon has now been observed in experiments. In a 2019 theoretical study using a similar approach, physicists also predicted the strong Mpemba effect in which, under carefully chosen parameters, the hottest liquid can cool exponentially faster compared to the initial cold liquid. This has also been observed experimentally.

However, the Mpemba effect for water remains unresolved, and we will have to wait a few more years for a definitive answer to this question.

**Siddhant Govardhan Agrawal*** He is a postdoctoral researcher at the ICMAT.*

**Coffee and Theorems**** ***is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”*

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