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

The reverse fact has been known since the time of Aristotle, at least 2,300 years ago. It was rediscovered by Erasto Mpemba, a Tanzanian high school student who, together with physicist Dennis Osborne, first investigated it in the 1960s. *physics education*The appropriate title wasCold?”(Cold?, in English). a few months ago she became viral trend on social networks, which involved throwing boiling water into the air to see how it quickly froze and turned into ice.

Used to be some controversy about this phenomenon, because it has been difficult to consistently repeat the result in the laboratory. The smallest details matter a lot, 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 generally accepted logic, all of the above phenomena 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. . one in 2017 study A general theoretical explanation of this phenomenon is offered, using the principle of unbalanced thermodynamics,

According to this principle, 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 it is difficult to quantify precisely what the state of the fluid is. requires an infinite number of numbers.

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

With greater precision, the state of the system can be modeled by means of a probability distribution function, which describes the probabilities of all possible states whose evolution is governed by a linear differential equation. For a system with finite states, this evolution is determined by properties of the so-called transition matrix. Now, for this type of system, whatever the initial probability distribution is, it will converge at some point to the equilibrium state. However, due to the shape of the transition matrix, there is a particular 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 for the hot liquid to be greater than that for the cold liquid after some time. This can happen when the PDF for the cold liquid is closer to this particular PDF than for the hot liquid. The result will be that after some time the colder liquid will transition to the equilibrium position more slowly than the hotter liquid.

The scientists predicted an inverse Mpemba effect: When two liquids are heated, the colder one can become hotter faster. Phenomenon now observed in experiments

Using this analysis, the paper’s authors predicted a reverse Mpemba effect, in which when two fluids are heated, the colder one can become hotter 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 the hottest liquid can cool faster than the initially cold liquid, under carefully chosen parameters. 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 Agarwal*** He is a postdoctoral researcher at ICMAT.*

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