The Mpemba paradox

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If you take two identical cups, fill one with warm water and one with cold and put them in the freezer, you’d expect the cooler one to freeze first, but it doesn’t always. In fact, in many circumstances, it is the warm water that freezes first.

This is the Mpemba paradox, named after Erasto Mpemba, who observed it as a schoolboy in Tanzania in the 1960s. When physicist Dr Denis G. Osbourne visited his school, Mpemba took the opportunity to ask about his strange observation. Although initially skeptical, Osbourne later reproduced the observations and several years later, in 1969, they jointly published the result. Since then, it has been reproduced in many experimental studies and its origins have been debated extensively.

Many people, when they want to cool water quickly, begin by putting it in the sun.

Mpemba wasn’t the first to observe this though. The effect has been discovered and rediscovered many times over at least two millennia and caught the attention of both the British polymath Francis Bacon and, independently, his contemporary, the French mathematician and philosopher René Descartes. The earliest known reference is from Aristotle, who wrote that ‘the fact that the water has previously been warmed contributes to its freezing quickly: for so it cools sooner. Hence many people, when they want to cool water quickly, begin by putting it in the sun.’ He thought it supported his idea of antiperistasis: that a quantity is intensified by being surrounded by its opposite.

Surprisingly, there is still no scientific consensus on the exact cause of the Mpemba effect. Many people have claimed that their explanation is the definitive answer but the root of the Mpemba effect is still regarded as an open problem. It’s a very interesting problem as well because it crosses disciplinary boundaries. Some explanations focus on chemistry, others look at physics. As with many problems, a clear route to the crux of this paradox is to examine it mathematically.

A mathematical view

We’re interested in the temperature T over time t. To find this we need to know the rate of change of temperature, $\mathrm{d} T/\mathrm{d} t$. A reasonable first model would be to say that this rate of change depends on the temperature, so we could write

\frac{\mathrm{d} T}{\mathrm{d} t} = f(T),

for some function f. But if this equation holds then the Mpemba effect cannot be real. Say the two cups start at 30°C and 5°C. When the warmer one cools down to 5°C, it will have to follow the same route to freezing that its rival has already begun, but without the headstart. This is the nub of the problem: the warm cup doesn’t just have to cool faster initially, it has to overtake its rival. When it reaches 5°C, it has to be cooling faster than the cooler cup was at that same temperature, so the equation above cannot be true. The rate of change of temperature can’t just depend on the current temperature T; it must also depend on $T_0$, the initial temperature. So in fact our model should be

One possibility for the temperatures of the water in the two cups over time

One possibility for the temperatures of the water in the two cups over time

\frac{\mathrm{d} T}{\mathrm{d} t} = f(T,T_0).

This is therefore a system with memory. What’s happening in the cup doesn’t just depend on its present temperature, but also on its past temperature, on its history. This is the key point. Any successful theory must explain not only how the warmer cup cools faster initially, but moreover how it can overtake. It must explain why the system has memory.

Hysteresis and systems with memory

The idea of a physical system having memory can be quite counterintuitive, but many examples do exist. Imagine hanging weights on an elastic band. When you add a new weight it might stretch from 10 cm to 12 cm, say. But if you remove that new weight the band doesn’t return all the way to 10 cm, it stays a bit more stretched. This is an example of hysteresis – when changing an input then reversing the change doesn’t bring the output back to the original level. Hysteresis is an interesting feature found in systems with memory.

Force vs extension for an elastic band. This loop-shaped graph is the signature of hysteresis.

Force vs extension for an elastic band. This loop-shaped graph is the signature of hysteresis.

Many other important examples of hysteresis can be found. Applying a magnetic field to iron magnetises it, but removing the field doesn’t reverse the process. This is known as magnetic hysteresis and is essential for hard disk drives to work. In flight, when an aeroplane flies up at too steep an angle, it starts to stall and lose lift. It can recover by decreasing its angle of attack, but it has to go down to a lower angle than that at which stalling started. And it’s not only physical systems that can exhibit hysteresis. During a recession, unemployment typically increases. But when the economy recovers, the employment rate doesn’t necessarily recover with it.

A worrying possibility is that our climate may exhibit hysteresis. Beyond certain tipping points, changes may be irreversible. This is obviously true for the extinction of animal species but might also apply to, for example, the melting of major ice sheets.


So if the Mpemba paradox requires the system to have memory, which current theories incorporate this feature?

One idea is that the warm water evaporates more, so with a lower volume it can cool quicker. If V is volume then we can write

\frac{\mathrm{d} T}{\mathrm{d} t} &= f(T,V)\\
\frac{\mathrm{d} V}{\mathrm{d} t} &= g(T,V)

for functions f and g. Thus memory is stored in the extra variable V. There is good evidence that this effect has a role but there are also control experiments with sealed lids or no loss of mass that show that it cannot be the sole explanation.

Alternatively, there are a whole group of explanations asserting that the environment makes the difference. One possibility is that the warmer cup might melt through frost at the bottom of the freezer until it touches the actual freezer surface, a better conductor of heat. Here, an additional variable E representing the environment stores memory. These explanations are certainly plausible but are also easy to control for in experiments. It seems unlikely that one of them could be the cause of the Mpemba paradox in every single case.

Perhaps the most successful explanation is related to supercooling. Supercooling is when a liquid passes below its freezing point, but doesn’t yet freeze because there are no places, such as dust particles, for the ice to starting forming. The liquid wants to freeze, but doesn’t know how to. Some experiments suggest that the cool water supercools more than the warm water. This would explain the Mpemba paradox, but replace it with a new puzzle: why does cooler water supercool more? This idea also highlights the need for a precise definition of the Mpemba paradox – does the warm water have to freeze first or reach 0°C first? The supercooling theory can only explain the former.


A mathematical simulation of a `Rayleigh-Taylor instability', an experiment with cold liquid (blue) starting above warm liquid (yellow). Note the beautiful spiral patterns, known as `Kelvin-Helmholtz instabilities', that form at the interface.

A mathematical simulation of a `Rayleigh-Taylor instability’, an experiment with cold liquid (blue) starting above warm liquid (yellow). Note the beautiful spiral patterns, known as `Kelvin-Helmholtz instabilities’, that form at the interface. (Source: Los Alamos National Laboratory. Terms.)

A final major theory that could explain the memory in the Mpemba experiment is based on convection. Convection is when a liquid or gas mixes because of density differences: hot air is light and rises to the top, cold air is heavy and sinks to the bottom. Convection is the key driver of our weather. The atmosphere is heated at the equator and cooled at the poles, which drives a huge conveyor belt of air.
A particularly beautiful example of convection can be seen if you take a pan of liquid such as oil and heat it from below at the right rate. Convection cells in the shape of hexagonal prisms known as ‘Rayleigh-Bénard cells’ form. Hot oil rises in the centre of each hexagon, and cooler oil sinks at its edges. It is remarkable that the molecules of oil arrange themselves into this ordered and efficient mathematical pattern. Although not the standard explanation, it has been speculated that this may be the origin of the Giant’s Causeway, a set of 40,000 polygonal basalt columns in County Antrim, Northern Ireland. These incredible shapes were formed as the molten basalt cooled.

Left: An experiment with oil heated from below. Hexagonal Rayleigh-Bénard cells have formed. Right: The Giant's Causeway in Northern Ireland.

Left: An experiment with oil heated from below. Hexagonal Rayleigh-Bénard cells have formed. (Source: Uni. Iowa, Physics and Astronomy) Right: The Giant’s Causeway in Northern Ireland.

A cartoon of the convection cells that drive the atmosphere

A cartoon of the convection cells that drive the atmosphere

A convecting liquid can lose heat faster, which may explain the Mpemba paradox. Convection moves hot water to the top so, if this is where the heat is lost, the cup would cool faster. This shows that the warmer cup could cool faster initially, but it may also explain how it can overtake. Convection is known to exhibit hysteresis. Water at 5°C in a −20°C freezer might not start convecting, but if it were already convecting, it could continue. So the memory is stored in the level of convection in the cup.

Understanding convection requires more complex equations than we have used so far. We would need to look not just at the average temperature, but at the temperature and velocity at every point in the cup. Interpreting and solving these equations requires more sophisticated mathematics, but is certainly possible.

A solution to the paradox?

It seems strange that the Mpemba paradox hasn’t yet been resolved. Surely a set of comprehensive experiments could test each one of the major theories? For example, you could test the role of convection by stirring both cups or the role of melting frost by placing them on an insulating mat. Modern experimental techniques can measure the heat and even the velocity at many points within the liquid. So we should be able to build a detailed picture of what is happening.

Perhaps the reason the paradox hasn’t been resolved is a practical one: it crosses so many disciplinary boundaries. A mathematician views it mathematically, whilst a chemist looks for chemical solutions. A full resolution requires the input of many disciplines.

Or perhaps there is no single solution. Maybe both supercooling and convection are strong enough effects to individually cause the Mpemba paradox. Then results from different experiments would seem incompatible and confusing to a scientist assuming a single cause. Either way, the Mpemba paradox remains a fascinating scientific mystery. It is simple, seemingly impossible and links together exciting science across the disciplines, from supercooling to convection to the bizarre property of hysteresis.


(Title image by Schnobby, licensed under Creative Commons CC BY–SA 3.0)

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Oliver Southwick is a PhD student from the UCL Department of Mathematics modelling large scale ocean currents.
Twitter  @OliverSouthwick    + More articles by Oliver

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