In 1969 Erasto Mpemba, a Tanzanian schoolboy, published an observation with Denis Osborne that he had first made making ice cream: a hot mix froze before a cool one. "Hot water freezes faster than cold" has been argued about ever since. In 2016 a careful study by Burridge and Linden found no meaningful evidence for it in pure water. Others have replied that they have seen it.
Read the argument and most of it is not about physics. It is about the word freezes. Faster to what? The water reaching zero, the first ice, or the whole cup solid? Measured with a thermometer where?
There is a quantum version of the same effect, found by Filiberto Ares and colleagues in chains of quantum spins, and nobody argues about that one, for the opposite reason. They defined the quantity first, a measure of how broken a symmetry is called entanglement asymmetry, and then found that the more broken it starts, the faster it heals. The effect is well posed because the finish line was drawn before anybody ran.
So I did it in that order. The engine now has a cooling module with three finish lines, and every answer it gives reports all three.
The part that is certain before any physics
Describe a cup by its temperature alone, and let how fast it cools depend only on that temperature. Then the hot cup cannot win. Not usually cannot. Cannot.
On the way down it has to pass through the cold cup's starting temperature. At that instant it is in exactly the state the cold cup started in, so from then on it does exactly what the cold cup did, late.
It does not matter how much faster hot water loses heat. With radiation switched on, a mug at 90 degrees sheds heat 5.2 times as fast as one at 20, where a straight-line cooling law would say 2.8. It still arrives at every finish line 57.6 minutes behind. The same 57.6 minutes at all three, which is exactly how long it takes to cool from 90 to 20.
The test suite checks this across 24 races, two cooling laws and three cup shapes. The hot cup is late by exactly its lag every time, to the hundredth of a second.
So any real effect needs a second variable: something the hot cup carries forward from having been hot, which the cold cup does not have. That is the question worth asking, and it turns every proposed explanation into a switch.
The switches
Each mechanism can be turned on and off, and each one says what kind of thing it is.
- Radiation is established physics and adds nothing but speed. It is there to show the rule above holding under a strongly nonlinear law.
- Evaporation is established physics and adds a second variable, which is mass. The hot cup loses more water on the way down, so it arrives at every temperature lighter than the cold cup did.
- Supercooling is marked assumed. Water often cools below zero before ice happens to form, by a few degrees and close to randomly. The claim that water which was heated first supercools less is disputed, so here each cup's freezing point is an input you set. Any win it produces is a win you chose. Every result the module returns carries that flag, and a test walks the results to make sure it is never missing.
The evaporation rate comes from the same air coefficient as the ordinary cooling, so the two cannot be tuned against each other. It lands within a factor of 1.7 of the standard pool-evaporation correlation, which shares no algebra with it. Energy balances to better than one part in a hundred billion, through cooling, the supercooled snap and freezing.
What came back
The classic claim: a mug at 90 against a mug at 20, in a normal freezer at minus 18, everything established switched on. Cold wins all three lines. It reaches zero 44 minutes sooner and freezes solid 41 minutes sooner. The hot mug lost 4.0 percent of its water to the air and the cold one 0.9 percent. That difference is real, and it closes about a quarter of the gap.
A shallow dish in a barely freezing freezer is where evaporation earns its keep. 200 grams, 95 against 40, at minus 5. The hot dish reaches zero 5.3 minutes later and freezes solid 8.1 minutes sooner. It lost 9.0 percent of its water against 3.9, so it had less left to freeze. That is a real Mpemba effect, with nothing assumed, at one finish line out of three.
In an ordinary freezer the hot cup never wins the race to zero. I ran 150 races at minus 18, over three cup shapes, with evaporation at its computed rate and at double it to cover the uncertainty in that rate. The hot cup reached zero first in none of them. Its best win at frozen solid was 2.6 percent. The race to zero is the line Burridge and Linden measured, and they found nothing either.
Supercooling does whatever you tell it to. Tell a mug at 40 that it will ice at minus 1, and a mug at 20 that it will ice at minus 8, and the hot one shows ice 8 minutes first. Then it finishes freezing solid 15 minutes last. The module reports that win as belonging to the assumption, because switching supercooling off removes it. Even told outright, with the same seven-degree advantage, a mug at 90 still shows its first ice 25 minutes behind a mug at 20.
What I believe now
In this model, for a mug in a kitchen freezer, the slogan is false at every finish line. As a precise claim it survives: a shallow dish, a mild freezer, a hot start near boiling, measured at frozen solid, wins by minutes on evaporation alone. Those are different claims, and an argument more than fifty years old is largely people holding one each.
"Hot water freezes faster" is a well formed answer to a question nobody finished asking. It is the same shape as the water post: an answer that is true at one place and was stated as though it were true everywhere. The fix is the same too. Say where.
Added after publishing: a cup with a top and a bottom
The first version of this post ended by calling stratification the next switch. It is built now, and it came with a question worth answering first: does it matter that water's viscosity changes with temperature? It changes a lot. Water is 6.3 times less viscous at boiling than at freezing.
On its own, it cannot matter. A viscosity that depends only on the cup's one temperature is still a one-temperature cup, and the rule at the top of this post already says what that cup can do. Viscosity can only matter by changing something the cup carries forward. The obvious something is how well the cup stirs itself, and that needs a cup with a top and a bottom.
So the new switch splits the cup into two layers. Each loses heat through its own surfaces, and they mix by buoyancy at a rate built from the water's viscosity, conductivity and density at their own temperature. Water is densest at 4 degrees. Above that, a top cooled by its surface sinks and the cup stirs itself. Below it, the cold water floats, the stirring stops, and ice forms on top of warmer water, as it does on a lake. The switch is marked modelled, a third status between established and assumed. The physics is established; two layers is the crudest cup that can show it, so it gives direction rather than magnitude.
It behaves like a cup should. The layers pull up to 3.5 degrees apart, and the ice forms on top while the bottom is still near 0.9 degrees. Then it changes almost nothing. Across 54 races no margin moved by more than 1.6 minutes, every changed verdict sat on the one-minute dead-heat line, and it created no hot win the well-mixed cup did not already have.
The reason is memory. A mug mixes away a difference between its top and bottom in about 3 minutes at 90 degrees, 6 minutes at 20, and 35 minutes just above 4. The hot cup forgets it was hot within minutes, in a race that takes hours, and the hottest water forgets fastest precisely because it is the least viscous. The effect of viscosity is real, and it points the wrong way for the slogan.
Two layers cannot hold everything a real cup does. The cold side walls drive a circulation that keeps a real hot cup warm on top, and a real cup keeps an ice lid over water near 4 degrees long after its first ice, where this one merges its layers at that moment. Those are the next places to look.
What is not in it
Also still missing:
- dissolved gas, which heating drives out;
- a hot cup melting the frost under it and sitting in better contact with the shelf;
- conduction through the growing ice, which is why both cups freeze solid faster here than they would in life;
- where the thermometer sits, beyond a top and a bottom. Burridge and Linden found that measuring the two cups a centimetre apart in height was enough to produce the appearance of an effect.
Every one of those is a candidate second variable, and the module says so rather than pretending to have covered it.
Race them yourself
This is the engine's own module running in your browser, switches and all. The top panel is temperature; with stratification on, the shaded band around each curve is the gap between that cup's top and bottom. The bottom panel is how much water is still liquid, and it is the one to watch. When the hot cup wins, both cups are sitting at exactly zero and a thermometer reads the same in each. The only place the win shows is in how much is left to freeze.