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Comment by tshaddox | Hacker News Reader

What still isn’t clear is what the the initially-hot system has “learned” at the moment it reaches the starting temperature of the initially-cool system. There must be some difference between the two states at that temperature if the initially-hot one is going to catch up with and surpass the initially-cool system.

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From my reading, the point is that the glass of water doesn't really "have a temperature" any more, since the cooling transition means it's no longer in thermal equilibrium.

That is, you could perhaps model the bulk of water as having a temperature field, and clearly every point in that field passes through the starting temperature of the initially-cool system, but the gradient landscape is vastly different.

But we are still dealing with energy? Every point should still transfer energy which should take time?

no, temperature is simply average kinetic energy and is always defined, even for systems far from equilibrium

The point is that the state of the system consists of more than its average temperature. It's not just "catching up" to the state of the initially-cold water, it's going through other configurations.

Would be interesting to know what the outcome of fluid dynamics and physics simulations of this setup is.

Heat transfer, convection, conduction, evaporation and so on should be available in useful implementations in state of the art simulation software.

The simple answer is that the cool system has a very uniform temperature (because it's been at that temperature for a long time), while the hot system that's cooling down is doing so with large spatial variations of temperature. In other words, all the water in the initially cool system is at 10°C, but the initially hot system has pockets of 5°C and pockets of 15°C.

It doesn't have to "learn" anything in order for there to be a substantial difference.

Interesting theory for a specific mechanism: "local temperature difference-induced convective cooling". Could be falsified by giving both initially-hot and initially-cold samples a stir rod, expecting the cooling performance of the initially-cold sample to improve to match the initially-hot sample.

Ho ho, guess what?

> The Scottish scientist Joseph Black investigated a special case of this phenomenon comparing previously-boiled with unboiled water; the previously-boiled water froze more quickly. Evaporation was controlled for. He discussed the influence of stirring on the results of the experiment, noting that stirring the unboiled water led to it freezing at the same time as the previously-boiled water, and also noted that stirring the very-cold unboiled water led to immediate freezing. https://en.wikipedia.org/wiki/Mpemba_effect

I'm surprised the article didn't mention this.

Wow! I guess that's the final answer then. Maybe to seal it, devise the inverse experiment where you somehow inhibit normal convection in boiled water and expect to see it take as long as the cold water.

Very interesting!

These temperature gradients obviously lead to convection.

Stronger convection means greater heat transfer, thus greater rate of cooling down.

But it'd be surprising if the inertia of the convection of the initially hot system didn't just gradually decline (because of friction) to almost exactly (little bit greater) the same level of convection (which the initially cool system had in the beginning) when it reaches the same average temperature.

Water currents can last a surprisingly long time. Pour one liquid into another of a slightly different color and the swirling mixing process can go on for minutes at least. Perhaps hours in the right circumstances.

Think annealing.

It's found it's way into the "minimal valley" whereas an arbitrary lukewarm state might be closer to a "ridge".

Clearly, yes, if you could start at the magic state that would be ideal. And there should be experiments that use a bunch of thermometers too compare the time from the same average temp (the initially warmer one just gets the "running start").

Mathematically it is not hard to abstractly characterize what is going on. (This is not saying the actual physics is easy!!) Temperature is an equivalence class on fluid states, but the average time to transition between those states does not form a metric space. The failure of the triangle property shows that the composition of transitions induces non-uniform distributions within the temperature equivalence classes that subvert the expected transition time by which we had attempted to build a metric space to begin with.

Just like non-euclidian space in the 19th century, this is the sort of thing where the mathematics can say "yeah sure seems legit" before the physics stops saying "wait wtaf",

That would imply that water once heated and then cooled is different from water never heated - at least for a time. I wonder if there is any evidence for that.

One difference could be that hot water is not as good a solvent for gas (think soda bubbles). So heated water would have shed most of its gas content which would lead to a different (faster?) freezing process, if freezing occurs faster than gas can dissolve back in.

Problem with that theory:

1. Temperature is monotonically related to energy content. A warmer system has more energy than a colder one, all else being equal.

2. To cool, a system must release energy. The rate at which a system can release energy is monotonically related to the difference in temperature between the cooling system and the cold sink to which its energy is being released. The bigger the temperature difference, the higher the rate of energy release (all else being equal).

For the Mbemba effect to be real, one of those two premises must be false. Which one is wrong?

> Temperature is monotonically related to energy content. A warmer system has more energy than a colder one, all else being equal.

In equilibrium, the temperature of the water is directly related to the energy content, with the heat capacity being the conversion factor.

But under the thermodynamic definition, the temperature of the system depends not only on its energy but also on its entropy. And one of the points the article is making is that even when we know the total amount of energy entering or leaving the system, its entropy may not be nearly so easy to measure or calculate when its state is far from equilibrium.

Sorry, but that is not true. The definition of the SI unit of temperature, the Kelvin, is given in terms of Boltzmann's constant, which has units of Joules (i.e. energy) per Kelvin.

IIRC (my thermodynamics is a long time ago), but 2 applies only when the two (cooling system and sink) are near equilibrium, which is not the case here.

I don't think either one is false. To me, 2 looks false if you consider the "system" in question to be the entire glass of water. But my interpretation is that in this case the glass of water is really many very small systems (pockets of unequal amounts of energy) all interacting with each other.

Let's say you're given a few hot potatoes and have to cool them down as fast as possible. You have a refrigerator but you can only keep one of them in it at a time. How do you decide which one to put in at any given time so they all reach temp the fastest? When the "system" involves many possible pairings of temperature differentials it feels very intuitive that some configurations would be better than others. So it removes the mind-bending thermodynamics law breaking aspect of it.

Admittedly, this isn't really what the article says! I honestly don't get how my above intuition squares with the whole energy minima thing, and so while useful as a thought exercise to at least help me entertain the idea, I'm not really sure it's correct?

The second one is false.

https://en.wikipedia.org/wiki/Leidenfrost_effect

Basically, heat transfer rate in a liquid goes down when a certain temperature is reached, because an insulating layer of vapor is formed.

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