# Mathematical study of Mpemba effect?

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It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named after a student who rediscoverd the effect in the sixties. Several theories have been proposed to explain the effect, but so far none of them seem to be generally accepted, see, e.g., this discussion on http://physics.stackexchange.com. In 2012 the Royal Society of Chemistry offered £1000 to the person or team producing the best and most creative explanation of the phenomenon, see http://www.rsc.org/mpemba-competition/. One problem is that many factors might play a role. The theories that try to explain the effect involve, for example, evaporation, convection, gas dissolved in the water, or interactions on molecular level, and it is difficult to design experiments that allow to isolate these factors.

Are there any mathematical studies (exact solutions for special cases, numerical analysis, simulations, etc.) based on the equations proposed to describe or explain the Mpemba effect? Do they allow to isolate different influences and to compare them with experiments, e.g. by simulating heat flow with convection and/or evaporation.

Does anybody here know of any such work? Or does anybody have a reference on simulations of similarly complex thermodynamical systems like a heat flow with convection and/or evaporation?

PS: I discovered two papers by a group of Chinese chemical physicists, seeO:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox and Mpemba Paradox Revisited -- Numerical Reinforcement. The second uses a finite element method to solve a one-dimensional model. I am not an expert in numerical analysis, but I believe modern mathematics should be able to go further than this.

PPS: I changed the formulation of the second paragraph, following Theo Johnson-Freyd's remark.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
asked Jan 5, 2014
retagged Oct 29, 2015
Surely this is not an appropriate question for MathOverflow. It belongs on a physics or chemistry forum instead.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
I know from experience that some people here seem to have a rather limited view of what mathematics is (I find very regrettable how mathoverflow.net/questions/80146/… was pushed back and forth between math and physics), but my question is mathematical. The question is whether mathematics can give sufficiently precise solutions to the complex systems of equations coming from the theories proposed by physicists to compare them with experiments and thus help to select the theory which gives the best predictions.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
Okay, I will give you the benefit of the doubt by reopening. Thus, I'll let the community decide this one. But I will note that Carlo Beenakker, who evidently has some expertise in both mathematics and physics, had cast a vote to close.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
Thanks! I find it deplorable that someone (not you, I think) had to down-vote Zurab Silagadze's reply just because he or she didn't like the question (or someone explain to me what is wrong with his answer).

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
No, I wasn't the downvoter on Zurab's answer; I just checked to make absolutely sure. (I'm in a slightly bad mood over some things seen on MO today, and did a spot of downvoting, but not on his answer which actually provided some relevant information.)

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
If I would give this problem to a student in physics, I would make sure this student has a good math background and programming skills. Nonlinear coupled differential equations require both to make progress, in addition to physical intuition to decide what terms in the governing equations need to be retained and which can be neglected to make the problem more tractable. These considerations are the essence of theoretical physics, and here at SE they are discussed in the physics forum. For MO or MSE a specific mathematical angle is needed. I do not see it here, which is why I voted to close.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Carlo Beenakker
I recently attended a beautiful talk by Pierre-Louis Lions on Analysis/PDEs, making models and simulating them (with a funny remark on the cheap simulation of the water in James Cameron's Titanic). I haven't worked in this area and don't know much about it, but for me it is part of research level mathematics. I would like to know where the boundaries of what is possible lie.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
The title of this question led me to think the question itself would be, in essence, the reference request "Do there exist mathematical studies of the Mpemba effect?" I would hope such a question could remain open, but I would expect the answer to be "No, such papers do not seem to exist in the literature". But the question as posed discusses instead the role mathematics has "to play in this discussion, by allowing a quantitative study" of some such things --- which is not about mathematics, but rather, as Carlo Beenakker points out, the essence of much of quantitative science.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Theo Johnson-Freyd
@Theo My question was exactly that reference request. I guess you are right, such papers do not seem to exist, or, even if they exist, the mainstream on this forum does not know them and would not be interested in them. Please excuse me for trying to explain in why I think such studies should exist and would be important. Btw, could you please explain to me what you mean by "quantitave science" and why none it is not part of mathematics, or where the borderline between mathematics and "quantitative science" lies? Do I have to ask the director of my department to fire several of my colleagues?

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
Do I have to ask the director of my department to fire several of my colleagues? *smile* You know, I work in a mathematics department, but focus on some corner of theoretical physics. Applied mathematics is a huge field, one that I don't know much about (and I know essentially no numerical analysis). I would love to attend colloquia on Mpemba effect if there is interesting mathematics there, and I really was hoping that this question would turn up some.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Theo Johnson-Freyd
I don't understand why this has been put on hold as off-topic. I know how the scope of this forum is defined in the help center, but I checked again. It doesn't say that we can only talk about pure mathematics here. The tags "simulation" and "numerical analysis" exist on this forum, they are part of applied mathematics, and simulating the non-linear partial equations that could describe the Mpemba effect would clearly be research level mathematics, IMHO. I wish those who voted to close this could explain their interpretation of the rules of the help center.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
@UwF Perhaps I've confused you by using the phrase "pure mathematics" under Zurab's answer. That was in my reply to guest, to try to explain why a mathematical question about Navier-Stokes would be squarely within MO territory; the same phrase was not meant to exclude "applied mathematics questions" from consideration at MO. But instead of arguing about it here, let's open a thread over at MO meta, where arguments about appropriateness should take place. This could get us started: meta.mathoverflow.net/questions/1374/….

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
@Todd Generally speaking I refuse to separate mathematics into pure and applied, I think such a distinction is not helpful. But I think my question concerns research level applied mathematics, and I am still waiting for those who voted to close this discussion to explain their reasons to me. I am looking forward to the discussion on MO Meta, thanks for opening it.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
@UwF Yeah, those phrases you don't like were put inside quotation marks for a reason. You are far from alone in your feelings about that.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
I certainly support and encourage questions of this type on MO.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Gil Kalai

+ 6 like - 0 dislike

Try this reference:

O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox, by Xi Zhang Yongli Huang, Zengsheng Ma and Chang Q Sun http://arxiv.org/abs/1310.6514

P.S. I see you have already found this reference. Some useful information about Mpemba effect can be found here http://math.ucr.edu/home/baez/physics/General/hot_water.html By the way it seems the competition already has a winner: http://www.rsc.org/mpemba-competition/mpemba-winner.asp

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Zurab Silagadze
answered Jan 5, 2014 by (255 points)
Thank you for this reference!

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
Please let us not answer questions that are clearly inappropriate for MO. I mean, this is not even borderline appropriate, however interesting it may be.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
Yes, unfortunately it is too late to participate in the competition. Somewhere on the webpages about the competition I found a remark that the articles submitted for the competition give the impression that experiment is more successful than theory in unravelling the mystery of the Mpemba. I think this status is unacceptable, a good explanation of the effect should lead to numerical models.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
And an explanation of the effect can only be accepted if there is a reasonable agreement between experiments and simulations of numerical models based that explanation.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
If a question following from the mathematical formulation of Navier-Stokes can be a Millenium mathematics question, then how are questions following from the mathematical formulation of the Mpemba effect so clearly inappropriately mathematical?

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user guest
@guest Right: even though the Navier-Stokes equation is motivated by physics, the Millenium Prize problem on the Navier-Stokes equation is carefully formulated to be a question of pure mathematics (see page 2 of claymath.org/sites/default/files/navierstokes.pdf). As Carlo Beenakker explained, there are physical judgments which enter the problem described here. So, while mathematics does play a role (as it does everywhere in physics), this in my opinion is not a strictly mathematical problem. (I won't argue this further, because after all I was the one who reopened the problem!)

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Todd Trimble
The Millenium Prize problem on the Navier-Stokes equation is a pure math problem, which my question isn't. Because, as far as I know, there is no generally accepted mathematical equation for the Mpemba effect, and because I don't expect that there are any exact solutions, existence or uniqueness results known, except for very special cases. But the rules in the help center say that this forum is about research level mathematics (not only pure mathematics). Could the people who voted to close this discussion explain to me why they think it is off-topic???

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user UwF
+ 4 like - 0 dislike

Try:

X. Zhang, Y. Huang, Z. Ma, Y. Zhou, J. Zhou, W. Zheng, Q. Jiang, and C.Q. Sun, Hydrogen-bond memory and water-skin supersolidity resolving the Mpemba paradox. PCCP, 2014. 16(42): 22995-23002.

X. Zhang, Y. Huang, Z. Ma, Y. Zhou, W. Zheng, J. Zhou, and C.Q. Sun, A common supersolid skin covering both water and ice. PCCP, 2014. 16(42): 22987-22994.

This post imported from StackExchange MathOverflow at 2015-10-29 18:25 (UTC), posted by SE-user Sun Changqing
answered Nov 23, 2014 by (40 points)

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