# On mathematical studies of the Mpemba effect

Since the days of Aristotle and Descartes, it has been known 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 rediscovered 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 Physics SE. In 2012, the Royal Society of Chemistry offered £1000 to the person or team producing the best and most creative explanation of the phenomenon, the Mpemba competition (Wayback Machine). 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, see O: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.

• Surely this is not an appropriate question for MathOverflow. It belongs on a physics or chemistry forum instead. Jan 5, 2014 at 19:25
• 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.
– UwF
Jan 5, 2014 at 19:34
• 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. Jan 5, 2014 at 19:43
• 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. Jan 5, 2014 at 20:08
• 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. Jan 6, 2014 at 2:09

Try this reference:

O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox, by Xi Zhang Yongli Huang, Zengsheng Ma and Chang Q Sun https://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 https://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 (Wayback Machine)

• 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. Jan 5, 2014 at 19:26
• 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.
– UwF
Jan 5, 2014 at 19:28
• 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? Jan 5, 2014 at 20:21
• @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!) Jan 5, 2014 at 21:16
• 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???
– UwF
Jan 7, 2014 at 13:37

Since this question is still open, I take the liberty of pointing to a recent survey of the status of the Mpemba effect, Pathological Water Science -- Four Examples and What They Have in Common, which draws the following conclusion:

If confounding factors (such as evaporation, dissolved gases, mixing by convective currents, inefficient thermal contacts) are removed, hot water may indeed freeze earlier than cold water, but not because it cools off more quickly: the hot water remains warmer than the cold water during the cooling process.

Nucleation sites in the container may lower the freezing temperature to anywhere between $$0$$ and $$-45^\circ$$ C and it may happen that the container filled with the cold water needs to be supercooled to a much lower temperature before the water freezes than the container filled with the hot water. The freezing temperature is a reproducible but unpredictable property of the container. If it would be possible to have two containers with identical nucleation sites, then cold water would freeze earlier than hot water.

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.