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](https://en.wikipedia.org/wiki/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](https://physics.stackexchange.com/q/32989) 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](http://www.rsc.org/mpemba-competition) ([Wayback Machine](https://web.archive.org/web/20150327030926/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, see [O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox](https://arxiv.org/abs/1310.6514) and [Mpemba Paradox Revisited — Numerical Reinforcement](https://arxiv.org/abs/1312.1014). 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.



On mathematical studies of the Mpemba effect
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](https://web.archive.org/web/20150203202503/http://www.rsc.org/mpemba-competition/mpemba-winner.asp))


