Kurtosis, Mpemba, Temperature and History

Just as a reminder what a couple of those words mean:

Kurtosis is the tendency for something to be a non-Gaussian distribution. Things where the sides of the curve are ‘fatter’ than usual; or similar non-random distributions. Think of a “bell curve” that has fatter ends with many more outlier data points.

Mpemba is the name stuck on the observation that sometimes warm water freezes faster than cold water. It is the name of a kid Tanzania who observed that he got ice cream faster if he used warm milk.

Kurtosis is important in the real world; but both things are widely ignored and misunderstood.

Substantially every intro stats class has a part where they stress that what they are teaching you applies only to a Gaussian distribution. (Later advanced stats relaxes that with other distributions). Most folks with only an intro to statistics rapidly forget that and try to apply those tools to all sorts of things without bothering to examine the distribution of their data first. That’s a fundamental error, and it is rampant in “Climate Science”.

Pretty much all we are taught about freezing water presumes a known standard heat of fusion, heat of vaporization, and specific heat. But the Mpemba Effect demonstrates that is not quite right. This assumption of known heats is also rampant in “Climate Science”, but it is not clear if it maters (where the statistics errors are known to be a big deal).

So instead of just poo-pooing the folks saying hot water freezes faster, some smart guys decided to actually investigate it. Water, being rather complex to study, they hit on the bright idea of studying a simpler material. They used a “granular gas”. What the heck is that? Well, turns out you can study things like sand as though it were a gas…

Now mix those three “edgy” thoughts together and what you get is insight and understanding.

https://phys.org/news/2017-10-counterintuitive-effect-hotter-cool-quickly.html

This link complained (politely) about my running an ad blocker (red stripe at the top of the page) and put up a similar warning stripe to advise me it uses cookies. Since I’m not fond of being nagged, I’m going to quote all the parts that I thought most interesting here for future reference. Bolding done by me.

Study may explain counterintuitive effect of why hotter systems can cool more quickly
October 23, 2017 by Lisa Zyga
[…]
In the last few decades, the Mpemba effect has been studied and observed in several physical systems besides water, including carbon nanotube resonators and ice-like water cages called clathrate hydrates. Despite these findings, the causes of the effect are not well-understood. Proposed explanations include the presence of impurities, hydrogen bonding, and supercooling. Even the mere existence of the Mpemba effect remains controversial, as one recent study found insufficient evidence to replicate a meaningful effect.

Now, their interest rekindled by a recent paper proposing a generic mechanism for similar effects, scientists Antonio Lasanta and coauthors from universities in Spain have returned to the question in a new study published in Physical Review Letters. In their work, the researchers theoretically demonstrate and investigate the Mpemba effect in granular fluids, such as those made of sand or other small particles.

Lisa writes really well, and it would be a nice thing to ‘hit the link’ and read the whole article. It’s a bit difficult to remove parts of the article from the quotes and not spoil the flow of the writing as she has already edited it to a tight text. In the original the word “paper” links to here:

http://www.pnas.org/content/114/20/5083

Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse

Zhiyue Lua,1,2 and Oren Razb,1,2

Author Affiliations

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 4, 2017 (received for review January 23, 2017)

Significance

It is commonly expected that cooling a hot system takes a longer time than cooling an identical system initiated at a lower temperature. Surprisingly, this is not always the case; in various systems, including water and magnetic alloys, it has been observed that a hot system can be cooled faster. These anomalous cooling effects are referred to as “the Mpemba effect”, and so far they lack a generic details-independent explanation. Based on recent developments in the theory of nonequilibrium thermodynamics, we propose a generic mechanism for similar effects, demonstrate it in various systems, and predict a similar anomalous behavior in heating.

Abstract

Under certain conditions, it takes a shorter time to cool a hot system than to cool the same system initiated at a lower temperature. This phenomenon—the “Mpemba effect”—was first observed in water and has recently been reported in other systems. Whereas several detail-dependent explanations were suggested for some of these observations, no common underlying mechanism is known. Using the theoretical framework of nonequilibrium thermodynamics, we present a widely applicable mechanism for a similar effect, the Markovian Mpemba effect, derive a sufficient condition for its appearance, and demonstrate it explicitly in three paradigmatic systems: the Ising model, diffusion dynamics, and a three-state system. In addition, we predict an inverse Markovian Mpemba effect in heating: Under proper conditions, a cold system can heat up faster than the same system initiated at a higher temperature. We numerically demonstrate that this inverse effect is expected in a 1D antiferromagnet nearest-neighbors interacting Ising chain in the presence of an external magnetic field. Our results shed light on the mechanism behind anomalous heating and cooling and suggest that it should be possible to observe these in a variety of systems.

Full text here:

http://www.pnas.org/content/114/20/5083.full

I’ll leave it for those interested to read the full text as it has a fair number of graphics in it and lots of formulas that don’t quote well / easily. (Lots of unicode needed to get it right). Back at the phys.org article…

Using simulations of granular systems and a simple kinetic theory approach, the researchers were able to determine that the initial conditions in which the system is prepared play a critical role in determining whether or not the system exhibits the Mpemba effect. Their analysis also enabled them to identify the initial conditions required in order for a granular system to exhibit the Mpemba effect.

“Our work shows that the existence of the Mpemba effect is very sensitive to the initial preparation of the fluid or, in other words, to its previous history,” coauthor Andrés Santos at the University of Extremadura in Badajoz, Spain, told Phys.org. “In our opinion, this may explain the elusiveness and controversy of the Mpemba effect in water, as a consequence of the lack of control on the detailed initial preparation of the sample.”

As the researchers showed, if a system is not prepared under certain initial conditions, then the colder system cools down more quickly than the warmer one, as expected, and there is no Mpemba effect.

“We theoretically showed, at least in the case of a gas, that a system’s temperature evolution and thus its cooling and/or heating rate do not depend on initial temperature alone, but also on the previous history of the system that control the initial value of the additional variables,” Santos said. “Therefore, it is perfectly possible that an initially heated system cools down quicker than a colder one with a different history.”

As the researchers explained further, the simplicity of the Mpemba effect in granular fluids compared to water and other systems enabled them to reach this conclusion.

“Our results show that the Mpemba effect is a generic non-equilibrium phenomenon that appears if the evolution of temperature depends on other physical quantities that characterize the initial state of the system,” Santos said. “In practice, such an initial state can be experimentally achieved if the system is taken by some physical procedure very far away from equilibrium (for instance, by a sudden heating impulse prior to cooling down). Our theoretical and computational work shows that the Mpemba effect is particularly simple in a granular gas, since, in practice, there is one single extra parameter controlling the Mpemba effect. This parameter is the kurtosis, which measures the deviation of the velocity distribution function from a Gaussian distribution.”

Guess what every climate model does NOT include… The Mpemba Effect and the history of the state of water in the system. They ignore kurtosis at all levels, but completely omit it at the level of water thermodynamics.

[…]
The results also support predictions of the existence of an inverse Mpemba effect: when heated, a colder sample may reach a hot target temperature sooner than a warmer sample. The researchers plan to investigate this area and others in the future.

“On the theoretical side, we plan to carry out a similar study in the case of a molecular solute (where collisions are fully elastic) suspended in a solvent that produces a nonlinear drag force on the solute particles,” Santos said. “Going back to granular fluids, we also want to analyze the impact of particle roughness and spin on the Mpemba effect. In the latter system, the simplest model would couple the temperature evolution to that of the parameter measuring the non-equipartition of energy between the translational and rotational degrees of freedom.

Gee, wonder if that might have anything to do with the sticky problem of cloud formation… are clouds a “molecular solute” suspended in a solvent (air) with nonlinear drag forces? What about dust in air and cloud formation?

So just how “settled” is this “settled science” for clouds, ice, our whole water world?

A technical managerial sort interested in things from Stonehenge to computer science. My present "hot buttons' are the mythology of Climate Change and ancient metrology; but things change...
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12 Responses to Kurtosis, Mpemba, Temperature and History

1. philjourdan says:

And here I thought it had to do with surface area exposed to the temperature gradient. But in essence they are saying that things return to their natural state faster than they change from it.

2. p.g.sharrow says:

When we measure the temperature of a volume, we are measuring the average temperature of the atoms contained in it. In a fluid, gas or liquid, the individual atomic/molecular temperatures may very greatly. Hot ones give up their energy loads quicker then the cool ones. This causes the volume average temperature to change faster…pg

3. A C Osborn says:

The classic example of this is when I make a cup of coffee, if I do not put any COLD milk in it, ie it is just hot water, it cools much quicker than if I put cold milk in it.
Even though the cold milk initially cools it.
I always thought it was to do with the milk making the fluid thicker and changing it’s surface tension.

4. jim2 says:

ACO – the milk’s oil may form a film through which the hot molecules can’t escape as readily.

5. jim2 says:

I’m thinking that the temperature in the core of an explosion cools much more quickly than an equivalent mass of, say, a ceramic heated to the same temperature.

6. oldbrew says:

There’s also a pressure factor…

It’s Scientifically Possible to Boil Water Until It Freezes Solid
As the video explains, you need a pressure chamber, which uses a vacuum pump to suck out all the air in the area you’re working with.

7. Part of the puzzle here is that in general the temperature of something is regarded as a single number that controls all the details of how energy is transferred, and that we can measure that temperature precisely by using a thermometer.

Whatever we’re using as a thermometer must give a give a reading that settles when the energy coming into it is equal to the energy going out of it (thermal equilibrium) so EM’s point about the “fat tail” is pretty important. The normal Maxwell-Boltzmann distribution of energy is assumed, and if that isn’t true then the thermometer will not give the correct number for the energy in the medium it is measuring. It may also mean that using a different method for measuring that temperature may give a different answer (for example using a stick thermometer, a thermocouple or an IR thermometer).

Saying that the temperature of something is a specific number is actually equivalent to saying that the amount of energy in each bin of an energy spectral analysis follows one specific curve. Although we pretend that this is a smooth curve, in practice (since matter is made of atoms) it must be stepped and we don’t measure the energy at a single point but over a small range. Count the number of atoms over that range of energy and produce a point that is itself an average. If that collection of average points falls on one curve of the Maxwell-Boltzmann set then that is the temperature. If it doesn’t follow that curve then the thermodynamic temperature is undefined, but a thermometer would still give you a single number as “the temperature”.

The underlying assumption (not often stated) is that the system has had long-enough to settle down and reach an equilibrium where the energy levels will follow that curve. When we’re heating or cooling some material then that isn’t necessarily true and normally won’t be.

The overall result is that the thermometer may lie to us when we’re measuring something whose total energy is changing too fast, and that how we measure that temperature is important. Temperature can be a tricky subject, mainly because we think it can be represented as a single number because we can feel it in our bodies and have grown up with thermometers that give a single number for the temperature at any point in time and space. It’s far more difficult to make something that will show us the actual quantities of atoms at each kinetic energy level range and the total energy contained in the sample.

As usual, using an average number (temperature) hides the details. For a lot of purposes, the average number is good enough. I’ve however been doing work on the details of the random interactions of thermal systems so found some counter-intuitive results. If we’re interested in how much energy is being transferred, then we have to work with the energy that is there and not use the measured temperature as a determinant of what will happen.

One explanation of the Mpemba Effect I read a while back was that the hotter ice-cream mix had more-pronounced circulation because of the increased heat-loss on the outside, and that these currents persisted as the mixture cooled. Testable by using a stirrer in both ice-cream samples, but I haven’t seen any tests of that being done. My bet is that such a stirring system would show that the cooler mix froze sooner. The hotter mix will of course lose energy faster initially, but the rate of loss of energy will most likely follow the expected curve (Newton’s law of cooling) provided the mixture is stirred enough that the outer layers (where the energy is lost) are moved fast-enough into the body of the mixture. Note that ice-cream mix is a bit thixotropic, so will provide a non-linear resistance to movement, and the mixture at the walls will thus be more-reluctant to move away from the walls and will form a bit of insulation unless there’s enough force applied to move it. If you have seen an old ice-cream churn, you’ll know that it has scrapers that pull the frozen ice-cream from the walls of the freezing-container – when this was designed people didn’t know the complexities of temperature, but found by experiment that such an arrangement gave better ice-cream faster than just freezing it without churning.

My explanation of the Mpemba effect would thus be dependent upon the actual measured temperatures of the “hotter” and “cooler” mixes. In this case, since the “cooler” mix was likely to be at around 25-30°C (since this was Africa in summer) then the “hotter” mix of maybe 40-50°C would gain circulation currents and cool faster because of those until it reached maybe 10-20°C, after which the circulation would slow or stop and the rate of cooling of both batches would be proportional to the measured temperature difference to the freezer. This may explain why some people confirmed the effect and some didn’t – it depends on what the mixture is, the temperatures of the two batches and the temperature of the freezer. In ideal conditions, it’s only going to be the initial temperature drop of the hotter one that will be faster because of the circulation currents, and whether these continue long-enough to overtake the cooler one depends on the details of the mix and the initial temperatures.

A lot of words, but thermodynamics (and temperature itself) is not as simple as is normally thought.

8. jim2 says:

OldBrew says: It’s Scientifically Possible to Boil Water Until It Freezes Solid

I’ve done this one. It works just fine.

9. jim2 says:

The Mpemba effect can be summarized as an improvement of heat flow by some means. In the case of ice cream, as Simon points out, it could be an initial viscosity change, or it could be that the initial hotter mixture becomes more well mixed than it would have otherwise and the effect on viscosity is persistent.

The alleged paradox is in the assumptions. For an even somewhat complex system, more than temperature or heat capacity are in play. You have to account for temperature, heat capacity, phase changes, chemical changes, thermal resistance, mass flows due to mixing … a whole host of variables come together to determine how quickly heat can move within, into, or out of the system.

10. J Martin says:

The 4th state of water may explain the mpemba effect. There’s an interesting YouTube video on the 4th state of water by a Washington ? University professor, though he didn’t cover the mpemba effect. It seems a close analysis of water demonstrates mysterious behaviour.

11. J Martin says:

12. pouncer. says:

Claims about mysterious states or properties of solutions that persist through changes in various other states and properties seems to me suspiciously like claims about “homeopathic” pharmacy. Doesn’t mean such claims are ‘wrong’, per se, but it does make me wonder ‘cui bono? ‘.

And THAT makes me wonder about English punctuation.