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January 2012 Archives

A colleague of mine in the Faculty of Education here at Waikato has drawn my attention to an elegantly titled paper by Andreas Quale: "On the Role of Mathematics in Physics"  Science and Education (2011), 20:359-372, for those of you who like references.

The paper is about the way mathematics relates to physics, highlighting the problem of teachers presenting the view (either explicitly or implicitly) that mathematics leads to physics. In other words, if you find a solution to an equation used in physics, that solution HAS to mean something. The one example quoted is Dirac's 'discovery', by theoretical means alone, of the positron, or anti-electron - here DIrac's equation, which he developed to obey certain conditions, had some unexpected solutions.

[1 Feb 12 - wrote this last night and got interrupted halfway through by a phone call - I now see I didn't finish this paragraph!  Dirac's equation, which described in mathematical terms a quantum theory consistent with special relativity, also had solutions that didn't seem to make sense. Dirac then thought through what the solutions would mean in term of physics, and came up with the interpretation of them being negative energy solutions - this in turn led to the idea of an anti-particle. Now, as it turned out, the anti-particles were later shown to exist, but did Dirac just get lucky here? Was there any reason per se that the strange solutions to the equation HAD to represent something physical? Probably not - for example, the average physicist is quite adept at picking the 'right' root from a quadratic equation and dismissing the other as 'unphysical'. Above all, physics is based on EXPERIMENT - i.e. the real world, not a theoretically constructed world (and remember, you're reading a theoretical physicist in this blog!]

Specifically, the author talks about the problem of presenting a realist ontology to students learning physics, whereas a better strategy would be to present a relativist ontology, as adopted by radical constructivism.

I have a couple of problems with this article. First of all, I'm not sure that the average physics teacher would recognize 'radical constructivism' if he got hit on the head with it or be able to tell at a glance a realist from a relativist. I certainly struggle with this education-speak.  But, most significantly, the author presents no evidence that the problem he describes is actually occuring. In other words, do physics teachers REALLY present the view that maths leads to physics - solve the equations and physics emerges no matter what. Maybe some do, but has anyone actually established this? If they have, could we have a reference or two for it.  In fact, there isn't a single reference for the first five pages of the article!- something that I have never, ever, seen in a physics paper.

You see, my recent interviews of physicists gives no evidence that this problem is happening - not in the tertiary sector at any rate. So, if there's a difference between what my data suggests and what the article suggests, I'd like to know some more details. And it's not there. Frustrating.

It will inspire me to do a bit of searching around, though.

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Yesterday I visited Auckland to interview a few people who work with physics in their industry-based jobs. This was part of my small research project on the role of maths within physics, which I've talked about earlier. One of the people I interviewed gave some really fascinating responses, perhaps in part because he had grown up in the former Soviet Union.

In the UK, and in NZ, doing science is equated by a great swath of people with being a geek. It pushes you to the margins of society; it's a real conversation killer to reveal that you do physics for a living. In short (and this is my opinion - I don't have a data source for this) science isn't properly valued because it's not understood what it achieves.

However, my interviewee described how being a scientist, and a physicist in particular, was one of the most admired positions that one could achieve. Scientists belonged to the 'elite' in the Soviet Union. They were respected, and science held great appeal. Those intellectually capable of doing it were motivated to do it.

Now, I'm not suggesting I'd rather have grown up in 70's and 80's Soviet Union as opposed to 70's and 80's United Kingdom (I'd rather be under the governance of Thatcher than Brezhnev any day), but perhaps we can learn something here. Why was it that scientists had such respect? Was it that people could see what scientists did and how they contributed to society? I'm speculating, but maybe it's worth probing at this one a bit more. In the meantime, I'll just have to work on the dumbed-down sanitized version of what I do at work for the next time someone asks me at a party.



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Our neighbours have a dog - one of those small, yappy, pointless pooches with an excessive bark-to-weight ratio. It runs about their garden, keeping our cat out and gets very loud when I venture over to the compost bin. The neighbours' garden is fortunately well-fenced, meaning the creature thankfully can't invade our side.

So, imagine my surprise, when I discover yesterday evening that one of our chickens is over on the dog's side of the fence. (It appeared the neighbours and the dog were out at this time). How did she get there? There's no obvious gap in the fence - there's even a mesh at the bottom of the fence presumably put up to stop the pesky canine from burrowing under (I mean, being the size of a rabbit it might start behaving like a rabbit). Hyacinth is only fourteen weeks old, and her ability to achieve flight is somewhat limited. She flaps her wings a lot, sometimes, but has shown little ability to rise more than a couple of centimetres in the air.

I had to go over to the other side  to rescue the very agitated chook, who was trying to squeeze through gaps in the fence that were way too small.

How did she get there? Sherlock Holmes remarked that, when you have eliminated the impossible, whatever remains, however improbable, must be the truth. In which case there is just one option: quantum tunnelling.

This is the phenomenon in which small particles can traverse through barriers that you wouldn't think they could get through on energy grounds alone - it arises from the probabilistic nature of quantum mechanics. It's a real effect, and can be seen with electrons. For example, the tunnel diode relies on this effect to function.

It's quite straightforward to estimate the probability of Hyacinth performing such a task. Using Schrodinger's equation, we can estimate the chance of an object tunneling as 'e' to the power of  minus (square root of (8 m V) ) times x divided by hbar), where 'e' is the base of natural logarithms, 2.71828..., m is the object's mass (say around 1 kg), V is the potential energy of the barrier, x is the thickness of the barrier and hbar is Planck's constant divided by 2pi, or about 10^(-34) Joule seconds.

Estimating the fence to be about a metre high, so the 1 kg chook needs a potential energy of about 10 J to get over it, 1 cm thick, gives us an estimate of 'e' to the power of minus 10 to the power of 32. Approximately. Simplifying a little, its about 10 to the power of minus 3 times 10 to the power 32, or:

0.000........0001, where the dots hide a few zeros.  Just how many zeros between decimal point and the '1'? Approximately 300 000 000 000 000 000 000 000 000 000 000 of them, which explains why I cheat and use the dots. It's a small number.

With no disrespect for Sherlock, I think that it might be prudent to think about other possibilities for Hyacinth's excursion next door. Such as maybe there is a gap in the fence that I've missed, or she's a better flyer than she lets on, or that those plums I had eaten were hallucinogenic and I've imagined the whole episode. Even that last one is more likely that quantum tunnelling.

So why does tunnelling work for electrons, but not for chickens? Fundamentally, it's because electrons are very, very, very small. You'd need about ten to the power 30 electrons to have the same mass as a chicken.

[2 February 2012 - Problem Solved? Last night I watched as the cat sneaked up on the chickens and then jumped at them from short range. In the general squarking and intense flapping of wings that followed, Hyacinth got at least a metre in the air. Brigitte, the other chook, stayed rather more terrestrial. His fun over, pussycat trotted away looking very smug.]





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At the end of last week, seeing a good(ish) weather forecast, we had a short break away in Wellington, taking the train both directions. Despite the lovely weather (Wellington isn't ALWAYS blowing a gale) we didn't get to see the volcanoes en-route, in either direction - they were shrouded in cloud as is often the case. The train journey from Auckland (or Otorohanga in our case) to Wellington is much hyped, and indeed it is very scenic in patches, and even spectacular in the occasional place (when the cloud lifts).  But it's not the most spectacular main-line train trip I've ever been on - e.g. the St Gotthard line in Switzerland beats it hands down. In my opinion, anyway.

One interesting feature on the line is the Raurimu spiral - a very loopy section of track where the line climbs about 200 meters up onto the top of the volcanic plateau in about 5 km. That's 5 km in a straight line - making an average gradient of about 1/25, (i.e. 1 metre up for every 25 m along), which, although fine on foot or a breeze in a car, is too much for a train to handle. Consequently the line does twice this distance, making a more manageable gradient of 1/50, achieving this through horseshoe bends and one complete loop, where the line crosses over itself. (Though note, a looped line is NOT unique to the Raurimu spiral, unlike the commentary implied, I know for sure, having travelled on it,  there are plenty of them on the St Gotthard line in Switzerland for example)

With ascending hills, its fairly obvious that, no matter how you do it, if you want to get from A to B you have to gain a specific amount of height. If B sits 200 metres above A, you need to climb 200 metres - whether you do this gradually or quickly, it is inescapable that you have to do it. That means moving 200 metres against a gravitational force. From a physics perspective, we can refer to this force as 'conservative', meaning that the amount of movement against it, from going from B to A, is the same no matter how you get there.

For example, you could climb a ladder that's 200 m high, straight upwards. In which case you move 200 m, and every step of it is in the exact opposite direction to the gravitational force. Or, you could walk up a 1/50 incline for 10 km (i.e. follow the train track). In which case you move 10000 m, but the gravitational force is almost at right-angles to your movement. For every 50 m you move horizontally, you only move 1 m against the force. By the time you do your 10 km walk, you've moved 200 m against the force, as you would climbing the ladder. Hopefully that's obvious.

In physics, gravity isn't the only force we consider. Take the electric force. If we have an electric field, a charged particle will feel it, and experience a force. The electric force is also conservative. That means if we move a charged particle from A to B, it will move the same distance against the force, no matter which route it goes in. The amount of potential energy it gains is the same, no matter the route. This is just like the gravity example. In gravitational fields (the earth) we consider lines of equal height as being 'contours' on a map - in an electric field the analogous lines are called 'equipotentials', meaning they are lines or surfaces of equal potential energy.

However, not all forces we come across are conservative. Consider a loop from A to B back to A again. Sometimes, when we do a loop, the total distance we move against the force is zero. That's the case in the gravity case - if we raise up 200 m then fall 200 m we end up at the same height we started with. We moved against the gravitational field on the way up (hard!), and with it (easy!) on the way down. But with the frictional force, for example, it doesn't work like that. No matter how you move, the frictional force acts exactly against you. If you push a box from A to B and back to A, it doesn't matter what route you take, friction acts against you all the time. Sometimes it might be more, sometimes less, depending on what surface you are on, but all the time it is against you. It's not a conservative force, and so it doesn't have equipotentials, or contours.

Of course, that doesn't mean friction is necessarily a bad thing - the train wouldn't be going anywhere if the wheels slid on the track. Conservative forces, however, are particularly neat from a physics perspective, and from a maths point of view the 'equipotentials', or 'contours', make them easy to handle. Working out how much energy the train uses to overcome the force of gravity on the ascent of the Raurimu spiral is easy - working how much energy the train uses to overcome friction or air resistance enroute is hard.

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I did an experiment last night on measuring the elastic constraint forces on light rigid bodies under extreme displacements. Or, in English, I tripped in the garden and stretched various ligaments in my right foot in ways in which they aren't supposed to be stretched.  Ouch.

I couldn't put any weight on the foot at all this morning, so had to use a chair as a walking frame and come downstairs on my bum. There's a lot to be said for one storey houses!

The X-rays show nothing is broken (though the doctor did find some evidence of a previous fracture that I didn't know I'd had!) but that doesn't stop it being very painful. Like with most stretchy things, there's a limit to a ligament's stretchability and I think I found it. Nothing that a course of leaches, some hot rocks or a dose of heavily diluted water can't solve, though ;-)





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On Saturday night my wife and I went down to Te Awamutu to watch 'Sherlock Holmes'   - there being no cinema in Cambridge :-(   It was a moderately naff piece of film - more James Bond than Sherlock Holmes - though quite suitable for some mindless Saturday night entertainment so long as you were prepared to put Sir Arthur's original creation to the back of your mind for a couple of hours.

Anyway, it inspired me to read again some of the stories, so I picked up the 'Complete Sherlock Holmes' from the shelf and, not wishing to drudge through A Study in Scarlet or The Sign of Four, I started with the first of the short stories, A Scandal in Bohemia.  N.B. I have read every one of the stories and novels - and could recount the plot lines for many of them - but A Scandal in Bohemia is one that escaped my memory.

There's a place in it that Holmes makes a fantastic comment on his methods. It's pretty well known - google it and lots comes up. Watson asks if Holmes has any theory about the case, and Holmes explains that he can't possibly have a theory, because he doesn't yet have any data.

 It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories instead of theories to suit facts.

 It's a great explanation of doing science. Let's start with what we know (the data) before coming up with an explanation. It's all too easy to have an explanation (something we'd like to be true - perhaps because it will bring in a nice big piece of industry funding if it is) and then go fishing for the data that supports it (and if we catch something else just through it back in the lake and wait until something more suitable turns up).  Not the way to go.

Kind of related to this, I too have been reading some of the 'alternative therapies' that have been reviewed recently by the NZ Herald. There's plenty of other blog comment on them, for example Alison's one here.  I loved Saturday's on Ganbanyoku - Japanese hot rocks. I'm sure lying on a hot rock can be very relaxing (so long as it's not too hot), but let's stick to what we know, rather than reporting total rubbish, shall we?

Unlike the sauna, stone bed [sic] heats through your body evenly and gently and not through the skin

says the hot rock expert. Let's think about that one. Doesn't heat you through the skin? Now, the last time I looked, most of my body was covered in skin. Are we meant to believe that somehow the heat gets in exclusively through our eyes, mouth and other unmentionable parts? Maybe a clue to the thinking comes later on in the article, with the journalist writing

As you lie on the stone bed, it gently heats you from the inside, gently spreading the heat around your body...

 Aha! So it's the old microwave 'heating from the inside out' myth, is it? The mysterious heat of the rocks somehow gets to your inside without first having travelled through your outside? Now, there's three ways heat moves - conduction, convection and radiation. Given the subject is lying blissfully in direct contact with the rocks, and there's not a great deal of air inside the human body, we can rule out convection. While the hot rocks will happily emit infra-red radiation, the chance of it getting inside you without being absorbed by the skin and other tissue on your periphery is pretty low. How do I know this? If it were otherwise, one could 'see' inside you using an infra-red camera. And I know from my use of them, as fun as imaging the human body is, you don't get to see inside, X-ray or MRI style. (At least, not directly. One can look for hot or cold patches, that may be related to tumors beneath the surface, but it isn't SEEING inside you.)  So, that leaves conduction. And, unfortunately conduction of heat from A to B involves going through all places in between. That means your skin.

So I think there has been a bit of twisting the facts to suit the old heat-from-the-inside myth. And I'm sure there are plenty more where that one came from.



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Last weekend we had an expedition out to rural Waikato somewhere beyond Morrinsville (in torrential rain of course) to a chicken farm and bought a couple of young chickens. The idea is that they'll give us an egg supply, and, if properly controlled (ha ha!) dig up the weeds, fertilize the garden etc. (Or, more likely, get into the vege patch and destroy it.)

Having watched their behaviour, it's clear there isn't much to a chicken. They basically do three things:: 1. Peck at the ground. 2. Walk 3. Make a 'puk puk puk' sound.  Less frequently they also wake up and sleep, drink, poo, and, just occasionally, acknowledge the existence of the cat.

And that's the lot. The first three things seem to be automatically programmed into the chicken - they are natural cycles the chicken does. Go 'puk', go 'puk', go 'puk' etc, until some major input disrupts this cycle.

They are rather like the neurons I model on the computer. Neurons have a few, fairly simple modes of behaviour, depending on their situation. A common mode is simply to fire electrical pulses regularly, unless something disrupts it. How quickly they fire depends on the extent and type of the electrical signals they receive - the more of them there are, the faster the neurons fire (if the signals are of an excitatory type).

The chickens tend to keep themselves close together, and it's interesting to note that the frequency of the 'puk puk puk' increases the closer they get.  (By this I mean the number of 'puks' per minute increases, not the pitch of the 'puks'.) One's noise is clearly encouraging the other to make more noise, and vice-versa. So, in physics terms, we have two coupled oscillators with a variable connection strength.  The overall puk-rate depends on the connection strength between them - the closer they are - the faster they oscillate. Just like neurons, really.

Furthermore, if there is an external event that preferentially takes their attention away from their friend (e.g. I try to grab one of them as she heads for the bok choy) their 'pukking' cycle is momentarily disrupted, other, less harmonic sounds appear, until the disrupting stimulus leaves and, after a time lag, the normal cycle of behaviour resumes. A set of coupled neurons can do just the same when you give it a sharp, disruptive stimulus.

Incidentally, it's also been interesting to watch the behaviour of Mizuna the cat towards the chickens. He started, predictably for a feline, with intense curiosity coupled with a bit of sheer terror (that's terror on the part of both the chickens and the cat), but the relationship has progressed through mutual ambivalence to what might be described as friendship. He now seems to spend time following the chickens about. Just so long as he doesn't start thinking he is a chicken.






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One of our recent Christmas gifts was an ice-cream maker. It's been very welcome, what with all those long hot sunny days we haven't had.

Making ice-cream is pretty straightforward. Just bung in the ingredients and stir the mixture while having it freeze. That's what the machine does. Stirring makes sure large ice crystals don't form, and also aerates the mixture to give it that gorgeous texture that every summer day needs - if we every get any.

Now, the difficult thing with a benchtop ice-cream maker is obviously getting the mixture to freeze. That means cooling the bowl, and cooling isn't an easy job for nature to do - the second law of thermodynamics tells us that. There are several solutions - one is to make the machine a mini-freezer in itself, and give it a pump, coolant and power supply. That's going the way of the commercial machines. You could put the machine in the freezer - but then you need a power supply to turn the mixing paddle - so you need some decent batteries or a power lead coming out of your freezer.

The most elegant solution in terms of keeping things tidy is what's done with our machine. The bowl has two layers - an inner container (which has the mixture in) placed in an outer container - and between the two is some kind of of liquid or gel that acts as a coolant. The two containers are sealed together, so the coolant doesn't escape. The idea is that you put the bowl in the freezer for several hours before you want to use it - this takes the bowl down to the correct temperature - then you bring it out.

Just how the coolant works I'm not sure, but it will have at least two purposes - first to be an insulator, slowing the passage of heat from the outside into the bowl when ice-cream making is in progress, and, second, to have a high thermal inertia. It will take a lot of heat for it to increase its temperature significantly - but also, it will need a long time in the freezer for it to cool down significantly. Also, the coolant may undergo a phase change when cooled and use latent heat to keep the bowl cold. In this case, the coolant requires heat to melt, and takes this from the contents. Thus it can bring the contents down quickly in temperature.  A mix of water and some impurity like salt would do this as it will freeze/melt at a lower temperature than zero degrees C, thus keeping the bowl below zero as it melts.

This kind of cooking is about taking heat away, rather than providing it, and is just as much to do with physics as is heating things up.



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I was back in the open-air university swimming pool yesterday. It was raining (inevitably), not that that really makes a difference. (I'm always amazed why the pool is so empty when its raining - as if it makes a difference to how damp you get).

With a nice rainstorm going overhead I got to see some unusual views of water droplet splashes - rather similar to the one I've pasted here from If you get your eyes close to water level, the view is pretty impressive, with little cylinders of water rising up a couple of centimetres from the surface for maybe a quarter of a second - just long enough to take it in before it vanishes.

Explaining just why a falling droplet makes such a tower when it hits the surface is something I won't try to do - since the governing physics is non-linear it's not a straightforward process.


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While waiting for my aged computer to boot-up on my return to work this morning, I was skimming through November's Physics World magazine, and noted an obituary to Rudolf Mossbauer.

He is best known in the physics world for observing 'resonance absorption' of gamma rays, and then developing the technique of Mossbauer spectroscopy.

When a nucleus undergoes gamma decay, it emits a gamma ray, which is a high-energy electromagnetic wave. (Light, radio, microwaves are other examples of electromagnetic waves). What happens is that the nucleus moves from a high energy state to a lower energy state, and this difference in energy is carried away by the gamma ray.

Well, not quite. That's because the nucleus will recoil when the gamma ray is emitted, like a gun will recoil when a bullet is fired. A small amount of the energy will be taken up by the recoil, so not quite all ends up in the gamma ray. This means that, if the gamma ray hits a similar nucleus in a low-energy state, it is not usual for the ray to excite the nucleus from the low-energy state back up to the high-energy state - 'resonance absorption'.

What Mossbauer did was to eliminate the recoil by embedding the nucleus in a crystal. With no energy being taken away by recoiling nuclei, all is available to the gamma ray, and resonance absorption can occur.

If you're not impressed by that clever bit of thinking, there's more. One can alter the energy of the gamma ray with the Doppler effect by having a moving source of gamma radiation. If we move the source towards a sample of material, the energy of the gamma rays hitting the material will be increased slightly; conversely, if we move the source away they will be reduced slightly. Since the exact energy of photons that will be absorbed by a sample of material will depend upon the chemical environment of the nuclei in the sample, this provides a very clever spectroscopic way of probing the chemical environment of a sample.

We have a set-up in our lab - Unfortunately it hasn't been used for a while since the cost of obtaining the Mossbauer sources (we use Cobalt-57 embedded into a crystal) is too great for it to be cost-effective for us. The source is mounted on a piston that can be driven at a very precise velocity towards a sample of material (that also contains Co-57 or Fe-57). By looking at the absorption at different source velocities we are effectively finding information out about the recoil of the Co/Fe-57 nuclei in the sample, which in turn tell us about the chemical environment of the Co/Fe-57. We don't need to move the piston very quickly - just a few millimetres a second is sufficient to provide the range of photon energies we need.

A pretty clever technique, really, but one that isn't greatly used today.

Rudolf Mossbauer received the Nobel Prize for Physics in 1961 for his work.

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