I’m frightfully late on following up on this, but ElKement has another entry in the series regarding quantum field theory, this one engagingly titled “On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace”. The objective is to introduce the concept of phase space, a way of looking at physics problems that marks maybe the biggest thing one really needs to understand if one wants to be not just a physics major (or, for many parts of the field, a mathematics major) and a grad student.

As an undergraduate, it’s easy to get all sorts of problems in which, to pick an example, one models a damped harmonic oscillator. A good example of this is how one models the way a car bounces up and down after it goes over a bump, when the shock absorbers are working. You as a student are given some physical properties — how easily the car bounces, how well the shock absorbers soak up bounces — and how the first bounce went — how far the car bounced upward, how quickly it started going upward — and then work out from that what the motion will be ever after. It’s a bit of calculus and you might do it analytically, working out a complicated formula, or you might do it numerically, letting one of many different computer programs do the work and probably draw a picture showing what happens. That’s shown in class, and then for homework you do a couple problems just like that but with different numbers, and for the exam you get another one yet, and one more might turn up on the final exam.

That’s all fine for building one’s ability to do the calculus involved, and maybe the numerical programming too, but it’s also kind of dull. If you can do this problem correctly once you can do it any number of times; we just have students do the problem a couple times over because it takes practice to get it correctly.

But what’s *interesting* isn’t the exact solution of the exact problem for a particular set of starting conditions. When your car goes over a bump, you’re interested in what the behavior is: is there a sudden bounce and a slide back to normal? Does the car wobble for a short while? Does it wobble for a long while? What’s the *behavior*?

And this is where phase space gets to be interesting and where you become a graduate student: instead of looking at how high the car is after one shock, look at what the important variables are. In a problem like this, that’ll ordinarily be the position and the momentum of the car. As time goes on, the position and the momentum of the car are going to change, in a way that chains the both of them together along with important physical properties like the energy of the system. You can imagine this — and you should, before your qualifiers — by imagining the position and the momentum as the different axes of a graph, and the system as a point that moves around in a path called either a trajectory or an orbit.

And where this gets really interesting is to imagine a whole bunch of points — positions and momentums for the car — and what happens to that collection as time evolves. Each one of these points represents a different shock that the car can get, and that the shock absorbers try to deal with; but each of them is going to react a little bit differently. The different points will spread out in a way that looks uncannily like bubbles in a rushing stream. Different points might go into different directions, and that represents the original system doing different things — bouncing the one time, for example, or bouncing twice, or bouncing for a long while.

After that, in grad school, you can start looking at a physics problem by trying to identify the different kinds of behavior, such as how many bounces it takes for the car to settle down again, and then working out what combinations of position and momentum put the car in each of those different behaviors, and where the dividing lines between those kinds of behavior are.

ElKement, as might be expected, discusses all this (so if *my* version doesn’t make sense, perhaps hers will, particularly as she’s got animated pictures to show it), as well as how it starts to change if you start to look into quantum mechanics.

Thanks again, Joseph. I guess your example concerned with a car makes more sense than my rather philosophical ramblings so I will reblog your post. From questions I have got on my article I conclude that the introduction of those large number of dimensions was probably not self-explanatory, so I am working on an update to this article with more illustrations related to hyperspace. Since I have not seen Liouville equation popularized (in the same way as ‘chaos theory’ is) I have to prepare some illustrations myself which is going to take a while.

I am also considering to pick a non-physics example as I have noticed that ‘many dimensions’ in the sense of these statistical sense (different states of a single system) got confused with those fancy dimensions related to string theory. I think I need to stress more that this is a space of ‘possibilities’ and not some space which is out there (and we are just those infamouse beetles on an inflating balloon that are not able to feel those dimensions).

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The car example was actually on my mind because I was thinking about writing a post regarding how I set up a differential equations laboratory project (where the students had to answer a set of problems both analytically and numerically, using a Mathematica-like program called Maple to work it out), and then I realized it was a pretty good setup to introducing phase space. I imagine trying to introduce them has to be done in either pendulums or masses-on-springs for want of other simple systems that are interesting and don’t have too many spatial dimensions.

Also I suspect you’re right in talk about dimensions confusing people because “dimension” is a word with so many meanings. I’m not sure of a good alternate word to use, though, and if I do carry on I might just give in and speak of “phase space dimension” instead. The construction is horrible but at least it makes the context clear enough.

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Reblogged this on Theory and Practice of Trying to Combine Just Anything and commented:

I am still working on a more self-explanatory update to my previous physics post … trying to explain that multi-dimensional hyperspace is really a space of all potential states a single system might exhibit – a space of possibilities and not those infamous multi-dimensional world that might really be ‘out there’ according to string theorists. In the meantime, please enjoy mathematician Joseph Nebus’ additions to my post which includes a down-to-earth example.

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(I would really like the typos in my reblog here ;-) )

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Aw, and I do want to thank you for kindly reblogging me. I think yours might be my first referral.

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It’s nice to see Elke’s influence spreading far and wide about the internet. D

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It is, and I’m glad to have found her blog.

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Thanks, Dave and Joseph! Flattering to hear people talk about me (in a nice way) :-)

I am also very grateful for those referrals of yours, Joseph. My blog is rather anti-popular – concluding from my numbers (views, followers) when comparing with other bloggers who blog for about the same time. Probably my blog is also not enough ‘niche’ and specialist and too much all over the place (Search term poetry, physics, random thoughts on society and the corporate world…)

Blog visitors constitute probably another hyperspace whose dynamics should be modeled. I guess we might find out that occasional spikes in popularity (Freshly Pressed, going viral) are dictated a chance.

But I believe it is better to have a small number of loyal followers you can have very interesting discussions with instead of 1000 people following and liking your blog because of its established popularity (winner-take-all effect in networks). I suppose if we would investigate the numbers of followers on all blogs in the world, assuming a reasonable time you need to ‘invest’ in reading as a follower… we would find out that most followers of those very popular blogs actually do not really follow because then they would need to spend 24/7 on speed reading.

This was probably quite random a comment. but I am re-reading Nassim Taleb’s books currently – so I am intrigued by anything Black-Swan-like …

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Ah, well, the blog is always greener … I worry about my own lack of popularity and rather admire the community you’ve got over at your site. I haven’t figured out the knack for drawing people out to making comments, yet. I might start giving out challenges.

As you say, though, a small group of people who’re actually listening is rewarding, and I just hope to get one new reader for every three spambots offering to get me to get rich by blogging.

Oddly I know that I got one of Taleb’s books from the library, as an audio book (I enjoy listening to them while commuting), but I didn’t get far into it. I’m not sure what happened; conceivably it was recalled back to the library.

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