Basic Thermodynamics Mathematical Witchcraft.

[Taikand]

Greetings forum.
I was (re)learning thermodynamics for my finals when I hit a problem. The teacher gave us a table with all sort of formulas for calculating the heat transfer and the mechanical work during a process (usually isobaric, isochoric and isothermic, sometimes adiabatic ) and I've been thinking that maybe there is a general formula that I can use on any arbitrary process?Maybe something related to the graphical representation (That is actually a 3 dimensional graph (4-dimensional if we take time into consideration) projected on a 2D surface ) ? My teacher knew not of any such mathematical wizardry and I thought of asking you.

[SU3SU2U1]

Well, from the first law work is always \int p dV, while heat flow is \int T dS, up to signs (heat flow in or out of a system, work done on the system or by the system). You should be able to use that, along with an equation of state to work out any formula you need.

[Taikand]

What's the purpose of classical thermodynamics if it's laws can be derived from statistical math and newtonian laws? I really hate it how it throws around all kinds of concepts without an explanation on why things happen as they do.

[mfb]

What's the purpose of chemistry, if its laws can be derived from physics?
Simple reason: You don't want to use the full physical description every time you look at some molecule.

In the same way, you don't want to talk about phase spaces and density matrices every time you have an expanding gas somewhere.

[starslayer]

Another possibility for why they haven't told you the true reason why a bunch of things in thermodynamics are true is that you don't have the tools to understand those reasons yet. Even among physics majors, many stat mech classes are famous for being confusing as all hell and rather impenetrable (I don't think I even kind of understood what the hell my textbook was saying until halfway through the course myself). And this is during upper level undergrad, after quite a bit more physics than you've had so far.

[SU3SU2U1]

Well, thermodynamics are the laws we actually see and measure with our instruments, which is clearly an important consideration. Also, the thermodynamic laws are very robust- it doesn't really matter what the underlying microscopic theory looks like, as long as it conserves energy and momentum you'll recover standard thermodynamics! Thats why the discoveries of quantum theory don't matter at all for thermodynamics problems.

[Minerva]

Another possibility for why they haven't told you the true reason why a bunch of things in thermodynamics are true is that you don't have the tools to understand those reasons yet. Even among physics majors, many stat mech classes are famous for being confusing as all hell and rather impenetrable (I don't think I even kind of understood what the hell my textbook was saying until halfway through the course myself). And this is during upper level undergrad, after quite a bit more physics than you've had so far.

True.

"Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously."

- David L. Goodstein

[Taikand]

What can be so hard about statistical mechanics?I mean it looks like pretty ordinary physics.

Anyway, back to my rant. "Why didn't physicists create a Unified Theory of Everything? That way I'd only have to learn that."

Now, back to being serious, I have the inability to accept things just because the math seems to look right, you should have seen me when I learned about the conservation of momentum.Am I the only one?

[Xanthir]

Now, back to being serious, I have the inability to accept things just because the math seems to look right, you should have seen me when I learned about the conservation of momentum.Am I the only one?

A great teacher would have told you, at least in hand-wavey terms, about Noether's Theorem, which explains *why* momentum is conserved.

Conservation of momentum is mathematically equivalent to saying that the laws of physics don't change in different locations. Similarly, conservation of angular momentum is mathematically equivalent to saying that the laws of physics don't change based on which direction you face, and conservation of energy is mathematically equivalent to saying that the laws of physics don't change over time.

Every conservation law is equivalent to a symmetry like this.

[SU3SU2U1]

I have the inability to accept things just because the math seems to look right, you should have seen me when I learned about the conservation of momentum

So get yourself some toy cars and some weights and do some experiments. The beauty of mechanics is you can create a lab on your table for pretty low cost. If you don't trust F=ma enough to trust conservation of momentum, then just check conservation of momentum directly.

Every conservation law is equivalent to a symmetry like this.

Unfortunately, you listed the only few intuitive symmetries. Charge conservation corresponds to the gauge invariance of electricity and magnetism- now do we make a gauge invariant theory because charge is conserved? OR is charge conserved because of gauge invariance?

Similarly, the symmetries corresponding to baryon number, lepton number, etc are very abstract.

[soggybomb]

Now, back to being serious, I have the inability to accept things just because the math seems to look right, you should have seen me when I learned about the conservation of momentum.Am I the only one?

As a previous poster mentioned, Noether's theorem is the most direct way to arrive at this conclusion. You can also look at the Lagrangian of any given system and consider the principle of least action.

[Taikand]

This has made me think that we may have a problem with how physics is taught. Do you agree?

[thoughtfully]

You'll be in school an awful long time if you must be taught all the fundamentals before you can learn the rules of thumb.

[eSOANEM]

You'll be in school an awful long time if you must be taught all the fundamentals before you can learn the rules of thumb.

Still, I think some discussion of Noether's theorem even in only broad, handwavey tones is probably appropriate at the level where you are expected to work quantitatively with conservation laws.

[Taikand]

You'll be in school an awful long time if you must be taught all the fundamentals before you can learn the rules of thumb.

I can't accept any notion that is not well defined unless I understand how it is useful. That's why I can't really learn formulas by heart without notions of it's inner working. Actually, I can, but I don't want to because I believe mistakes are usually made when people skip steps in the logical train of thought. That's why I always dive right into problems and only when I require a certain notion do I do my own research.
To explain what I mean the ordinary chain could be :
A-B-C-D
but teachers usually go
A-G
and the explanation would be because on the AB edge we consider X to be negligible therefore B-E and Y is considered an ideal situation therefore we can ignore C.
For me this looks like taking formulas out of the S.
I believe that by taking the most general case and only afterwards seeing what happens in certain conditions that simplify the situation is better than limiting the student to a set of theoretical knowledge that fails in certain cases that are kinda on the edge.
This kind of thinking I believe helps me because I can explain why I do certain things better than the people that simply accept those rules, and I think I'm not the only one.

[moiraemachy]

You'll be in school an awful long time if you must be taught all the fundamentals before you can learn the rules of thumb.

Still, I think some discussion of Noether's theorem even in only broad, handwavey tones is probably appropriate at the level where you are expected to work quantitatively with conservation laws.

I only know Noether's theorem in "broad and handwavey" way, but I strongly disagree. A good talk on Galilean relativity seems far more intuitive, on topic and easy to grasp, conveying the relevant symmetries. Noether's theorem, if my knowledge is not mistaken, is about a much more abstract and general case.

Also, I doubt that the knowledge of the existence of Noether's theorem will satisfy someone who questions "why is momentum conserved?", since a student asking this kind of question could ask "where does the symmetry come from?". My opinion is that this kind of question should be answered by a talk about the scientific method: we assume momentum is conserved because that's what we consistently verify, and oh, also, every interaction that follows Newton's laws of motion automatically follows this principle. We do not assume momentum is conserved because we can somehow deduce it from pure logic.

[eSOANEM]

I certainly wouldn't discuss Noether's theorem without Galilean relativity.

What I was proposing was a system whereby it is shown mathematically that the Galilean transforms do not affect the mechanics of a Newtonian system (this is fairly easy to show), then bring up Noether's theorem (in terms along the lines of "symmetries in the system lead to conservation laws" and maybe even giving some way of telling if a given symmetry and quantity could be paired this way (their product must have units of angular momentum)), if they ask where Noether's theorem comes from, tell them the truth that's it's a far from easy piece of work to derive, tell them what maths they'd need and possibly give a slightly handwavey argument through the proof if they're still not satisfied.

Teachers should not be afraid I think to introduce concepts which cannot fully be understood at their students level. I am reminded of something I read a while ago about a school in Germany which teaches complex numbers shortly after it teaches negative numbers. I doubt very much those children can grasp the full majesty of the complex world, but their strong grounding in complex arithmetic will undoubtedly help them when they need to do proper maths with complex numbers.

[Xanthir]

Teaching complexes alongside negatives actually makes some historical sense, too - at the time we invented complexes, there were still legitimate mathematicians who regarded "negative" numbers as equally imaginary.

Plus, there's a clever way you can teach them to be nearly identical. Just define some new number "e" where e != 1 but e² = 1. Go through the standard constructions and whatnot, before finally revealing that you were just teaching them about negative numbers all along. Then say that complex numbers are identical, but i² = -1.

[moiraemachy]

eSOANEM, I'd certainly get behind a class like that... it's just that it sounded to me that you were suggesting that Noether's theorem is as something important, while I believe that, in that context, it's a mere curiosity, and likely to do more harm than good if the teacher isn't vey competent and fully grasps the concept which is being talked about.

Teachers should not be afraid I think to introduce concepts which cannot fully be understood at their students level. I am reminded of something I read a while ago about a school in Germany which teaches complex numbers shortly after it teaches negative numbers. I doubt very much those children can grasp the full majesty of the complex world, but their strong grounding in complex arithmetic will undoubtedly help them when they need to do proper maths with complex numbers.

This really goes straight to the point I am trying to make: mentioning extra material to stir curiosity and encourage further research is great, but I believe that teachers must, first, clearly define what the students must be expected to learn, and what is flavor knowledge. And this core of what students should learn has to, in my opinion, be fully understood - or at least have all the essential assumptions laid out clearly.

I feel really divided on the example you gave about complex numbers: it'll probably help the students with their algebraic manipulations later, but is is also dangerous - it may enforce the view of mathematics as the manipulation of strange symbols with arbitrary rules. The fact that it took some time for the mathematical community to fully accept complexes counts as evidence against teaching them: it suggests that a student that accepts that complexes work may be only submitting to his teacher's authority. In my experience, when introduced to complexes, it's the best students who squirm, and try to game the new system to make inconsistencies arise. After all... if you don't provide a good explanation for why complexes are ok, they'll begin to wonder why it's not valid to just define problems away.

[eSOANEM]

eSOANEM, I'd certainly get behind a class like that... it's just that it sounded to me that you were suggesting that Noether's theorem is as something important, while I believe that, in that context, it's a mere curiosity, and likely to do more harm than good if the teacher isn't vey competent and fully grasps the concept which is being talked about.

Teachers should not be afraid I think to introduce concepts which cannot fully be understood at their students level. I am reminded of something I read a while ago about a school in Germany which teaches complex numbers shortly after it teaches negative numbers. I doubt very much those children can grasp the full majesty of the complex world, but their strong grounding in complex arithmetic will undoubtedly help them when they need to do proper maths with complex numbers.

This really goes straight to the point I am trying to make: mentioning extra material to stir curiosity and encourage further research is great, but I believe that teachers must, first, clearly define what the students must be expected to learn, and what is flavor knowledge. And this core of what students should learn has to, in my opinion, be fully understood - or at least have all the essential assumptions laid out clearly.

I agree entirely. Having had the good fortune of being in sets of high-performing students for sciences my entire career, I often find myself accidentally assuming that the majority of an arbitrary class "get" the basics. My suggestion should therefore be thought of more as a means of stretching the bright students or a response to the question "why is energy/momentum etc. conserved" (which will usually indicate that at least one student wants to be stretched) and, if necessary should be done as a more one-on-one discussion whilst the rest of the class does some problems to become fluent with the basics.

I feel really divided on the example you gave about complex numbers: it'll probably help the students with their algebraic manipulations later, but is is also dangerous - it may enforce the view of mathematics as the manipulation of strange symbols with arbitrary rules. The fact that it took some time for the mathematical community to fully accept complexes counts as evidence against teaching them: it suggests that a student that accepts that complexes work may be only submitting to his teacher's authority. In my experience, when introduced to complexes, it's the best students who squirm, and try to game the new system to make inconsistencies arise. After all... if you don't provide a good explanation for why complexes are ok, they'll begin to wonder why it's not valid to just define problems away.

True, although the same argument could be easily applied to teaching negative numbers.

Anyway, as I say, I read the article a long time ago, it's quite possible the school no longer exists let alone has the same syllabus. Personally, I think that the benefits outweigh the potential cost but it's certainly not a clear cut thing.

As a more "horizontal" example of teaching different syllabi: my school has a lot of links to a couple of schools in Ethiopia (one of the ex-teachers founded a charity providing aid including a lot of school-related stuff) and so quite a few of the teachers at my school have paid visits over there and seen some of their lessons even if they haven't taken any themselves. Now that we've started doing a tiny bit of group theory in our final year before uni, the teacher teaching it to us told us that, in Ethiopia, some really basic group theory is taught along with basic arithmetic with the idea being that, without an understanding of groups, you can't really understand how arithmetic works. That seems a bridge too far to me, but I'm not sure if that's because it really is, or just because I wasn't taught that way.

On the subject of trying to game the system, I remember that I didn't really have any problem with complex numbers when they were introduced, I just accepted them, and played around seeing where it led; likewise with taking the derivative and the whole "are you dividing by 0" debate we must have had a hundred times in our lessons. Anyway, I didn't try to find contradictions in complex numbers, instead, I took the idea of "take unsolvable calculation and define answer as "blah" and run with it" a bit too far.

I tried to come up with a consistent real system of dividing by 0 (the thread's probably still floating around the maths forum somewhere). It didn't work very well although, by defining it on the Riemann sphere instead of the real field, I'm pretty sure it would yield something a bit like the surreal numbers but complexified.

[SU3SU2U1]

The problem with reaching for Noether is that its both unnecessary and unlikely to be enlightening. You replace one thing they don't understand with a second thing they also don't understand.

You can prove from Newton's second law and Newton's third law that momentum is conserved in two short lines. You can verify it quantitatively in a number of simple labs. If someone asked 'why momentum is conserved?' you should point to the validity of Newton's laws- IF the equations we are studying are true, than momentum conservation simply must be true.

Now, if someone asks "why do Newton's laws have the form they do?" This might be a good time to say that we don't really know, but we have some ideas. This is a good time for a discussion of Galilean relativity- maybe the laws are formulated so that different constant velocity observers can agree. Asserting that momentum conservation is the same thing as translation symmetry without having some argument the students have a hope of following isn't useful.