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  • Joseph Nebus 8:50 pm on Monday, 8 July, 2013 Permalink | Reply
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    On exact and inexact differentials 


    The CarnotCycle blog recently posted a nice little article titled “On Exact And Inexact Differentials” and I’m bringing it to people’s attention because its the sort of thing which would have been extremely useful to me at a time when I was reading calculus-heavy texts that just assumed you knew what exact differentials were, without being aware that you probably missed the day in intro differential equations when they were explained. (That was by far my worst performance in a class. I have no excuse.)

    So this isn’t going to be the most accessible article you run across on my blog here, until I finish making the switch to a full-on advanced statistical mechanics course. But if you start getting into, particularly, thermodynamics and wonder where this particular and slightly funky string of symbols comes from, this is a nice little warmup. For extra help, CarnotCycle also explains what makes something an inexact differential.

    carnotcycle

    From the search term phrases that show up on this blog’s stats, CarnotCycle detects that a significant segment of visitors are studying foundation level thermodynamics  at colleges and universities around the world. So what better than a post that tackles that favorite test topic – exact and inexact differentials.

    When I was an undergraduate, back in the time of Noah, we were first taught the visual approach to these things. Later we dispensed with diagrams and got our answers purely through the operations of calculus, but either approach is equally instructive. CarnotCycle herewith presents them both.

    – – – –

    The visual approach

    Ok, let’s start off down the visual track by contemplating the following pair of pressure-volume diagrams:

    visual track

    The points A and B have identical coordinates on both diagrams, with A and B respectively representing the initial and final states of a closed PVT system, such as an…

    View original post 782 more words

     
  • Joseph Nebus 5:43 am on Thursday, 21 March, 2013 Permalink | Reply
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    Gibbs’ Elementary Principles in Statistical Mechanics 


    I had another discovery from the collection of books at archive.org, now that I thought to look for it: Josiah Willard Gibbs’s Elementary Principles in Statistical Mechanics, originally published in 1902 and reprinted 1960 by Dover, which gives you a taste of Gibbs’s writings by its extended title, Developed With Especial Reference To The Rational Foundation of Thermodynamics. Gibbs was an astounding figure even in a field that seems to draw out astounding figures, and he’s a good candidate for the title of “greatest scientist to come from the United States”.

    He lived in walking distance of Yale (where his father and then he taught) nearly his whole life, working nearly isolated but with an astounding talent for organizing the many complex and confused ideas in the study of thermodynamics into a neat, logical science. Some great scientists have the knack for finding important work to do; some great scientists have the knack for finding ways to express work so the masses can understand it. Gibbs … well, perhaps it’s a bit much to say the masses understand it, but the language of modern thermodynamics and of quantum mechanics is very much the language he spoke a century-plus ago.

    My understanding is he published almost all his work in the journal Transactions of the Connecticut Philosophical Society, in a show of hometown pride which probably left the editors baffled but, I suppose, happy to print something this fellow was very sure about.

    To give some idea why they might have found him baffling, though, consider the first paragraph of Chapter 1, which is accurate and certainly economical:

    We shall use Hamilton’s form of the equations of motion for a system of n degrees of freedom, writing q_1, \cdots q_n for the (generalized) coördinates, \dot{q}_1, \cdots \dot{q}_n for the (generalized) velocities, and

    F_1 q_1 + F_2 q_2 + \cdots + F_n q_n [1]

    for the moment of the forces. We shall call the quantities F_1, \cdots F_n the (generalized) forces, and the quantities p_1 \cdots p_n , defined by the equations

    p_1 = \frac{d\epsilon_p}{d\dot{q}_1}, p_2 = \frac{d\epsilon_p}{d\dot{q}_2}, etc., [2]

    where \epsilon_p denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates, and on this account we denote it by \epsilon_p . This will not prevent us from occasionally using formulas like [2], where it is sufficiently evident the kinetic energy is regarded as function of the \dot{q}‘s and q‘s. But in expressions like d\epsilon_p/dq_1 , where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the p’s and q’s.

    (There’s also a footnote I skipped because I don’t know an elegant way to include it in WordPress.) Your friend the physics major did not understand that on first read any more than you did, although she probably got it after going back and reading it a touch more slowly. And his writing is just like that: 240 pages and I’m not sure I could say any of them could be appreciably tightened.


    Also, I note I finally reached 9,000 page views! Thank you; I couldn’t have done it without at least twenty of you, since I’m pretty sure I’ve obsessively clicked on my own pages at minimum 8,979 times.

     
    • Peter Mander 8:05 pm on Thursday, 21 March, 2013 Permalink | Reply

      Fully agree with your assessment of Gibbs’ greatness. The US should be immensely proud of him.

      Like

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