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On the relation between quantum and statistical mechanics

  1. Jan 22, 2015 #1

    Demystifier

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    It is well known that quantum mechanics in the path-integral form is formally very similar to equilibrium statistical mechanics formulated in terms of a partition function. In a relatively recent, very readable and straightforward paper
    http://lanl.arxiv.org/abs/1311.0813
    John Baez (a well known mathematical physicist) and Blake Pollard develop this formal analogy further by introducing a quantum analogy of entropy, which they call quantropy. I feel that this paper might be interesting and illuminating for many people on this forum.

    Another reason for posting it is to make a concealed critique of AdS/CFT correspondence. This example demonstrates that, just because there is a mathematical correspondence between two theories, doesn't mean that the two theories really describe the same physics.
     
    Last edited: Jan 22, 2015
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  3. Jan 23, 2015 #2
    Similar considerations may apply between information theory and statistical mechanics ? The same mathematical expression occurs both in statistical mechanics (thermodynamic entropy) and in information theory (information-theory entropy) does not in itself establish any connection between these fields ?

    Patrick
     
  4. Jan 23, 2015 #3

    Demystifier

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    In this case the relation is much more than merely formal. Both entropies are really examples of entropy in a semantic sense, which cannot be said for quantropy.
     
  5. Jan 23, 2015 #4
    E.T Jaynes wrote in his book "Probability Theory The Logic of Science"

    Patrick
     
  6. Jan 23, 2015 #5

    Demystifier

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    I think Jaynes wanted to say that within thermal physics there are two semantically different notions of entropy: one statistical and the other thermodynamical. If this is what he wanted to say, then I agree. But statistical Gibbs entropy in thermal physics and statistical Shannon entropy are semantically very much related.
     
  7. Jan 23, 2015 #6
    May be, yet his goal seem to show that statistical mechanics, communication theory, and a mass of other applications are all instances of a single method of reasoning.

    A viewpoint from which thermondynamic entropy and information-therory entropy appear as the same concept ( http://bayes.wustl.edu/etj/articles/theory.1.pdf )

    Isn't it the case for the concept of quantropy ?

    Patrick
     
    Last edited: Jan 23, 2015
  8. Jan 23, 2015 #7

    Demystifier

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    No. Entropy is derived from probability, which is a positive quantity. Quantropy is derived from probability amplitude, which is not a positive quantity.
     
  9. Jan 23, 2015 #8

    TeethWhitener

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    Interesting idea. I haven't read the entire paper too closely, but I'm wondering about the correspondence [itex]T \mapsto i\hbar[/itex]. In stat mech, you have a lot of quantities that are derivatives with respect to [tex]\beta \propto \frac{1}{T}[/tex]
    For instance: [tex]\langle E \rangle = - \frac{d}{d\beta}ln Z[/tex]
    How much sense does it make to have a derivative with respect to a constant (Planck's constant)?
     
  10. Jan 23, 2015 #9

    Demystifier

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    Mathematically, there is no problem with it. Physically, you can replace Planck constant with a true variable y, perform a derivative with respect to y, and then evaluate all quantities at y=h. The result is the same.
     
  11. Jan 23, 2015 #10

    TeethWhitener

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    Right, but I'm not sure what insight it adds. Reading the paper, it seems like the authors aren't really sure either. Although, glancing over it again, the partition function has a suggestive form [tex]Z=\frac{2\pi \hbar i \Delta t}{m (\Delta x)^2}[/tex]
    so that if we choose [tex]\frac{\Delta t}{(\Delta x)^2}=\frac{1}{c} \frac{1}{ \Delta \tilde{x}}[/tex]
    we get [tex]Z=\frac{ih}{mc\Delta \tilde{x}}[/tex]
    so that the length parameter [itex]\Delta \tilde{x}[/itex] and the partition function [itex]Z[/itex] define the Compton wavelength of the particle.
     
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