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Where does exp(-4pi^2) appear in physics?

  1. Aug 13, 2009 #1
    This is a somewhat unusual question.
    I am looking for any physics problem
    in which the number or factor exp(-4pi^2)
    appears.

    If you know one, I'd like to hear about it!

    François
     
  2. jcsd
  3. Aug 14, 2009 #2

    Born2bwire

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    Given a potential described as -exp(-x), what is the work done against the potential when moving a unit mass from infinity to x = 4\pi^2 ?
     
  4. Aug 14, 2009 #3

    arildno

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    Introduce the new mass unit, [itex]\hat{m}\equiv{m}{e}^{4\pi^{2}}[/itex], where m is the standard unit of mass.

    Then, Newton's second law of motion reads:
    [tex]F=e^{-4\pi^{2}}\hat{m}a[/tex]
     
  5. Aug 14, 2009 #4

    Mapes

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    Here's one that's less arbitrary: the solution of

    [tex]\frac{\partial u(x,t)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}[/tex]

    is

    [tex]u(x,t)=\sum_{n=1}^\infty A_n\exp^{-(n\pi/L)^2t}\sin\frac{n\pi x}{L}[/tex]

    where [itex]A_n[/itex] is calculated from the initial conditions. But the upshot is, because the above equation governs heat diffusion by conduction, if you had a bar of material with length 0.5 m, thermal diffusivity 1 m2 s-1, end temperatures of 0°C (i.e., [itex]u(0,t)=u(L,t)=0[/itex]), and an initial sinusoidal temperature distribution with maximum temperature 1°C (i.e., [itex]u(x,0)=\sin(\pi x/L)[/itex]), the temperature at the midpoint after 1 second is predicted to be [itex]\exp(-4\pi^2)[/itex], which is the term you're looking for.
     
  6. Aug 15, 2009 #5

    Vanadium 50

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    How is that less arbitrary? You still have to pick a material with the right size and properties.
     
  7. Aug 15, 2009 #6

    Mapes

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    It's less arbitrary because the exponential function and the [itex]\pi^2[/itex] term come out of the physics rather than being input as variables or functions, as in the other two cases (although we do need to specify a sinusoidal initial temperature distribution). And the values aren't fixed; it could be a [itex]10\,\mu m[/itex] long microfabricated silicon beam (thermal diffusivity [itex]8\times 10^{-5}\,\mathrm{m^2}\,\mathrm{s^{-1}}[/itex]) after [itex]5\,\mu s[/itex], for example. Don't you think it's interesting that the [itex]\pi^2[/itex] emerges naturally here?
     
  8. Aug 15, 2009 #7

    Born2bwire

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    Not really. Pretty much any time-harmonic or complex number system you can probably easily massage out an exp(\pi) and exp(\pi^2) dependence of some kind. I could specify a transmission line and given a certain length and loss I could get you an attenuation of exp(-4\pi^2).

    The OP is just posing a really bad question. It is completely arbitrary because, like arildno shows, when it comes to picking out a constant you can get it from just about any kind of equation from judicious choice of your units, scale, or choice of parameters.
     
  9. Aug 15, 2009 #8

    Mapes

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    Go for it! That's what the poster seems to be looking for: physical circumstances in which the term arises naturally. Maybe he or she saw the term on a blackboard once, or a poster (or a tattoo!), and wants to know what the context might have been. I don't know. But I don't think it's a bad question; in fact, I was looking forward to comparing the responses to get a sense of the... grand interconnectedness of physics. Consider it a challenge: what's the least amount of massaging needed to get [itex]\exp(-4\pi^2)[/itex], without inputting it directly?
     
  10. Aug 15, 2009 #9

    arildno

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    "physical circumstances in which the term arises naturally"

    What is natural, or unnatural about a particular choice of length scale??

    Yet, they are also part of physics..:smile:
     
  11. Aug 15, 2009 #10

    Vanadium 50

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    Getting a pi^2 is not horribly difficult; have a 4 or 5 dimensional volume in phase space. You'll have to turn a 1/2 or an 8/15 into a 4, but that shouldn't be hard. Then find a reason to exponentiate it...e.g. as a partition function.

    This is contrived, of course, but no more so than a length of one meter, time of one second, capacitance of one farad, etc.

    As far as the context, I fear it's not a blackboard, poster or tattoo. A search of other messages here will perhaps provide some enlightenment.
     
  12. Aug 15, 2009 #11

    Mapes

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    Ah, got it. I was originally thinking it was something innocuous like https://www.physicsforums.com/showthread.php?t=257304".
     
    Last edited by a moderator: Apr 24, 2017
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