Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Direct correlation function

  1. Jul 16, 2015 #1
    Hi all.
    I'm learning something about critical phenomena and I have one problem.

    I'm bad with Fourier transforms so I don't know how from 7.37 we have 7.38.
    I have tryed everything I knew, but fruitless. I have attached picture of my problem.

    Does anybody has any idea how I can solve this?
     

    Attached Files:

  2. jcsd
  3. Jul 16, 2015 #2

    DrDu

    User Avatar
    Science Advisor

    Multiply 7.37 with the denominator of the RHS and then Fourier transform using that a product in Fourier space becomes a convolution in direct space (and vice versa), i.e. the integral on the RHS of eq. 7.38.

    https://en.wikipedia.org/wiki/Convolution_theorem
     
  4. Jul 16, 2015 #3
    Thanks!
    Is the Fourier transform of C (with hat) just C without hat, taking into account deffinition of C?
    What make me confused are the arguments of functions in 7.38. Why r-r', and so on...?
     
  5. Jul 18, 2015 #4
    Anybody? To help with this manybody? :)
     
  6. Jul 18, 2015 #5

    DrDu

    User Avatar
    Science Advisor

    That C and ##\hat{C}## are Fourier transforms of each other is clearly stated in the article.
    r1-r2 is as valid a variable as e.g. x or y. In a homogeneous system, translation invariant quantities cannot depend on both r1 and r2 but via their difference r1-r2. I suppose this has been discussed in your text previously
     
  7. Jul 18, 2015 #6
    Yes, I know, it was discussed earlier. But that what confused me was arguments of those functions under integral. Why are they different? And why they differ from those arguments of function outside of integral?
     
  8. Jul 18, 2015 #7

    DrDu

    User Avatar
    Science Advisor

    I think you first have to convince yourself that the integral represents a convolution.
     
  9. Jul 18, 2015 #8
    I have used convolution theorem, but I didn't get exactly same solution as author. Look how I calculated that. Arguments of my functions are considerably different. And that is what confuses me...
     

    Attached Files:

  10. Jul 19, 2015 #9

    DrDu

    User Avatar
    Science Advisor

    Rename your variables:
    r->r-r'
    And
    r''->r''-r'
     
  11. Jul 19, 2015 #10
    Thank you very much Doc!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Direct correlation function
  1. Correlation function (Replies: 10)

Loading...