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Show that this goes to zero.

  1. Jul 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the total time derivative wrt time of this function

    [tex]S=-k\int_{r} \int_{v}f \ln(f)d^{3}vd^{3}r[/tex]

    vanishes.


    2. Relevant equations

    [tex]\frac{DS}{Dt}=\frac{\partial S}{\partial t}+\vec{v} \bullet \nabla S+ \vec{a}\bullet\nabla_{v}S[/tex]

    where [tex]\nabla_{v}[/tex] is the gradient in velocity space.

    3. The attempt at a solution


    After observing the partial wrt time = 0, dotting out the gradients/vectors, and gathering 'like' terms (variable-wise) I have this

    [tex]\frac{DS}{Dt}=(v_{x}\frac{\partial S}{\partial x}+a_{x}\frac{\partial S}{\partial v_{x}})+(v_{y}\frac{\partial S}{\partial y}+a_{y}\frac{\partial S}{\partial v_{y}})+(v_{z}\frac{\partial S}{\partial z}+a_{z}\frac{\partial S}{\partial v_{z}})[/tex]

    For the first term on the LHS we have

    [tex]v_{x}\frac{\partial S}{\partial x}+a_{x}\frac{\partial S}{\partial v_{x}}=\frac{dx}{dt}(-k\int_{y,z}\int_{v}f\ln(f)\!d^{3}vdydz)+\frac{dv}{dt}(-k\int_{r}\int_{v_{y},v_{z}}f\ln(f)dv_{y}\!dv_{z}d^{3}r)[/tex]

    due to the fundamental theorem of calculus. ([tex]\frac{d}{dx} \int (F) dx=F[/tex])

    To be honest I don't know if this path is going anywhere or if there is a more obvious way, but I am stumped on where to go next. I do not want to have the answer just plainly given to me because this is a graded homework assignment but I also have no clue on how to complete it sufficiently. I am hoping I can get a little direction.
     
    Last edited: Jul 7, 2009
  2. jcsd
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