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Statistical mechanics. Partition function.

  1. Mar 15, 2014 #1
    1. The problem statement, all variables and given/known data
    If ##Z## is homogeneous function with property
    ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##
    and you calculate Z(T,V,N). Could you calculate directly ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##.


    2. Relevant equations
    ##Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k T)^{\frac{3N}{2}}\int...\int_{V} e^{-\frac{U}{kT}}##



    3. The attempt at a solution
    When I have for exaple
    ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)=\frac{1}{h^{3N}N!}(2\pi m k \alpha T)^{\frac{3N}{2}}\int...\int_{\alpha^{-\frac{3}{\nu}}V} e^{-\frac{U}{k\alpha T}}##
    I'm not sure how to integrate this ##e^{-\frac{U}{k\alpha T}}## in exponent.
    [
     
  2. jcsd
  3. Mar 19, 2014 #2
    From this if I understand well
    ##\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}##
    need to be equal
    ##\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}=\alpha^{-\frac{3N}{\nu}}...##
     
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