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Maxwell-Boltzmann distribution for transport equations

  1. May 18, 2013 #1
    I have to calculate the transport coefficients for the Maxwell-Boltzmann distribution. But I'm not sure what distribution I have to use.
    As far as I know it should not be the MB distribution for [itex]v[/itex]-space (Velocity) or [itex]E[/itex]-axis (Energy), since that will get me the wrong dimensions in the end. I have to use the distribution per state.

    But I'm not sure how this looks. The integral I have to solve, for me getting the electrical conductivity (1st transport coefficient) I need, is given by:

    [tex]{{\mathcal{L}}^{\,\left( 0 \right)}}={{\left( \frac{2m}{{{\hbar }^{2}}} \right)}^{3/2}}\frac{{{e}^{2}}\tau }{{{\pi }^{2}}m}\int{\left( -\frac{\partial {{f}_{MB}}}{\partial \varepsilon } \right)}\,{{\varepsilon }^{3/2}}d\varepsilon,[/tex]

    at least, again, when trying to calculate the electrical conductivity, which in the end should end up being Drudes formula [itex]\sigma =\frac{n{{e}^{2}}\tau }{m}[/itex].

    So basically, not hard. But I have to get the distribution function right.

    As far as I know the MB-distribution is given by:

    [tex]{{f}_{MB}}\left( \varepsilon \right)=C{{e}^{-\varepsilon /{{k}_{B}}T}},[/tex]

    where [itex]C[/itex] is what I need to figure out, since that will determine the dimensions of my coefficients.

    According to my book the normalized MB distribution function is:

    [tex]\bar{n}=\frac{{\bar{N}}}{{{Z}_{1}}\left( T,V \right)}{{e}^{-\varepsilon /{{k}_{B}}T}},[/tex]

    where:

    [tex]\frac{{{Z}_{1}}\left( T,V \right)}{{\bar{N}}}=\frac{V}{{\bar{N}}}\left( \frac{2\pi m{{k}_{B}}T}{{{h}^{2}}} \right){{Z}_{\operatorname{int}}}\left( T \right),[/tex]

    and [itex]{{Z}_{\operatorname{int}}}\left( T \right) = 1[/itex] in my case.

    But I'm not quite sure how to about this? As far as I can see, it's not just inserting the reversed term of this in [itex]C[/itex] - at least not from what I can see. Maybe it's the [itex]V/N[/itex] I'm not sure about.

    So, anyone who can give me a clue, or...?
     
  2. jcsd
  3. May 19, 2013 #2
    Your normalization factor should be :

    [tex]\frac{{{Z}_{1}}\left( T,V \right)}{{\bar{N}}}=\frac{V}{{\bar{N}}}\left( \frac{2\pi m{{k}_{B}}T}{{{h}^{2}}} \right)^{3/2}[/tex]


    The n in the drude law is the number density. i.e. the number of electrons per unit volume. In your normalization constant what is \bar{n}, \bar{N} ? :wink:
     
  4. May 19, 2013 #3
    The bar over [itex]n[/itex] and [itex]N[/itex] means the "mean" of whatever it is...

    But do I know what this is ?
    Or at least V/N, or...?
     
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