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Finding beta for the boltzman distribution.

  1. Oct 26, 2011 #1
    Hello! I'm trying to do a satisfactory derivation of the boltzmann distribution. By using lagrange multipliers I've come as far as to prove that

    [tex]P(i) = \frac{1}{Z} e^{-\beta E(i)}[/tex]
    [tex]Z = \sum_i e^{-\beta E(i)},[/tex]

    but how does one actually establish that
    [tex]\beta = 1/kT?[/tex]
  2. jcsd
  3. Oct 26, 2011 #2


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    Take a monatomic ideal gas and derive the mean energy,

    [tex]U=-\frac{\partial \ln Z}{\partial \beta}[/tex]

    and compare with the definition of the temperature,

    [tex]U=\frac{3}{2} N k T.[/tex]
  4. Oct 26, 2011 #3
    I think that Leonard Susskind does this derivation in one of his lectures on statistical mechanics that is available on youtube.

    Last edited by a moderator: Apr 26, 2017
  5. Oct 26, 2011 #4
    Ah, yes that is certainly a way to go. But how could that result possibly be general? Doesn't the distribution apply to any combination of systems who shares a total energy E and a number of particles N?
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