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Normal (probability) distribution and Partition function.

  1. Jan 22, 2007 #1
    Let be the continous partition function:

    [tex] Z(\beta)=(N!)^{-1}\int_{V}dx_1 dx_2 dx_3 dx_4 .....dx_N exp(-\beta H(x_1, x_2 , x_3 , .... ,x_n,p_1 , p_2 , ......., p_N [/tex]

    if the Hamiltonian is 'quadratic' in p's are q's do this mean that the particles in the gas solid or whatever follow a Normal distribution (is Maxwell distribution under a quadratic potential or with U=0 potential a Normal distribution??)
  2. jcsd
  3. Jan 22, 2007 #2

    Physics Monkey

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    Hi Kevin,

    First a notational issue, you need to also integrate over the momenta, not just the positions, to get the partition function.

    Now, to answer your question, a normal distribution is synonymous with a Gaussian distribution. A Gaussian or normal random variable has a probability distribution which is an exponential of an expression quadratic in the variable. Hence, since the probability distribution for your particles is proportional to [tex] \exp{(-\beta H)} [/tex], the positions and momenta of your particles are normal random variables if the Hamiltonian is quadratic in those positions and momenta.
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