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Mean energy of system with ## E = \alpha |x|^n ##

  1. Apr 21, 2015 #1
    1. The problem statement, all variables and given/known data
    If the energy ##E## of a system behaves like ## E = \alpha |x|^n##, where ## n =1, 2, 3, \dots ## and ## \alpha > 0##, show that ## \langle E \rangle = \xi k_B T ##, where ##\xi## is a numerical constant.

    2. Relevant equations
    $$ \langle E \rangle = \frac{ \int_{- \infty}^{ \infty} Ee^{-\beta E}}{\int_{-\infty}^{ \infty} e^{-\beta E}},$$ where ## \beta = \frac{1}{k_BT}.##

    3. The attempt at a solution
    Since the integral is even, it can be written as $$\frac{ \int_{0}^{ \infty} \alpha x^n e^{-\beta \alpha x^n}}{\int_{0}^{ \infty} e^{-\beta \alpha x^n}},$$

    It can also be written as

    $$ \frac{ \frac{ \partial}{ \partial \beta} \left ( \int_{0}^{ \infty} - e^{-\beta \alpha x^n} \right ) } {\int_{0}^{ \infty} e^{-\beta \alpha x^n}}$$

    where the partial derivative was taken outside the integral.

    I have no idea how to solve the integral. Mathematica didn't draw up anything useful.
     
  2. jcsd
  3. Apr 21, 2015 #2
    Have you tried a change of variable e.g. y=xn?
     
  4. Apr 21, 2015 #3
    I think I did try it, but it didn't look like it could result in anything useful.
     
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