Fluctuations for a harmonic trap

*What is the coefficient of the differential of work in the case of onedimensional gas atoms in a harmonic trap? 
  Solution: We remember that the Hamiltonian for this problem is

   
 Then, we obtain the partition function,

and the logarithm of it


 To get the coefficient required (which is a thermodynamic variable and an observable) we use the expression mentioned on page 43 from course's notes. This is

 



*Use the formula (87) to calculate the fluctuations of this coefficient. Does the result make sense?

Solution: The formula (87) is


Using the equipartion theorem we have that*




Replacing the results we have


Finally 



Note: I use this page to get the equations on images, using latex language: https://www.codecogs.com/latex/eqneditor.php?lang=es-es

Comentarios

  1. Thomas30 de marzo de 2020 a las 7:43 p.m.

    You cannot use derivations with respect to parameters in the equipartition theorem. The equations of the type " = kB T are only valid for derivatives with respect to the canonical variables. But you can realize that the second derivative of H with respect to w is equal to
    "sum_j m q_j^2" and that is equal to 1/(2w^2) times the potential energy. And we already saw, how to compute the average potential energy using the equipartition theorem.
    Can you correct your calculation?

    ResponderBorrar

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