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Thermodynamics Boltzmann Statistics

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Homework Statement


Consider a classical particle moving in a one-dimensional potential well u(x). The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics. Show that the average position of the particle is given by [tex]\overline{x}=\frac{\int xe^{-(\beta)u(x)}\,dx}{\int e^{-(\beta)u(x)}\,dx}[/tex]

Homework Equations


Partition function, equipartition theorem


The Attempt at a Solution


I don't know where they get the integrals from, the partition function is a sum.
 
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Answers and Replies

  • #2
quasar987
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It's a sum over all possible states. In the classical case, the states of a system is determined by the positions and momenta of all particles in the system. But position and momenta are continuous variables, so the sum becomes an integral.
 
  • #3
dextercioby
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He says right there: "classical particle". So you have to use the "classical" ensembles, namely the canonic one.

Daniel.
 
  • #4
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Do you ever look at a problem for a really long time, not having any clue how to do it, and then all of a sudden something just clicks? This is one of those problems.
 

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