Thermodynamics Boltzmann Statistics

[ultimateguy]

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 $$\overline{x}=\frac{\int xe^{-(\beta)u(x)}\,dx}{\int e^{-(\beta)u(x)}\,dx}$$

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.

 

[quasar987]

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.

 

[dextercioby]

He says right there: "classical particle". So you have to use the "classical" ensembles, namely the canonic one.

Daniel.

 

[ultimateguy]

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.