# Thermodynamics Boltzmann Statistics

1. Nov 29, 2006

### ultimateguy

1. The problem statement, all variables and given/known data
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 $$\overline{x}=\frac{\int xe^{-(\beta)u(x)}\,dx}{\int e^{-(\beta)u(x)}\,dx}$$

2. Relevant equations
Partition function, equipartition theorem

3. The attempt at a solution
I don't know where they get the integrals from, the partition function is a sum.

Last edited: Nov 29, 2006
2. Nov 30, 2006

### 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.

3. Nov 30, 2006

### dextercioby

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

Daniel.

4. Nov 30, 2006

### 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.