# Thermodynamics Boltzmann Statistics

## 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.

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quasar987
Homework Helper
Gold Member
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