# Partition sum of particle, high/low temperature limits

1. Feb 29, 2012

### SoggyBottoms

1. The problem statement, all variables and given/known data
We have a single particle that can be in one of three different microstates, $\epsilon_0$, $\epsilon_1$ or $\epsilon_2$, with $\epsilon_0 < \epsilon_1 < \epsilon_2$. The particle is in thermal equilibrium with a heat bath at temperature T.

1) Calculate the canonical partition function.

2) Give the probabilities $P_j$ that this particle is in state $j$ for $j = 0, 1, 2$.

3) What is the average energy in the high and low temperature limit?

3. The attempt at a solution

1) $$Z(\beta) = e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2}$$

2)

$$P(0) = \frac{e^{- \beta \epsilon_0}}{Z(\beta)}$$

$$P(1) = \frac{e^{- \beta \epsilon_1}}{Z(\beta)}$$

$$P(2) = \frac{e^{- \beta \epsilon_2}}{Z(\beta)}$$

3)

$$\langle E \rangle = \epsilon_0 \cdot P(0) + \epsilon_1 \cdot P(1) + \epsilon_2 \cdot P(2) \\ = \frac{\epsilon_0 e^{- \beta \epsilon_0} + \epsilon_1 e^{- \beta \epsilon_1} + \epsilon_2 e^{- \beta \epsilon_2}}{e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2}}$$

So then I just plug in $\beta = \frac{1}{k_B T}$ and see what happens when T gets very low or very high, and then for high temperatures I get $U = \frac{\epsilon_0 + \epsilon_1 + \epsilon_2}{3}$ and for low temperatures it goes to 0. Is this correct?

Edit: I meant partition function in the title.

Last edited: Feb 29, 2012