Questions

• What thermodynamic potential can be calculated from the canonical partition function? How is this done and how is this result derived?
• Give an expression that allows one to calculate the ensemble average, $\langle A \rangle$, for the observable $A$. You may assume that this quantity can be calculated based on the positions, $\mathbf{x}$, and momenta, $\mathbf{p}$, of the atoms using a function $A( \mathbf{x},\mathbf{p})$.
• Explain why $1 = \sum_j e^{-\beta H(\mathbf{x}_j,\mathbf{p}_j )- \Psi}$
• Now calculate the first derivative of $1 = \sum_j e^{-\beta H(\mathbf{x}_j,\mathbf{p}_j) - \Psi}$ with respect to $\beta$ and hence show that $\langle E \rangle = - \frac{ \partial \Psi }{\partial \beta }$
• Calculate the second derivative of $1 = \sum_j e^{-\beta H(\mathbf{x}_j,\mathbf{p}_j) - \Psi}$ with respect to $\beta$ and hence show that $\langle (H - \langle E \rangle )^2 \rangle = \frac{ \partial^2 \Psi }{\partial \beta^2 }$
• Explain (in your own words) why $\langle (H - \langle E \rangle )^2 \rangle = - \frac{ \partial \langle E \rangle }{\partial \beta }$.
• Use the chain rule to show that: $\frac{\partial \langle E \rangle }{\partial \beta} = k_B T^2 \frac{\partial \langle E \rangle }{\partial T}$
• Use the result you have just arrived at to write an expression that tells you how the heat capacity can be calculated from the fluctutations in the total energy $\langle (H - \langle E \rangle )^2\rangle$