\(N+1\) states with \(0, 1, \ldots , N\) links open from the left \[E_n = N \epsilon _0 + n(\epsilon _1-\epsilon _0) = N\epsilon _0 + n\epsilon \qquad \textrm {with} \quad \epsilon = \epsilon _1 - \epsilon _0\] \[Z = \sum _{n=0}^N e^{-\beta (N\epsilon _0 + n\epsilon )} = e^{-\beta N\epsilon _0 } \sum _{n=0}^N e^{-\beta n\epsilon } \qquad \textrm {finite geometric series}\] \[Z = e^{-\beta N\epsilon _0 } \, \frac {1- e^{-\beta \epsilon (N+1)}}{1-e^{-\beta \epsilon }} \]
\[\langle E \rangle = N\epsilon _0 +\langle n \rangle \epsilon ~, \qquad \langle n \rangle = (\langle E \rangle - N \epsilon _0)/\epsilon \] \[ \langle E \rangle = -\frac {\partial }{\partial \beta } \ln Z = N\epsilon _0 -\frac {\partial }{\partial \beta } \ln \frac {1- e^{-\beta \epsilon (N+1)}}{1-e^{-\beta \epsilon }} \] \[ \langle E \rangle = N \epsilon _0 - \frac {(N+1)\epsilon }{e^{\beta (N+1)\epsilon } -1} + \frac {\epsilon }{e^{\beta \epsilon }-1}\] \[\langle n \rangle = - \frac {N+1}{e^{\beta (N+1)\epsilon } -1} + \frac {1}{e^{\beta \epsilon }-1}\]
\(T\to 0\) (\(\beta \to \infty \)): \(\langle n \rangle \to 0\)
\(T\to \infty \) (\(\beta \to 0\)): Taylor expansion in \(\beta \)
\[ \langle n \rangle = \frac {1}{\beta \epsilon + (\beta \epsilon )^2/2} - \frac {N+1}{(N+1)\beta \epsilon + (N+1)^2(\beta \epsilon )^2/2}\]
\[ \langle n \rangle = \frac {1}{\beta \epsilon } \left [ 1-\beta \epsilon /2 -1 + (N+1) \beta \epsilon /2 \right ]\]
\[ \langle n \rangle = N/2\]
semiclassical approach, work in cylindrical coordinates (\(r, \Theta , z\)) w.r.t. rotation axis
single-particle Hamiltonian \(H= \mathbf {p}^2/(2m) -m\omega ^2 r^2 /2\)
\[Z_N=\frac 1 {N!} Z_1^N~, \qquad Z_1 = \int \frac {d^3p\,d^3r}{h^3} e^{-\beta \mathbf {p}^2/(2m) +\beta m\omega ^2 r^2/2} =Z_{kin} Z_{pot}\]
\[Z_{kin} = \frac {1}{h^3} (2\pi m k_B T )^{3/2} \qquad \textrm { as in class}\]
\[Z_{pot} = \int _0^R r\, dr \int _0^{2\pi } d\Theta \int _0^H dz \, e^{\beta m\omega ^2r^2/2} = \frac {2\pi H}{m\beta \omega ^2} \left ( e^{\beta m \omega ^2 R^2/2} - 1 \right )\]
\[ U = -\frac {\partial }{\partial \beta } \ln Z_N = -N\frac {\partial }{\partial \beta } \ln Z_1 = \frac 32 N k_B T + k_B T - \frac {m\omega ^2 R^2 / 2}{1- e^{-\beta m \omega ^2 R^2 /2}} \]
\[n(r) \sim e^{\beta m \omega ^2 r^2/2} \qquad \textrm {(Normalize!)}\] \[n(r) = \frac {\beta m \omega ^2}{e^{\beta m \omega ^2 R^2/2} -1 }\, e^{\beta m \omega ^2 r^2/2} \]
\[Z_N=\frac 1 {N!} Z_1^N~, \qquad Z_1 = \int \frac {d^3p\,d^3r}{h^3} e^{-\beta c|\mathbf {p}|} = \frac V{h^3} \int d^3p \, e^{-\beta c|\mathbf {p}|}\] use spherical coordinates \[Z_1 = \frac V{h^3} 4\pi \int dp\, p^2 e^{-\beta c p} = \frac {8 \pi V }{h^3 \beta ^3 c^3}\] \[ F = -k_B T \ln Z_N = -N k_B T \left [ \ln \left ( \frac {8 \pi V}{N h^3 \beta ^3 c^3} \right ) + 1 \right ]\]
\[p= - \left ( \frac {\partial F}{\partial V}\right )_T = N k_B T / V \]
\[U = -\frac {\partial }{\partial \beta } \ln Z_N = 3 N k_B T\] \[ C_V = \left ( \frac {\partial U}{\partial T}\right )_V = 3 N k_B\]
\[C_p = \left ( \frac {\delta Q}{\partial T}\right )_p = \left ( \frac {\partial U}{\partial T}\right )_p + p\left ( \frac {\partial V}{\partial T}\right )_p = 3N k_B + N k_B = 4 N k_B\] \[ \gamma = C_p / C_V = 4/3\]