Pb 6.2: Two-state system Consider the statistical mechanics of a tiny object with only two discrete states: one of energy $E_1$ and the other of higher energy $E_2 > E_1$.

(a) Boltzmann probability ratio. Find the ratio of the equilibrium probabilities $\rho_2/\rho_1$ to find our system in the two states, when weakly coupled to a heat bath of temperature $T$ . What is the limiting probability as $T\rightarrow \infty$? As $T \rightarrow 0$?

(b) Probabilities and averages. Use the normalization of the probability distribution (the system must be in one or the other state) to find $\rho_1$ and $\rho_2$ separately. What is the average value of the energy E?

Pb 6.8: Euler (a) Using the fact that the entropy $S(N, V, E)$ is extensive for large systems, show that $$ N \left.\frac{\partial S}{\partial N}\right|_{V.E} + V \left.\frac{\partial S}{\partial V}\right|_{N.E} + E \left.\frac{\partial S}{\partial E}\right|_{V.N} = S.$$ Show from this that in general $ S = (E + PV − \mu N)/T $ (eqn 6.74) and hence $E = TS − PV + \mu N$. (eqn 6.75) This is Euler’s relation.

(b) Test this explicitly for the ideal gas. Use the ideal gas entropy $$ S(N,V,E) = \frac{5}{2} Nk_B + Nk_B \ln \left[ \frac{V}{Nh^3} \left( \frac{2\pi mE}{2N} \right)^{3/2} \right],$$ to derive formula for $T$ , $P$, and $\mu$ in terms of $E$, $N$, and $V$, and verify eqn 6.74.

Pb 6.9: Gibbs-Duhem As a state function, $E$ is supposed to depend only on $S$, $V$, and $N$. But eqn 6.75 seems to show explicit dependence on $T$ , $P$, and $\mu$ as well; how can this be?

Another answer is to consider a small shift of all six variables. We know that $dE = T dS − P dV + \mu dN$, but if we shift all six variables in Euler’s equation we get $dE = T dS − P dV + \mu dN + S dT − V dP + N d\mu$. This implies the Gibbs–Duhem relation $0 = S dT − V dP + N d\mu$. (eqn 6.77). This relation implies that the intensive variables $T$ , $P$, and $\mu$ are not all independent; the change in $\mu$ is determined given a small change in $T$ and $P$.

(a) Write $\mu$ as a suitable derivative of the Gibbs free energy $G(T, P,N)$. This makes $\mu$ a function of the three variables $T$, $P$, and $N$. The Gibbs–Duhem relation says it must be independent of $N$.

(b) Argue that changing the number of particles in a large system at fixed temperature and pressure should not change the chemical potential. (Hint: Doubling the number of particles at fixed $T$ and $P$ doubles the size and energy of the system as well.) The fact that both $G(T, P,N)$ and $N$ are extensive means that $G$ must be proportional to $N$. We used this extensivity to prove the Euler relation in Exercise 6.8; we can thus use the Euler relation to write the formula for $G$ directly.

( c) Use the Euler relation (eqn 6.75) to write a formula for $G = E −TS +PV$ . Is it indeed proportional to $N$? What about your formula for $\mu$ from part (a); will it be dependent on $N$?

Pb 6.16: Rubber band free energy This exercise illustrates the convenience of choosing the right ensemble to decouple systems into independent subunits. Consider again the random-walk model of a rubber band in Exercise 5.12 – $N$ segments of length $d$, connected by hinges that had zero energy both when straight and when bent by 180◦. We there calculated its entropy $S(L)$ as a function of the folded length $L$, used Stirling’s formula to simplify the combinatorics, and found its spring constant $K$ at $L = 0$.

Here we shall do the same calculation, without the combinatorics. Instead of calculating the entropy $S(L)$ at fixed $L$, we shall work at fixed temperature and fixed force, calculating an appropriate free energy $\chi(T, F)$. View our model rubber band as a collection of segments $s_n =\pm 1$ of length $d$, so $L = d \sum_{n=1}^N s_n$. Let $F$ be the force exerted by the band on the world (negative for positive $L$).

(a) Write a Hamiltonian for our rubber band under an external force. Show that it can be written as a sum of uncoupled Hamiltonians $H_n$, one for each link of the rubber band model. Solve for the partition function $X_n$ of one link, and then use this to solve for the partition function $X(T, F)$. (Hint: Just as for the canonical ensemble, the partition function for a sum of uncoupled Hamiltonians is the product of the partition functions of the individual Hamiltonians.)

(b) Calculate the associated thermodynamic potential $\chi(T, F)$. Derive the abstract formula for $\langle L\rangle$ as a derivative of $\chi$, in analogy with the calculation in eqn 6.11. Find the spring constant $K = \partial F/\partial L$ in terms of $\chi$. Evaluate it for our rubber band free energy at $F = 0$.