Physics 4311: Thermal Physics – Homework 11

due date: Wednesday, April 22, 2026

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Problem 1: System of 3-level atoms (12 points)

Consider a system of $N$ independent atoms, each having three states $| 1 ⟩, | 2 ⟩, | 3 ⟩$ with energies $𝜖_{1} = 𝜖_{2} = 0$ , $𝜖_{3} = 𝜖 > 0$ . The system is in equilibrium with a heat bath at temperature $T$ .

a): Calculate the partition function $Z_{1}$ of a single atom and the partition function $Z_{N}$ of the $N$ -atom system.
b): Find the Helmholtz free energy and the internal energy.
c): Calculate the entropy and discuss its behavior for $T \to 0$ . Where does the excess entropy come from?

Problem 2: Generalized equipartition theorem (8 points)

Consider a classical degree of freedom $q$ that makes a contribution to the Hamiltonian of the form $\frac{1}{2} A | q |^{n}$ where $n$ and $A$ are positive constants. Find the average internal energy stored in this degree of freedom as a function of temperature.

Problem 3: Polymer under tension (20 points)

A model of a polymer consists of a large number of identical rods (“monomers”) of length $ℓ$ , attached end to end as shown in the figure. The connections between the rods are completely flexible so that the orientation of each rod (characterized by the spherical angles $Θ_{i}$ and $ϕ_{i}$ w.r.t. the $z$ -axis) is independent of the neighboring rods. One end of the polymer is fixed, the other end is attached to a weight of mass $M$ . Neglecting the mass of the monomers, the energy of a given state of the polymer can be written as

E (Θ_{1}, ϕ_{1}, \dots, Θ_{N}, ϕ_{N}) = - Mg L_{z} = - \sum_{i = 1}^{N} Mgl cos Θ_{i}

The spherical angles take values $0 < Θ_{i} < π$ and $0 < ϕ_{i} < 2 π$ . The system is in thermal equilibrium at temperature $T$ .

a): Calculate the single-monomer partition function $Z_{1}$ .
b): Find the partition function of the entire polymer and its free energy.
c): Compute the average vertical distance $L_{z}$ between the fixed end of the polymer and the mass $M$ as a function of $M$ , $g$ , $ℓ$ , and $T$ .
d): Analyze your result for $L_{z}$ in the limits of $T \to 0$ and $T \to \infty$ . Compare the resulting values of $L_{z}$ to your physical expectation.
e): At high temperatures, $k_{B} T ≫ Mgℓ$ , the polymer acts like a spring, i.e., there is a linear relation between between the applied force $Mg$ and its vertical length $L_{z}$ . Find this relation from your result in part c). [Hint: Taylor expansion of $L_{z}$ to the lowest non-vanishing order]