Physics 6311: Statistical Mechanics - Homework 8

due date: Tuesday, Oct 21, 2025

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Problem 1: Particle number fluctuations in grand-canonical ensemble (10 points)

Consider a many-particle system in the grand-canonical ensemble characterized by a chemical potential $μ$ and temperature $T$ . Compute the variance of the particle number $N$ and relate it to $(∂N ∕ ∂μ)$ . Use the result to discuss how the particle number fluctuations behave in the thermodynamic limit.

Problem 2: Atoms on a lattice (15 points)

Consider a lattice having $N$ regular lattice sites as well as $N$ interstitial lattice sites. Each site can be occupied by either 0 or 1 atoms. An atom on a regular site has energy 0 while an atom on an interstitial site has energy $𝜖$ . The whole lattice is now occupied by $N$ atoms and coupled to a heat bath at temperature $T$ . Use the grand-canonical ensemble to analyze this system.

a): Find the number $N_{r}$ of atoms on the regular sites as a function of the temperature and the chemical potential.
b): Find the number $N_{i}$ of atoms on the interstitial sites as a function of the temperature and the chemical potential.
c): Find the value of $μ$ for which the total particle number $N_{r} + N_{i}$ equals $N$ .
d): Compute the average energy and express it as a function of $T$ and $N$ .

Problem 3: Identical particles in two-level system (15 points)

A quantum mechanical system has two single-particle states $| a ⟩$ and $| b ⟩$ with energies
$𝜖_{a} = - 𝜖_{b} = 𝜖$ .

a): The system is occupied by two identical particles. Write down all possible states, the corresponding energies and the canonical probabilities for these states for bosons (S=0) and for fermions (S=1/2, but both particles being in the $↑$ state). Using the canonical ensemble calculate the Helmholtz free energy, the entropy, the internal energy and the specific heat as functions of temperature.
b): Consider an additional term in the Hamiltonian, viz, an interaction between the particles of the form $U n_{a} n_{b}$ . where $U$ is the interaction energy and $n_{a}$ and $n_{b}$ are the particle numbers of the two single-particle states. How do the canonical probabilities for the two-boson states from a) change as a result of $U$ ? Discuss the limits $U \to \infty$ and $U \to - \infty$ .