Physics 4311: Thermal Physics

Physics 4311: Thermal Physics – Test 1

Friday, Feb 27, 2026

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Problem 1: Short questions (15 points each = 45 points)

a): A particle performs a random walk consisting of 4 independent steps. In each step it moves a distance $a$ either to the left (with probability 3/4) or to the right (with probability 1/4). Find the probability that the particle ends up at the starting position at the end of the walk.
b): A box contains an ideal gas consisting of $N$ Neon atoms (atomic mass 20) and $N$ Argon atoms (atomic mass 40). The gas is in equilibrium at temperature $T$ . A small hole is made in the wall of the container, and the gas starts escaping. Find the ratio of the number of Neon atoms and the number of Argon atoms in the the flux of particles escaping through the hole.
c): A box contains an ideal gas of atoms of radius $a$ , number density $N ∕ V = n$ , and temperature $T$ . The temperature is now lowered from $T$ to $T ∕ 2$ . Does the mean free path of the atoms (i) increase, (ii) decrease, or (iii) stay the same? (circle one!)

Problem 2: Molecules on surface (60 points)

Consider $N$ diatomic molecules absorbed on a surface which forms a square lattice. Each molecule has three possible positions. It can either lie flat on the surface or “stand up” perpendicular to the surface. If the molecule lies flat, it aligns with either the $x$ -direction or the $y$ -direction of the lattice (see picture). The energy of a molecule standing up is $𝜖 > 0$ , whereas molecules lying flat have zero energy. The molecules are independent of each other, and the system is in equilibrium at temperature $T$ .

a): Determine the probability of a given molecule to stand up.
b): Determine the probability of a given molecule to lie flat in the $x$ -direction.
c): Determine the probability of a given molecule to lie flat in the $y$ -direction.
d): What is the zero-temperature limit of the probability of standing up?
e): What is the infinite-temperature limit of the probability of standing up?
f): Compute the average energy as a function of the temperature.
g): What is the zero-temperature limit of the average energy?
h): What is the infinite-temperature limit of the average energy?

Problem 3: Two-dimensional ideal gas (45 points)

Consider an ideal gas of particles of mass $m$ in equilibrium at temperature $T$ , restricted to move in two dimenisons.

a): Write down the two-dimensional Maxwell velocity distribution $g (v_{x}, v_{y})$
b): Compute the average kinetic energy of a particle $⟨ E ⟩ = ⟨ m {\vec{v}}^{2} ∕ 2 ⟩$ .
c): Starting from the Maxwell distribution, derive the (properly normalized) probability density $f (E)$ for the kinetic energy $E = m {\vec{v}}^{2} ∕ 2$ . (Hint: Go to polar coordinates.)

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\int_{- \infty}^{\infty} 𝑑𝑥 e^{- \frac{1}{2} x^{2} ∕ σ^{2}} = {(2 π σ^{2})}^{1 ∕ 2} \int_{- \infty}^{\infty} 𝑑𝑥 x^{2} e^{- \frac{1}{2} x^{2} ∕ σ^{2}} = {(2 π σ^{2})}^{1 ∕ 2} σ^{2}

\int_{0}^{\infty} 𝑑𝑥 x e^{- 𝑎𝑥} = 1 ∕ a^{2}, \int_{0}^{\infty} 𝑑𝑥 x^{2} e^{- 𝑎𝑥} = 2 ∕ a^{3}, \int_{0}^{\infty} 𝑑𝑥 x^{3} e^{- 𝑎𝑥} = 6 ∕ a^{4}, \int_{0}^{\infty} 𝑑𝑥 x^{4} e^{- 𝑎𝑥} = 24 ∕ a^{5}