Questions: Statistical Mechanics: Ensembles and the Boltzmann Distribution

Question 1

What does the partition function Z = Σ exp(−Eᵢ/kT) represent in the canonical ensemble?

• A. A normalization factor that sums Boltzmann weights over all microstates, from which thermodynamic properties are derived
• B. The probability of finding the system in its lowest-energy microstate
• C. The total number of accessible microstates at temperature T
• D. The average energy of the system at temperature T

Answer: (A)

Z is not itself a probability. Dividing any individual Boltzmann weight exp(−Eᵢ/kT) by Z gives the probability pᵢ. Because Z encodes the entire weighted sum over microstates, every thermodynamic quantity (energy, entropy, free energy) can be derived from it via appropriate derivatives.

Question 2

At very high temperatures (T → ∞), the Boltzmann distribution predicts that all accessible microstates have approximately equal probability.

• T. True
• F. False

Answer: True

As T → ∞, the exponent −Eᵢ/kT → 0 for every state, so exp(−Eᵢ/kT) → 1 regardless of Eᵢ. All Boltzmann weights become equal, and pᵢ = 1/Ω for every accessible microstate. This is the high-temperature, classical limit where energy differences become negligible compared to thermal fluctuations.

Question 3

Why does the statistical-mechanical formula S = k ln Ω give entropy a molecular-level meaning?

Model answer: Ω counts the number of microstates consistent with a given macrostate. Because all microstates are equally probable, more microstates means the system is more disordered and harder to predict microscopically. Entropy measures this spread: the larger Ω is, the more ways the system can be arranged while looking the same macroscopically, and the higher the entropy.

This connects directly to the equal-probability postulate: a system with more microstates is more likely to be found in arrangements we cannot distinguish from each other. The logarithm makes entropy additive for independent subsystems (since Ω_total = Ω₁ × Ω₂, and ln(Ω₁Ω₂) = ln Ω₁ + ln Ω₂), matching the thermodynamic requirement that entropy is extensive.