Questions: Fundamental Principles of Statistical Mechanics

Question 1

A canonical ensemble system is in thermal contact with a heat bath at temperature T. Microstate A has energy 2k_BT and microstate B has energy 4k_BT. What is the ratio of their probabilities P(A)/P(B)?

• A. 1 — all microstates are equally probable at thermal equilibrium
• B. e² ≈ 7.4 — lower-energy microstates are more probable by the Boltzmann factor
• C. 1/2 — probability is proportional to energy
• D. 2 — higher-energy microstates are preferred at elevated temperature

Answer: (B)

In the canonical ensemble, the probability of a microstate with energy E is proportional to e^{−E/k_BT}. P(A)/P(B) = e^{−2k_BT/k_BT} / e^{−4k_BT/k_BT} = e^{−2}/e^{−4} = e² ≈ 7.4. Lower-energy microstates are more probable. Equal probability applies only to the microcanonical ensemble (isolated system at fixed energy), not to canonical systems in thermal contact with a heat bath.

Question 2

Why is the partition function so central to statistical mechanics?

• A. It identifies which specific microstate the system occupies at equilibrium
• B. It counts the total number of particles in the system
• C. All thermodynamic quantities — energy, entropy, free energy, heat capacity — can be derived from it by taking appropriate derivatives
• D. It determines the rate at which the system transitions between microstates

Answer: (C)

The partition function Z = Σ e^{−E_i/k_BT} (sum over all microstates) encodes all thermodynamic information. Average energy: U = −∂(ln Z)/∂β. Helmholtz free energy: F = −k_BT ln Z. Entropy: S = −∂F/∂T. Heat capacity: C = ∂U/∂T. Once Z is known, the entire edifice of classical thermodynamics is determined — the partition function is the single bridge between molecular energy levels and all macroscopic observables.

Question 3

The fundamental postulate of statistical mechanics states that an isolated system at equilibrium is equally likely to be found in any of its accessible microstates.

• T. True
• F. False

Answer: True

This is the single foundational assumption from which all of thermodynamics follows. A system with more accessible microstates is overwhelmingly more likely to be found in a high-microstate-count macrostate — which is exactly what Boltzmann's entropy formula S = k_B ln W captures. The second law, equilibrium, and temperature equalization all emerge from this one postulate combined with counting.

Question 4

Temperature is a microscopic property of individual molecules that statistical mechanics identifies as their average kinetic energy.

• T. True
• F. False

Answer: False

Temperature is a macroscopic property that emerges from collective behavior — individual molecules do not have a temperature. Statistical mechanics shows that for an ideal gas, temperature is related to the *average* kinetic energy of the ensemble as a whole, but this is a statistical quantity, not a property of any single molecule. Saying a single molecule 'has a temperature' is a category error that statistical mechanics explicitly corrects.

Question 5

Explain in your own words how the fundamental postulate of equal probability of microstates connects to the macroscopic concept of entropy increasing toward equilibrium.

Model answer: If every accessible microstate is equally probable, then macrostates corresponding to more microstates are overwhelmingly more likely to be observed. A system initially in a low-entropy state (few microstates) will evolve toward higher-entropy states simply because there are vastly more ways to be disordered than ordered. 'Entropy increases' is not a separate fundamental law — it is the statistical consequence of equal microstate probability combined with the enormous number of particles (~10²³). The equilibrium state is the macrostate with the most microstates, and S = k_B ln W quantifies exactly how many ways there are to achieve each macrostate.

The second law of thermodynamics is not a fundamental dynamical law but a statistical near-certainty. For macroscopic systems, the most probable macrostate is so overwhelmingly more likely than alternatives that deviations are never observed in practice. Statistical mechanics replaces the phenomenological statement 'entropy increases' with the statistical explanation 'systems move toward states that can be realized in more ways.'