Questions: Statistical Ensembles and Probability Distributions

Question 1

A physicist needs to compute the thermodynamic properties of a gas in a box with a fixed, known energy E. For mathematical convenience, she uses the canonical ensemble (fixed T, not fixed E) rather than the microcanonical ensemble. In the thermodynamic limit, her results will be:

• A. Wrong, because the canonical ensemble assumes energy can fluctuate, contradicting the fixed-energy constraint.
• B. Identical to the microcanonical result, because all ensembles give equivalent predictions for macroscopic quantities when N is very large.
• C. A good approximation only if the temperature is very low.
• D. Slightly off because the canonical ensemble is defined for open systems.

Answer: (B)

Ensemble equivalence in the thermodynamic limit is a foundational result of statistical mechanics. Energy fluctuations in the canonical ensemble are of order 1/√N relative to the mean. For N ~ 10²³, this fraction is ~10⁻¹¹ — utterly negligible. The 'fixed E' constraint and the 'fixed T' constraint become indistinguishable for macroscopic quantities. Physicists routinely exploit this: use whichever ensemble makes the math tractable, and the physics comes out the same.

Question 2

In the microcanonical ensemble (fixed E, V, N), the fundamental postulate assigns equal probability to all compatible microstates. What thermodynamic quantity does this define, and how?

• A. Temperature T = E/N, the average energy per particle.
• B. Entropy S = k ln Ω, where Ω is the total number of microstates compatible with the macrostate.
• C. Pressure P = NkT/V from the ideal gas law.
• D. Free energy F = E − TS, minimized at equilibrium.

Answer: (B)

The equal-probability postulate directly defines entropy through Boltzmann's formula S = k ln Ω, where Ω counts the number of microstates consistent with the macroscopic constraints (E, V, N). This is Boltzmann's famous equation — it connects the microscopic multiplicity of states to the macroscopic thermodynamic entropy. All other thermodynamic quantities (temperature, pressure, chemical potential) then follow by differentiating S with respect to E, V, and N respectively.

Question 3

In the thermodynamic limit (N → ∞), the canonical ensemble and microcanonical ensemble yield identical predictions for macroscopic thermodynamic quantities because energy fluctuations in the canonical ensemble become negligible relative to the mean energy.

• T. True
• F. False

Answer: True

Energy fluctuations in the canonical ensemble scale as √N (the standard deviation of energy), while the mean energy scales as N. The relative fluctuation is ~1/√N, which approaches zero as N → ∞. For a macroscopic system with N ~ 10²³ particles, this means the canonical ensemble's energy is effectively fixed at its mean value — indistinguishable from the microcanonical constraint of truly fixed energy. This equivalence justifies choosing ensembles purely for mathematical convenience.

Question 4

To correctly analyze a system at fixed temperature in contact with a heat bath, you should use the canonical ensemble; using the microcanonical ensemble would give incorrect thermodynamic predictions.

• T. True
• F. False

Answer: False

This is the most common misconception about ensembles. In the thermodynamic limit, all three ensembles (microcanonical, canonical, grand canonical) give identical predictions for macroscopic quantities. A physicist may use the microcanonical ensemble for a system at fixed temperature — or the canonical ensemble for an isolated system — and obtain the same thermodynamic results either way. The choice of ensemble is purely a matter of mathematical convenience, not physical accuracy. The ensemble that makes the calculation tractable is always the right one to use.

Question 5

Why can a physicist choose whichever statistical ensemble is mathematically most convenient, even when it doesn't exactly match the physical constraints of their system?

Model answer: Because all ensembles give identical predictions for macroscopic thermodynamic quantities in the thermodynamic limit (N → ∞). In the canonical ensemble, for instance, energy fluctuations are proportional to √N while the mean energy is proportional to N — so the relative fluctuation ~1/√N vanishes as N grows. For a macroscopic system with ~10²³ particles, this fraction is ~10⁻¹¹: the canonical ensemble's energy is effectively fixed at its mean, making it physically indistinguishable from the microcanonical ensemble's truly fixed energy. This ensemble equivalence means the choice is purely about which partition function is easier to compute.

The canonical ensemble's partition function Z = Σ e^(−βEᵢ) is particularly tractable: all thermodynamic properties follow from F = −kT ln Z by differentiation. This mathematical convenience — rather than any physical correspondence to a heat bath — is the primary reason physicists often default to the canonical ensemble even when studying isolated systems.