Questions: Partition Function and Thermodynamic Properties

Question 1

A system of N independent, identical harmonic oscillators has total partition function Z_total. If you double the number of oscillators to 2N, what happens to Z_total and to ln Z_total?

• A. Z_total doubles; ln Z_total increases by ln 2
• B. Z_total doubles; ln Z_total also doubles
• C. Z_total is squared (Z²); ln Z_total doubles
• D. Z_total is squared (Z²); ln Z_total increases by ln 2

Answer: (C)

For independent subsystems, partition functions multiply: Z_total = Z^N, so doubling N gives Z^{2N} = (Z^N)² — Z_total is squared. Taking the logarithm: ln(Z²) = 2 ln Z — it doubles. This multiplicative-to-additive conversion is exactly why ln Z appears in every thermodynamic formula. Extensive properties (energy, entropy, Helmholtz free energy) must add when combining identical subsystems. Since Z multiplies but thermodynamic quantities add, the logarithm is the natural bridge. This is why A = −k_BT ln Z, not A = −k_BT Z.

Question 2

A two-level system has partition function Z = 1 + e^{−βε}. As temperature T → ∞ (β → 0), which statement correctly describes the thermodynamic behavior?

• A. Z diverges to infinity, making the thermodynamic description break down at high temperature
• B. Z approaches 2 and ln Z approaches ln 2; both energy levels become equally populated and entropy approaches its maximum value k_B ln 2
• C. Z approaches 1 because e^{−βε} → 0 at high temperature, collapsing the system to its ground state
• D. Z approaches e^{−βε} and all derived properties approach zero

Answer: (B)

As β → 0, e^{−βε} → 1, so Z → 1 + 1 = 2, and ln Z → ln 2. Both energy levels have equal Boltzmann weight — equal population — which is maximum disorder. The entropy S → k_B ln 2 (one bit of entropy for a two-level system), and the internal energy approaches ε/2 (the average of the two level energies). The system doesn't break down; rather, Z reaches a finite ceiling equal to the number of states — the limit where thermal energy far exceeds all energy level splittings and every state is equally accessible.

Question 3

The reason all thermodynamic properties are derived from ln Z rather than Z itself is that extensive properties of independent subsystems must add, and the logarithm converts the multiplicative combination of partition functions into an additive one.

• T. True
• F. False

Answer: True

This is the core structural reason. For N independent subsystems, Z_total = Z₁ × Z₂ × … × Z_N, so ln Z_total = ln Z₁ + ln Z₂ + … + ln Z_N — the additive structure matches thermodynamics. Internal energy, entropy, and Helmholtz free energy are all extensive and must add when identical systems are combined. Using Z directly would give products, not sums. The logarithm is not arbitrary mathematical convenience; it reflects the deep connection between the multiplicative probability structure of statistical mechanics and the additive extensive structure of thermodynamics.

Question 4

The partition function Z generally equals the total number of quantum states available to the system.

• T. True
• F. False

Answer: False

Z = Σ e^{−βE_i} is a Boltzmann-weighted sum — high-energy states are exponentially suppressed. Only in the limit T → ∞ (β → 0) do all factors equal 1, making Z equal to the number of states. At any finite temperature, Z is less than the total number of states and represents the 'effective number of thermally accessible states.' A better description is that Z is a generating function, not a counter. For a harmonic oscillator with infinitely many energy levels, Z is finite at any finite temperature even though the number of states is infinite — because high levels are exponentially excluded.

Question 5

Explain why the partition function can be called a 'generating function' for thermodynamics, and what role ln Z specifically plays in extracting thermodynamic properties.

Model answer: The partition function Z encodes all equilibrium thermodynamic information in a single quantity. Successive derivatives of ln Z with respect to β (at constant V) yield thermodynamic observables: U = −(∂ ln Z/∂β)_V for internal energy; another derivative gives heat capacity; ∂ ln Z/∂V gives pressure (up to factors of k_BT). Entropy and Helmholtz free energy follow from A = −k_BT ln Z and S = (U − A)/T. The logarithm is essential because extensive properties must add when combining independent subsystems — Z_total = Z₁ × Z₂ multiplies, so ln Z_total = ln Z₁ + ln Z₂ adds. Statistical mechanics thus reduces all of equilibrium thermodynamics to computing Z and differentiating.

The analogy to probability generating functions is precise: just as successive derivatives of a moment-generating function yield statistical moments, successive derivatives of ln Z yield thermodynamic observables. This is not coincidental — statistical mechanics is fundamentally a probabilistic theory, and Z is the normalization constant for the Boltzmann probability distribution over energy states.